Recurrence Relations of the Multi-Indexed Orthogonal Polynomials VI : Meixner-Pollaczek and continuous Hahn types

DPSU-19-2 Recurrence Relations of the Multi-In dexed Orthogonal P olynomials VI : Meixner-P ollacze k and contin uous Hahn t yp es Satoru Odak e F acult y of Science, Shinsh u Univ ersit y , Matsumoto 390-862 1, Japan Abstract In p revious pap ers, w e discussed the r ecurrence relations of the multi-indexed or- thogonal p olynomials of the Laguerre, J acobi, Wilson, Ask ey-Wilson, Rac ah and q - Racah t yp es. In this pap er we explore those of the Meixner-Pol laczek and contin uous Hahn t yp es. F or the M -indexed Meixner-P ollaczek and contin uous Hahn p olynomials, w e p resen t 3 + 2 M term recurrence relations with v ariable dep end en t co efficien ts and 1 + 2 L term ( L ≥ M + 1) recurren ce relations with constan t co efficien ts. Based on the latter, the generalized closure relations and the creation/annihilation op erators of the quan tum mec hanical systems describ ed by the m ulti-indexed Meixner-Po llaczek and con tin uous Hahn p olynomials are obtained. 1 In tro duction Orthogonal p olynomials in the Ask ey sc heme of the (basic) h yp ergeometric orthogonal p o ly- nomials [1], whic h satisfy second or der differen tial or difference equations, can b e successfully studied in the quan tum mec hanical formulation: ordinary quan tum mec hanics ( o QM), dis- crete quantum mec hanics with pure imaginary shifts (idQM) [2]–[5] and discrete quantum mec hanics with real shifts (rdQM) [6]– [8]. By defor ming the exactly solv able quan tum sys- tems describ ed b y the orthogonal p o lynomials in the Ask ey sc heme, new types of ort ho gonal p olynomials are obtained [9]–[24]. They are called exceptional or m ulti-indexed orthogo- nal p olynomials {P n ( η ) | n ∈ Z ≥ 0 } (the range of n is finite fo r finite rdQM systems). They satisfy second order differen tial or difference equations a nd form a complete set of o rthog- onal basis in an appropriat e Hilb ert space in spite of missing degrees. This degree missing is a c haracteristic f eature of them, and thereb y the Bo chne r’s t heorem and its generaliza- tions [25, 26 ] are a v oided. W e distinguish the follo wing tw o cases; t he set of missing degrees I = Z ≥ 0 \{ deg P n | n ∈ Z ≥ 0 } is case-(1): I = { 0 , 1 , . . . , ℓ − 1 } , or case-(2): I 6 = { 0 , 1 , . . . , ℓ − 1 } , where ℓ is a p ositiv e in teger. The situation of case-(1) is called stable in [13]. Ordinary orthog onal p olynomials in one v ariable are c haracterized b y the three term recurrence relations [26]. Since the multi-indexe d orthogonal p olynomials are not or dina r y orthogonal p olynomials, they do not satisfy the three term recurrence relations. Instead of the three term recurrence relations, they satisfy some recurrence relations with more terms [27]–[35]. In previous papers [28, 31, 33, 34, 35 ], the recurrence relations for t he case-(1) m ulti- indexed orthogo nal p olynomials (L aguerre (L) and Jacobi (J) types in oQM [15], Wilson (W) and Ask ey-Wilson (A W) types in idQM [17], Racah (R) a nd q -Racah ( q R) t yp es in rdQM [19]) w ere studie d. There are t w o kinds of recurrence relations: with v ariable dep enden t co efficien ts and with constan t co efficien ts. Recen tly , based on the idQM systems whose phy sical range of the co ordinate is the whole real line, the case-(1) mu lti-indexed orthogonal p olynomials of Meixner-P ollaczek (MP) and con tin uous Hahn (cH) t yp es are constructed [36]. W e remark that the MP a nd cH p oly- nomials r educe to the Hermite (H) p olynomial in certain limits but there are no case-(1) m ulti-indexed Hermite orthogona l p olynomials. In this pap er w e explore the recurrence relations for the case-(1) m ulti-indexed ortho gonal p olynomials of MP and cH ty p es. By similar metho ds used in W a nd A W cases, we deriv e t w o kinds of recurrence relations. The recurrence relatio ns with constan t co efficien ts are closely related t o the generalized closure relations [34]. Th e generalized closure relations prov ide the exact Heisen b erg op erator solution o f a certain op erator, from whic h the creation and annihilation op erators o f the sys tem are obtained. W e presen t the creation and annihilation op erators of the defor med MP and cH systems. The deformed Hamiltonian H D is determined b y the denominato r p olynomial Ξ D ( η ), where D is an index set of the virtual state w av efunc tions used in M -step Darb oux trans- formations. The degree of Ξ D ( η ) is given b y ℓ D (A.26) and there is no restriction on ℓ D for L, J, W and A W cases. Ho w ev er, for MP and cH cases, the degree ℓ D m ust b e ev en in o rder that the Hamiltonian H D is hermitian. The multi-index ed MP and cH orthogonal p olynomials a re constructed for ev en ℓ D (and some conditions) [36]. In this pa p er w e define 2 the multi-index ed MP and cH p olynomials for any index set D , namely ℓ D ma y b e o dd and they ma y not b e orthogo nal p olynomials. W e conjecture that the recurrence relations for the m ulti-indexed MP and cH p olynomials hold for non-orthogona l case. This pap er is organized as fo llo ws. In section 2 the case-(1) m ulti-indexed MP and cH p olynomials are recapitulated. The definitions of the m ulti-indexed MP and cH p olynomials are extended to non-ortho g onal case. In section 3 the recurrence relations of the case-(1 ) m ulti-indexed MP and cH or t ho gonal p olynomials are deriv ed. The recurrence relations with v ariable dep enden t co efficien ts are presen ted in section 3.1 , and those with constan t co effi- cien ts in section 3.2. Some explicit examples are presen ted in section 3.3. In section 4 the generalized closure relations and the creation/a nnihilat io n op erators are presen ted. Section 5 is fo r a summary a nd commen ts. In App endix A formulation of idQM and deformed systems are recapitulated. Some prop erties of the m ulti-indexed MP p olynomials ar e presen ted in App endix B, and those of the m ulti-indexed cH p olynomials are presen ted in App endix C. In App endix D more examples for section 3 are presen ted, which corresp ond to non-orthogonal case. 2 Multi-ind e xed Meixne r - P oll acz e k and Contin uous Hahn p olynomials In this section w e recapitulate the case-(1) m ulti-indexed Meixner-P ollaczek and con tinuous Hahn p olynomials [36]. Generalizing the result of [36], w e first define the m ulti-indexed MP and cH p o lynomials for an y D , and then consider the condition that they b ecome orthogona l p olynomials. The case-(1) m ulti-indexed orthogona l p olynomials [15, 17, 19, 2 2, 36] a nd those of case- (2) [3, 4, 7, 37, 38, 39] are constructed based on the quan tum mec hanical form ulations [5]. F or the Meixner-P ollaczek (MP) and Con tin uous Hahn (cH) p olynomials, we use the discrete quan tum mec hanics with pure imag inary shifts (idQM) [2, 5]. The form ulation of idQM is presen ted in App endix A and we follo w the notation there. F or MP and cH cases, the lo w er b ound x 1 , upp er b ound x 2 , the para meter γ , the sin usoidal co ordinate η ( x ) and the auxiliary function ϕ ( x ) are x 1 = −∞ , x 2 = ∞ , γ = 1 , η ( x ) = x, ϕ ( x ) = 1 . (2.1) Namely , the phys ical range o f the co ordinate x is the whole real line. It is not necess ary to 3 distinguish ˇ P n and P n since η ( x ) = x , but w e will use b oth notations to compare with other cases in [17]. 2.1 Meixner-P ollaczek and con tin uous Hahn p olynomials First w e tak e a set of para meters λ a s follows: MP : λ = ( a, φ ) , a, φ ∈ R , cH : λ = ( a 1 , a 2 ) , a 1 , a 2 ∈ C . (2.2) The fundamen tal data are the follo wing [2]: ( n ∈ Z ≥ 0 ) V ( x ; λ ) =  e i ( π 2 − φ ) ( a + ix ) : MP ( a 1 + ix )( a 2 + ix ) : cH , (2.3) E n ( λ ) =  2 n sin φ : MP n ( n + b 1 − 1) : cH , b 1 def = a 1 + a 2 + a ∗ 1 + a ∗ 2 , (2.4) ˇ P n ( x ; λ ) = P n  η ( x ); λ  = ( P ( a ) n  η ( x ); φ  : MP p n  η ( x ); a 1 , a 2 , a ∗ 1 , a ∗ 2  : cH (2.5) =        (2 a ) n n ! e inφ 2 F 1  − n, a + ix 2 a    1 − e − 2 iφ  : MP i n ( a 1 + a ∗ 1 , a 1 + a ∗ 2 ) n n ! 3 F 2  − n, n + b 1 − 1 , a 1 + ix a 1 + a ∗ 1 , a 1 + a ∗ 2    1  : cH (2.6) = c n ( λ ) η ( x ) n + (lo w er or der terms) , c n ( λ ) = ( 1 n ! (2 sin φ ) n : MP 1 n ! ( n + b 1 − 1) n : cH , (2.7) δ =  ( 1 2 , 0) : MP ( 1 2 , 1 2 ) : cH , κ = 1 , f n ( λ ) =  2 sin φ : MP n + b 1 − 1 : cH , b n − 1 ( λ ) = n. (2.8) (Although the notation b 1 conflicts with b n − 1 ( λ ), we think this do es not cause any confusion.) Here P ( a ) n ( η ; φ ) and p n ( η ; a 1 , a 2 , a 3 , a 4 ) in (2.5) are the Meix ner-Pollacz ek and con tin uous Hahn p olynomials of degree n in η , resp ectiv ely [1]. Note that ˇ P ∗ n ( x ; λ ) = ˇ P n ( x ; λ ). The Meixner-P ollaczek p o lynomial with φ = π 2 has a definite par ity , MP : ˇ P n ( − x ; λ ) = ( − 1) n ˇ P n ( x ; λ ) for λ = ( a, π 2 ) . (2.9) The p olynomials ˇ P n ( x ) satisfy the forward and bac kw ard shift relations (A.10), hence the second order difference equation (A.1 4). Next let us restrict a set of parameters λ as follow s: MP : a > 0 , 0 < φ < π , cH : Re a i > 0 ( i = 1 , 2) . (2.10) 4 Then the Hamiltonian o f idQM system is w ell-defined and hermitian. Additional fundamen- tal da ta are the fo llo wing [2]: φ n ( x ; λ ) = φ 0 ( x ; λ ) ˇ P n ( x ; λ ) , (2.11) φ 0 ( x ; λ ) = ( e ( φ − π 2 ) x p Γ( a + ix )Γ( a − ix ) : MP p Γ( a 1 + ix )Γ( a 2 + ix )Γ( a ∗ 1 − ix )Γ( a ∗ 2 − ix ) : cH , (2.12) h n ( λ ) =          2 π Γ( n + 2 a ) n ! (2 sin φ ) 2 a : MP 2 π Q 2 j,k =1 Γ( n + a j + a ∗ k ) n ! (2 n + b 1 − 1)Γ( n + b 1 − 1) : cH . (2.13) Note that φ ∗ 0 ( x ; λ ) = φ 0 ( x ; λ ). The eigenfunctions φ n ( x ) are orthogonal eac h other (A.4), whic h giv es the orthogona lity relation of ˇ P n ( x ) (A.7). The MP and cH idQM systems hav e shape in v ariance prop ert y (A.8), whic h give s the forw ard and bac kw ard shift relat io ns (A.10). The second order difference equation (A.14) is a rewrite of the Sch r¨ odinger equation (A.3). 2.2 Multi-indexed Meixner-P ollaczek and con tin uous Hahn p oly- nomials The case-(1) multi-indexe d Meixne r-Pollaczek and con tin uous Ha hn p olynomials w ere con- structed b y deforming the idQM systems in § 2.1 [36]. The index set D = { d 1 , . . . , d M } ( d j : m utually distinct) lab els the virtual state wa v efunctions used in t he M -step D arb oux transformations. F or cH system, there are tw o types of virtual states (t yp e I and I I) and D is D = { d 1 , . . . , d M } = { d I 1 , . . . , d I M I , d II 1 , . . . , d II M II } ( M = M I + M II , d I j : m utually distinct, d II j : mutually distinct). First w e define m ulti-indexed MP and cH p olynomials for any λ and D , whic h ma y not b e o rthogonal p olynomials. Then we presen t a sufficien t condition for them to b ecome orthogonal p olynomials. 2.2.1 definitions (for an y λ and D ) A set of pa r ameters λ is tak en as (2.2) and d j ’s are non-negat ive in tegers. The twis t op era- tions t and constants ˜ δ are defined b y MP : t ( λ ) def = (1 − a, φ ) , ˜ δ def = ( − 1 2 , 0) , (2.14) cH : t yp e I : t I ( λ ) def = (1 − a ∗ 1 , a 2 ) , ˜ δ I def = ( − 1 2 , 1 2 ) , 5 t yp e I I : t II ( λ ) def = ( a 1 , 1 − a ∗ 2 ) , ˜ δ II def = ( 1 2 , − 1 2 ) . (2.15) F or cH case, corresp onding t o the t yp e I and type I I, w e add superscripts I and I I. The virtual state energies ˜ E v ( λ ) and virtual state p olynomials ξ v ( η ; λ ) are defined b y MP : ˜ E v ( λ ) def = − 2(2 a − v − 1) sin φ, ˇ ξ v ( x ; λ ) def = ξ v  η ( x ); λ  def = ˇ P v  x ; t ( λ )  = P v  η ( x ); t ( λ )  , (2.16) cH : ˜ E I v ( λ ) def = − ( a 1 + a ∗ 1 − v − 1)( a 2 + a ∗ 2 + v) , ˜ E II v ( λ ) def = − ( a 2 + a ∗ 2 − v − 1)( a 1 + a ∗ 1 + v) , ˇ ξ I v ( x ; λ ) def = ξ I v  η ( x ); λ  def = ˇ P v  x ; t I ( λ )  = P v  η ( x ); t I ( λ )  , (2.17) ˇ ξ II v ( x ; λ ) def = ξ II v  η ( x ); λ  def = ˇ P v  x ; t II ( λ )  = P v  η ( x ); t II ( λ )  . The virtual state p olynomials ξ v ( η ; λ ) are po lynomials of degree v in η and satisfy e H ( λ ) ˇ ξ v ( x ; λ ) = ˜ E v ( λ ) ˇ ξ v ( x ; λ ). Note that ˇ ξ ∗ v ( x ; λ ) = ˇ ξ v ( x ; λ ). The functions r j ( x ( M ) j ; λ , M ) ( j = 1 , 2 , . . . , M ) are defined b y MP : r j ( x ( M ) j ; λ , M ) def = ( − 1) j − 1 i 1 − M ( a − M − 1 2 + ix ) j − 1 ( a − M − 1 2 − ix ) M − j , (2.18) cH : r I j ( x ( M ) j ; λ , M ) def = ( − 1) j − 1 i 1 − M ( a 1 − M − 1 2 + ix ) j − 1 ( a ∗ 1 − M − 1 2 − ix ) M − j , r II j ( x ( M ) j ; λ , M ) def = ( − 1) j − 1 i 1 − M ( a 2 − M − 1 2 + ix ) j − 1 ( a ∗ 2 − M − 1 2 − ix ) M − j , (2.19) where x ( n ) j def = x + i ( n +1 2 − j ) γ . The auxiliary function ϕ M ( x ) introduced in [4] is ϕ M ( x ) = 1 in the presen t case. Let us define the denominator p olynomial Ξ D ( η ; λ ) and the m ulti-indexed polynomial P D ,n ( η ; λ ) : ˇ Ξ D ( x ; λ ) def = Ξ D  η ( x ); λ  , ˇ P D ,n ( x ; λ ) def = P D ,n  η ( x ); λ  . (2.20) Here ˇ Ξ D ( x ; λ ) and ˇ P D ,n ( x ; λ ) are giv en by determinan ts as follows . F or MP , they a re i 1 2 M ( M − 1) det  ˇ ξ d k ( x ( M ) j ; λ )  1 ≤ j,k ≤ M = ϕ M ( x ) ˇ Ξ D ( x ; λ ) , (2.21) i 1 2 M ( M +1)           ˇ ξ d 1 ( x ( M +1) 1 ; λ ) · · · ˇ ξ d M ( x ( M +1) 1 ; λ ) r 1 ( x ( M +1) 1 ; λ , M + 1 ) ˇ P n ( x ( M +1) 1 ; λ ) ˇ ξ d 1 ( x ( M +1) 2 ; λ ) · · · ˇ ξ d M ( x ( M +1) 2 ; λ ) r 2 ( x ( M +1) 2 ; λ , M + 1 ) ˇ P n ( x ( M +1) 2 ; λ ) . . . · · · . . . . . . ˇ ξ d 1 ( x ( M +1) M +1 ; λ ) · · · ˇ ξ d M ( x ( M +1) M +1 ; λ ) r M +1 ( x ( M +1) M +1 ; λ , M + 1 ) ˇ P n ( x ( M +1) M +1 ; λ )           = ϕ M +1 ( x ) ˇ P D ,n ( x ; λ ) . (2.22) 6 F or cH, they are i 1 2 M ( M − 1)    ~ X ( M ) d I 1 · · · ~ X ( M ) d I M I ~ Y ( M ) d II 1 · · · ~ Y ( M ) d II M II    = ϕ M ( x ) ˇ Ξ D ( x ; λ ) × A, (2.23) i 1 2 M ( M +1)    ~ X ( M +1) d I 1 · · · ~ X ( M +1) d I M I ~ Y ( M +1) d II 1 · · · ~ Y ( M +1) d II M II ~ Z ( M +1) n    = ϕ M +1 ( x ) ˇ P D ,n ( x ; λ ) × B , (2.24) where A and B are A = M I − 1 Y j =1 ( a 2 − M − 1 2 + ix, a ∗ 2 − M − 1 2 − ix ) j · M II − 1 Y j =1 ( a 1 − M − 1 2 + ix, a ∗ 1 − M − 1 2 − ix ) j , (2.25) B = M I Y j =1 ( a 2 − M 2 + ix, a ∗ 2 − M 2 − ix ) j · M II Y j =1 ( a 1 − M 2 + ix, a ∗ 1 − M 2 − ix ) j , (2.26) and ~ X ( M ) v , ~ Y ( M ) v and ~ Z ( M ) v are  ~ X ( M ) v  j = r II j ( x ( M ) j ; λ , M ) ˇ ξ I v ( x ( M ) j ; λ ) , ( j = 1 , 2 , . . . , M ) ,  ~ Y ( M ) v  j = r I j ( x ( M ) j ; λ , M ) ˇ ξ II v ( x ( M ) j ; λ ) ,  ~ Z ( M ) n  j = r II j ( x ( M ) j ; λ , M ) r I j ( x ( M ) j ; λ , M ) ˇ P n ( x ( M ) j ; λ ) . (2.27) (F or the cases of type I only ( M II = 0) or type I I o nly ( M I = 0) , the expressions (2.23) and (2.24) a re rewritten as (2.21) and (2.2 2), see [36 ].) The denominator p o lynomial Ξ D ( η ; λ ) a nd the multi-indexe d p olynomial P D ,n ( η ; λ ) a re p olynomials in η and their degrees are ℓ D and ℓ D + n , resp ectiv ely ( w e assume c Ξ D ( λ ) 6 = 0 and c P D ,n ( λ ) 6 = 0, see (B.1)– (B.2) and (C.1)– (C.2)). Here ℓ D is given by ( A.2 6). Note tha t ˇ Ξ ∗ D ( x ; λ ) = ˇ Ξ D ( x ; λ ) and ˇ P ∗ D ,n ( x ; λ ) = ˇ P D ,n ( x ; λ ). Under the p erm utation of d j ’s, ˇ Ξ D ( x ) and ˇ P D ,n ( x ) change their signs. The par ity pro p ert y of the Meixner-P ollaczek p olynomial (2.9) is inherited by the m ulti-indexed p olynomials of ev en ℓ D , MP : ˇ P D ,n ( − x ; λ ) = ( − 1) n ˇ P D ,n ( x ; λ ) for λ = ( a, π 2 ) and eve n ℓ D . (2.28) The deformed p oten tial functions V D ( x ; λ ) are defined b y ( A.21) a nd (A.24). The multi- indexed p olynomials ˇ P D ,n ( x ; λ ) satisfy the forward and bac kw ard shift relations (A.34) , hence the second order difference equation (A.3 8). 7 2.2.2 orthogonal polynomials The deformed idQM systems should b e w ell-defined, na mely the deformed Hamiltonian H D ( λ ) should b e hermitian. W e restrict λ a s (2.10) and imp ose the f ollo wing conditions, MP : max j { d j } < 2 a − 1 , (2.29) cH : max j { d I j } < a 1 + a ∗ 1 − 1 , max j { d II j } < a 2 + a ∗ 2 − 1 , (2.30) under whic h the virtual state energies b ecome negativ e, MP : ˜ E v ( λ ) < 0 ⇔ 2 a > v + 1 , cH : ˜ E I v ( λ ) < 0 ⇔ a 1 + a ∗ 1 > v + 1 , ˜ E II v ( λ ) < 0 ⇔ a 2 + a ∗ 2 > v + 1 . The deformed Ha milto nian H D ( λ ) is hermitian, if t he condition (A.27) is satisfied [36]. Since the rectangular domain D γ con tains the whole real axis, the degree of Ξ D ( η ; λ ), ℓ D (A.26), should b e ev en. Although w e hav e no analytical pro of that there exists a range o f parameters λ satisfying the condition (A.27), w e can verify that there exists suc h a ra nge of λ b y num erical calculation (for small M and d j ). W e ha v e observ ed v ario us sufficien t conditions fo r the parameter range satisfying (A.27), see [36]. In the rest o f this section we assume that the condition (A.27) is satisfied. The eigenfunctions of the deformed Hamilto nia n H D ( λ ) hav e the form (A.22). No t e that ψ ∗ D ( x ; λ ) = ψ D ( x ; λ ). The eigenfunctions φ D n ( x ) are orthog onal eac h other, whic h giv es the orthogonality relation of ˇ P D ,n ( x ), (A.28)–(A.29). The m ulti-indexed orthogonal p o lynomial P D ,n ( η ; λ ) has n zeros in the phy sical region η ∈ R ( ⇔ η ( x 1 ) < η < η ( x 2 )), whic h interlace the n + 1 zeros o f P D ,n +1 ( η ; λ ) in the phy sical region, and ℓ D zeros in the unphy sical region η ∈ C \ R . These pro p erties and (A.28) can b e v erified b y n umerical calculation. Since the deformed Ha milto nian H D ( λ ) ( A.1 9) is expre ssed in terms of the p oten tial function V D ( x ; λ ) (A.21), H D ( λ ) is determined by t he denominator p olynomial ˇ Ξ D ( x ; λ ), whose normalizatio n is irrelev an t. Under the p ermu tatio n of d j ’s, the deformed Hamiltonia n H D is in v arian t. The deformed MP and cH idQM systems ha ve also shap e inv ariance prop ert y (A.30), whic h give s the forward and bac kw ard shift relations (A.34). The second o r der difference equation (A.38) is a rewrite of the Schr¨ odinger equation (A.18). The prop erties (B.4) a nd (C.4)–(C.5) are also the consequences of the shap e in v aria nce. By (B.3) and (C.3), the similar pro p ert y holds for the denominato r p olynomial ˇ Ξ D ( x ; λ ). The M -step deformed 8 system lab eled by D with 0 is equiv alen t to the ( M − 1)- step deformed system lab eled b y D ′ with shifted parameters λ + ˜ δ . F or the m ulti-indexed W a nd A W p o lynomials, there are equiv alence a mong the index sets D [40]. The multi-index ed MP and cH p olynomials ha v e also equiv alence in the same form as W and A W cases, whic h is deriv ed from the prop erties (B.4) and (C.4)–(C.5). 3 Recurrence Relation s In this section w e presen t the recurrence relatio ns of the case-(1) multi-indexe d Meixner- P ollaczek and con tin uous Hahn p olynomials. The re are t w o t yp es of recurrence relations: v ariable dep enden t co efficien ts and constan t co efficien ts. The three term recurrence relations of the Meixne r-Pollaczek and con tinuous Hahn p oly- nomials are [1] η P n ( η ; λ ) = A n ( λ ) P n +1 ( η ; λ ) + B n ( λ ) P n ( η ; λ ) + C n ( λ ) P n − 1 ( η ; λ ) , (3.1) where A n , B n and C n are MP : A n ( λ ) = n + 1 2 sin φ , B n ( λ ) = − ( n + a ) cot φ, C n ( λ ) = n + 2 a − 1 2 sin φ , (3.2) cH : A n ( λ ) = ( n + 1)( n + b 1 − 1) (2 n + b 1 − 1)(2 n + b 1 ) , B n ( λ ) = i  a 1 − ( n + b 1 − 1)( n + a 1 + a ∗ 1 )( n + a 1 + a ∗ 2 ) (2 n + b 1 − 1)(2 n + b 1 ) + n ( n + a 2 + a ∗ 1 − 1)( n + a 2 + a ∗ 2 − 1) (2 n + b 1 − 2)(2 n + b 1 − 1)  , (3.3) C n ( λ ) = ( n + a 1 + a ∗ 1 − 1)( n + a 1 + a ∗ 2 − 1)( n + a 2 + a ∗ 1 − 1)( n + a 2 + a ∗ 2 − 1) (2 n + b 1 − 2)(2 n + b 1 − 1) . F or simplicit y of pres en tatio n, w e set P n ( η ) = 0 for n ∈ Z < 0 and define A n , B n and C n for n ∈ Z < 0 b y (3.2)–(3.3). Then, (3.1) hold for n ∈ Z . Similarly w e set P D ,n ( η ) = 0 for n ∈ Z < 0 . 3.1 Recurrence relations with v ariable dep enden t co efficien ts W e presen t 3 + 2 M term recurrence relations with v ariable dep endent co efficien ts. F or the case-(1) multi-indexed Wilson a nd Ask ey-Wilson p olynomials, suc h recurrence relatio ns are 9 sho wn in [28]. Since the deriv a t io n can b e applied in the presen t case without difficult y , w e presen t only the r esults here. Let us define ˇ R [ s ] n,k ( x ) ( n, k ∈ Z , s ∈ Z ≥− 1 ) as follo ws: ˇ R [ s ] n,k ( x ) = 0 ( | k | > s + 1 or n + k < 0) , ˇ R [ − 1] n, 0 ( x ) = 1 ( n ≥ 0) , ˇ R [ s ] n,k ( x ) = A n ˇ R [ s − 1] n +1 ,k − 1 ( x + i γ 2 ) +  B n − η ( x − i s 2 γ )  ˇ R [ s − 1] n,k ( x + i γ 2 ) (3.4) + C n ˇ R [ s − 1] n − 1 ,k +1 ( x + i γ 2 ) ( s ≥ 0) . Here A n , B n and C n are giv en b y (3.2)–(3.3). Note that A n ( n < − 1), B n ( n < 0) and C n ( n < 0) do not app ear, b ecause A − 1 = 0 (and we regard A − 1 × ( · · · ) = 0). F or example, non-trivial ˇ R [ s ] n,k ( x ) ( n + k ≥ 0) for s = 0 , 1 are s = 0 : ˇ R [0] n, 1 ( x ) = A n , ˇ R [0] n, 0 ( x ) = B n − η ( x ) , ˇ R [0] n, − 1 ( x ) = C n , s = 1 : ˇ R [1] n, 2 ( x ) = A n A n +1 , ˇ R [1] n, 1 ( x ) = A n  B n + B n +1 − η ( x − i γ 2 ) − η ( x + i γ 2 )  , ˇ R [1] n, 0 ( x ) = A n C n +1 + A n − 1 C n +  B n − η ( x − i γ 2 )  B n − η ( x + i γ 2 )  , ˇ R [1] n, − 2 ( x ) = C n C n − 1 , ˇ R [1] n, − 1 ( x ) = C n  B n + B n − 1 − η ( x − i γ 2 ) − η ( x + i γ 2 )  . It is easy to see that ˇ R [ s ] n,k ( x ) is a p o lynomial in x = η ( x ). W e define R [ s ] n,k ( η ) as follows : ˇ R [ s ] n,k ( x ) = R [ s ] n,k  η ( x )  ( | k | ≤ s + 1) : a p olynomial of degree s + 1 − | k | in η ( x ) . (3.5) Note that ˇ R [ s ] ∗ n,k ( x ) = ˇ R [ s ] n,k ( x ). Then we ha v e the following result. Theorem 1 The multi-indexe d Meixner-Pol laczek an d c ontinuous Hahn p ol yno m ials satisfy the 3 + 2 M term r e curr enc e r elations wi th variable dep endent c o efficients: M +1 X k = − M − 1 R [ M ] n,k ( η ) P D ,n + k ( η ) = 0 , (3.6) which h o l d s for n ∈ Z . Remark 1 The deriv ation in [28] uses only the algebraic prop ert y of the M - step Da rb oux transformations. Hence Theorem 1 ho lds fo r P D ,n ( η ; λ ) with any λ and D (namely P D ,n ( η ; λ ) ma y not b e orthogona l p olynomials). Remark 2 The m ulti-indexed p olynomials P D ,n ( η ) ( n ≥ M + 1) ar e determined b y the 3 + 2 M term recurrence r elations (3.6) with M + 1 “initial data ” [28], P D , 0 ( η ) , P D , 1 ( η ) , . . . , P D ,M ( η ) . (3.7) 10 After calculating the initial data (3.7) b y (2.22) and (2.2 4), w e can obtain P D ,n ( η ) through the 3 + 2 M term recurrence relations (3.6). The calculation cost of this metho d is mu c h less than the original determinan t expression (2.22) and (2.24) for large M . 3.2 Recurrence relations with constan t co efficien ts W e presen t 1 + 2 L term recurrence relations with constant co efficien ts. F o r the case-(1) m ulti- indexed Wilson and Ask ey-Wilson p olynomials, suc h recurrence relations are presen ted in [31] and sho wn in App endix B of [35]. Since the deriv ation can b e applied in the presen t case without difficult y , w e presen t only t he results here. W e wan t to find the following recurrence relations, X ( η ) P D ,n ( η ) = L X k = − n r X, D n,k P D ,n + k ( η ) , where r X, D n,k ’s are constan ts and X ( η ) is some p olynomial of degree L in η . The ov erall normalization and the constan t term of X ( η ) a r e not imp ortant, b ecause t he c hange of the former induces that of the o v erall normalization of r X, D n,k and the shift of t he latter induces that of r X, D n, 0 . The sin usoidal co ordinat e η ( x ) ( η ( x ) = x in the presen t case) satisfies [41, 31] η ( x − i γ 2 ) n +1 − η ( x + i γ 2 ) n +1 η ( x − i γ 2 ) − η ( x + i γ 2 ) = n X k =0 g ′ ( k ) n η ( x ) n − k ( n ∈ Z ≥ 0 ) , (3.8) where g ′ ( k ) n is giv en b y [42] g ′ ( k ) n = θ ( k : ev en ) ( − 1) k 2 2 − k  n + 1 k + 1  . (3.9) Here θ ( P ) is a step function for a prop o sition P ; θ ( P ) = 1 for P : true, θ ( P ) = 0 for P : f a lse. F or a p o lynomial p ( η ) in η , let us define a p olynomial in η , I [ p ]( η ), as follows : p ( η ) = n X k =0 a k η k 7→ I [ p ]( η ) = n +1 X k =0 b k η k , (3.10) where b k ’s are defined b y b k +1 = 1 g ′ (0) k  a k − n X j = k +1 g ′ ( j − k ) j b j +1  ( k = n, n − 1 , . . . , 1 , 0 ) , b 0 = 0 . (3.11) 11 The constant term of I [ p ]( η ) is c hosen to b e zero. It is easy to show that this p o lynomial I [ p ]( η ) = P ( η ) satisfies ˇ P ( x − i γ 2 ) − ˇ P ( x + i γ 2 ) η ( x − i γ 2 ) − η ( x + i γ 2 ) = ˇ p ( x ) , (3.12) where ˇ P ( x ) = P ( η ( x )) and ˇ p ( x ) = p ( η ( x )). F or the denominator p olynomial Ξ D ( η ) and a p olynomial in η , Y ( η )( 6 = 0), w e set X ( η ) = X D ,Y ( η ) as X ( η ) = I  Ξ D Y  ( η ) , deg X ( η ) = L = ℓ D + deg Y ( η ) + 1 , (3.13) where Ξ D Y means a p olynomial (Ξ D Y )( η ) = Ξ D ( η ) Y ( η ). Note that L ≥ M + 1 b ecause of ℓ D ≥ M . The minimal degree o ne, whic h cor r espo nds to Y ( η ) = 1, is X min ( η ) = I  Ξ D  ( η ) , deg X min ( η ) = ℓ D + 1 . (3.14) Then w e ha v e the followin g theorem. Theorem 2 F or any p olynomia l Y ( η ) ( 6 = 0) , we take X ( η ) = X D ,Y ( η ) as ( 3.13) . Then the multi-indexe d Meixner-Pol laczek and c ontinuous Hahn ortho gonal p olynom ials P D ,n ( η ) satisfy 1 + 2 L term r e curr enc e r elations with c onstant c o efficien ts: X ( η ) P D ,n ( η ) = L X k = − L r X, D n,k P D ,n + k ( η ) , (3.15) which h o l d for n ∈ Z ≥ 0 . Her e r X, D n,k ’s ar e c onstants. Remark 1 By defining r X, D n,k = 0 for n < 0, (3.15) holds for n ∈ Z . Remark 2 Any p olynomial X ( η ) giving the recurrence relations with constant co efficien ts m ust hav e the form (3.13) [31]. Remark 3 Man y parts of the deriv ation in [31] and App endix B of [35] are done algebraically , but some parts use the orthogonality . So w e can not conclude that Theorem 2 holds for P D ,n ( η ; λ ) with an y λ and D . How ev er, explic it calculation for small M , d j , n and deg Y suggests the follo wing conjecture. Conjecture 1 The or em 2 holds for P D ,n ( η ; λ ) with any λ and D (na mely P D ,n ( η ; λ ) may not b e ortho gonal p olynomials). Remark 4 Direct ve rification o f this theorem is rather straigh tforw ard fo r lo w er M and smaller d j , n and deg Y , b y a computer algebra system , e.g. Mathematica. The co efficien ts 12 r X, D n,k are explicitly obtained for small d j and n . Ho w ev er, t o obtain the closed expression of r X, D n,k for general n is not an easy task ev en for small d j , and it is a differen t kind of problem. W e presen t some examples in § 3 .3 and App endix D. Remark 5 Since Y ( η ) is arbitrary , w e obtain infinitely many recurrence relations. How ev er not all o f them a re indep enden t. The relations among them are unclear. F or ‘ M = 0 case’ (namely , ordinary orthogonal p olynomials), it is trivial that recurrence relations obtained from a rbitrary Y ( η ) (deg Y ≥ 1) are deriv ed b y the three term recurrence r elat io ns. Let us consider some prop erties o f the co efficien t r X, D n,k . By using the orthogonality relation (A.28) and the recurrence relatio ns ( 3 .15), w e obtain the relations among them, r X, D n + k , − k = h D ,n + k h D ,n r X, D n,k (1 ≤ k ≤ L ) . (3.16) Explicit forms of h D , n + k h D , n with k ≥ 0 are MP : h D ,n + k h D ,n = ( n + 2 a ) k ( n + 1) k M Y j =1 n + 2 a − d j − 1 + k n + 2 a − d j − 1 , (3.17) cH : h D ,n + k h D ,n = ( n + a 1 + a ∗ 1 , n + a 1 + a ∗ 2 , n + a 2 + a ∗ 1 , n + a 2 + a ∗ 2 ) k ( n + 1 , n + b 1 − 1) k 2 n + b 1 − 1 2 n + b 1 − 1 + 2 k × M I Y j =1 ( n + a 1 + a ∗ 1 − 1 − d I j + k )( n + a 2 + a ∗ 2 + d I j + k ) ( n + a 1 + a ∗ 1 − 1 − d I j )( n + a 2 + a ∗ 2 + d I j ) × M II Y j =1 ( n + a 2 + a ∗ 2 − 1 − d II j + k )( n + a 1 + a ∗ 1 + d II j + k ) ( n + a 2 + a ∗ 2 − 1 − d II j )( n + a 1 + a ∗ 1 + d II j ) . (3.18) The v a lues of P D ,n ( η ) at some sp ecific v alues η 0 are kno wn explicitly as (B.6) and (C.7). By substituting η 0 for η in (3.15), w e ha v e X ( η 0 ) P D ,n ( η 0 ) = L P k = − L r X, D n,k P D ,n + k ( η 0 ), whic h giv es r X, D n, 0 = X ( η 0 ) − L X k = − L k 6 =0 P D ,n + k ( η 0 ) P D ,n ( η 0 ) r X, D n,k . (3.19) Therefore it is sufficien t to find r X, D n,k (1 ≤ k ≤ L ). The top co efficien t r X, D n,L is easily obtained b y comparing the highest degree terms, r X, D n,L = c X c P D ,n c P D ,n + L , (3.20) where c X is the co efficien t of the highest term of X ( η ) = c X η L + (lo w er order terms) and c P D ,n is giv en by (B.2) and (C.2). F o r later use, w e provide a conjecture ab out r X, D n, 0 . 13 Conjecture 2 As a function of n , the c o effi c i e nts r X, D n, 0 has the fol lowing form, r X, D n, 0 = − I ( z ) Q L j =1 α j ( z ) α 2 L +1 − j ( z )     z = E n , I ( z ) : a p olynomial in z , (3.21) wher e α j ( z ) α 2 L +1 − j ( z ) wil l b e given in (4.1 0) . The de gr e e of I ( z ) is deg I = L for MP, deg I ≤ 2 L for cH. Note that t his p olynomial I ( z ) is not hing to do with the map I [ · ] in (3.1 0). 3.3 Examples F or illustration, w e presen t some examples of the co efficien ts r X, D n,k of the recurrence relations (3.15) for m ulti- indexed orthogonal p olynomials. See App endix D for non-or t ho gonal case. Except for Ex.1 in § 3 .3.1, we presen t only r X, D n,k (1 ≤ k ≤ L ), b ecause r X, D n,k ( − L ≤ k ≤ 0) are obtained b y (3.16) –(3.19). 3.3.1 m ulti-indexed Meixner-P ollaczek p olynomials Ex.1 D = { 2 } , Y ( η ) = 1 ( ⇒ ℓ D = 2 , X ( η ) = X min ( η ) , L = 3) : 7-term recurrence relations X ( η ) = η 12  8 sin 2 φ · η 2 − 6(2 a − 3) sin 2 φ · η + 12 a 2 − 24 a + 13 + (12 a 2 − 36 a + 23) cos 2 φ  , r X, D n, 3 = ( n + 1) 3 12 sin φ 2 a + n − 3 2 a + n , r X, D n, − 3 = (2 a + n − 3) 3 12 sin φ , r X, D n, 2 = − 1 2 ( n + 1) 2 (2 a + n − 3) cot φ, r X, D n, − 2 = − 1 2 (2 a + n − 3) 3 cot φ, r X, D n, 1 = ( n + 1)(2 a + n − 3) 4 sin φ  4 a + 3 n + 2(2 a + n − 1) cos 2 φ  , r X, D n, − 1 = (2 a + n − 3)(2 a + n − 1) 4 sin φ  4 a + 3 n − 3 + 2(2 a + n − 2 ) cos 2 φ  , r X, D n, 0 = − 1 6  (2 a − 3)(2 a − 1)(7 a − 1) + 2(3 6 a 2 − 60 a + 19) n + 6(8 a − 7) n 2 + 10 n 3  cot φ + 1 24 (2 a + 2 n − 1)  4 a (7 a + 5 n − 16) + 33 − 22 n + 4 n 2  sin 2 φ. The p olynomial I ( z ) (3.21) is I ( z ) = − 48 sin φ sin 2 φ  (4 + cos 2 φ ) z 3 + 6  6 a − 5 + 2( a − 1) cos 2 φ  sin φ · z 2 + 4  4(6 a 2 − 9 a + 2) + (12 a 2 − 24 a + 11) cos 2 φ  sin 2 φ · z + (2 a − 3)(2 a − 1)  14 a + 7 + (14 a − 11) cos 2 φ  sin 3 φ  . 14 Ex.2 D = { 1 , 2 } , Y ( η ) = 1 ( ⇒ ℓ D = 2 , X ( η ) = X min ( η ) , L = 3): 7- term recurrence relat io ns X ( η ) = 2 sin φ 3 η  2 sin 2 φ · η 2 − 3( a − 1) sin 2 φ · η + 3 a 2 − 9 a + 7 + (3 a 2 − 6 a + 2) cos 2 φ  , r X, D n, 3 = 1 6 ( n + 1) 3 (2 a + n − 3) 2 (2 a + n ) 2 , r X, D n, 2 = − ( n + 1) 2 (2 a + n − 3) 2 2 a + n cos φ, r X, D n, 1 = 1 2 ( n + 1)(2 a + n − 3)  4 a + 3 n − 4 + 2(2 a + n − 2 ) cos 2 φ  . The p olynomial I ( z ) (3.21) is I ( z ) = − 96 sin 2 φ sin 2 φ  (4 + cos 2 φ ) z 3 + 12( a − 1)(3 + cos 2 φ ) sin φ · z 2 + 4  4(6 a 2 − 12 a + 5) + (12 a 2 − 24 a + 11) cos 2 φ  sin 2 φ · z + 8( a − 1)  a (7 a − 11) + (7 a 2 − 14 a + 6) cos 2 φ  sin 3 φ  . Ex.3 D = { 2 } , Y ( η ) = η ( ⇒ ℓ D = 2 , L = 4): 9-term recurrence relations X ( η ) = η 24  12 sin 2 φ · η 3 − 8(2 a − 3) sin 2 φ · η 2 + 3  4 a 2 − 8 a + 5 + (4 a 2 − 12 a + 7) cos 2 φ  η − 2( 2 a − 3) sin 2 φ  , r X, D n, 4 = ( n + 1) 4 32 sin 2 φ 2 a + n − 3 2 a + n + 1 , r X, D n, 3 = − ( n + 1) 3 cos φ 12 sin 2 φ 2 a + n − 3 2 a + n (5 a + 3 n ) , r X, D n, 2 = ( n + 1) 2 8 sin 2 φ (2 a + n − 3)  2(2 a + 2 n + 1) + (4 a + 3 n ) cos 2 φ  , r X, D n, 1 = − ( n + 1) cos φ 4 sin 2 φ (2 a + n − 3)  4 a ( a + 1) + (11 a + 1) n + 5 n 2 + 2( a + n )(2 a + n − 1) cos 2 φ  . The p olynomial I ( z ) (3.21) is I ( z ) = − 192 sin 2 φ  3(18 + 16 cos 2 φ + cos 4 φ ) z 4 + 4  36(4 a − 3) + 4(34 a − 27) cos 2 φ + (10 a − 9) cos 4 φ  sin φ · z 3 + 12  2(86 a 2 − 119 a + 30) + 8(22 a 2 − 33 a + 10) cos 2 φ + (16 a 2 − 28 a + 11) cos 4 φ  sin 2 φ · z 2 + 16  3(56 a 3 − 98 a 2 + 39 a − 9) + 4(4 8 a 3 − 96 a 2 + 50 a − 9) cos 2 φ + (24 a 3 − 60 a 2 + 44 a − 9) cos 4 φ  sin 3 φ · z + (2 a − 3)  3(2 a + 1)(68 a 2 − 4 a + 1) + 4(2 a − 1)(2 a + 1 ) (34 a − 3) cos 2 φ + (2 a − 1)(68 a 2 − 80 a + 15) cos 4 φ  sin 4 φ  . 15 Ex.4 D = { 4 } , Y ( η ) = 1 ( ⇒ ℓ D = 4 , X ( η ) = X min ( η ) , L = 5) : 11-term recurrence relations X ( η ) = η 960  128 sin 4 φ · η 4 − 320(2 a − 5) cos φ sin 3 φ · η 3 + 160  4 a 2 − 16 a + 17 + (4 a 2 − 20 a + 23) cos 2 φ  sin 2 φ · η 2 − 80(2 a − 5)  4 a 2 − 8 a + 5 + (4 a 2 − 20 a + 19) cos 2 φ  cos φ sin φ · η + 240 a 4 − 1440 a 3 + 3160 a 2 − 3000 a + 1067 + 4(80 a 4 − 560 a 3 + 1360 a 2 − 1380 a + 511) cos 2 φ + (80 a 4 − 800 a 3 + 2760 a 2 − 3800 a + 1689) cos 4 φ  , r X, D n, 5 = ( n + 1) 5 240 sin φ 2 a + n − 5 2 a + n , r X, D n, 4 = − 1 24 ( n + 1) 4 (2 a + n − 5) cot φ, r X, D n, 3 = ( n + 1) 3 48 sin φ (2 a + n − 5)  8 a + 5 n + 4(2 a + n − 1) cos 2 φ  , r X, D n, 2 = − 1 6 ( n + 1) 2 (2 a + n − 5)(2 a + n − 1) cot φ  2 a + 2 n + 1 + (2 a + n − 2) cos 2 φ  , r X, D n, 1 = n + 1 24 sin φ (2 a + n − 5)(2 a + n − 1)  12 a 2 + 6 a (4 n − 1) + 10 n ( n − 1) + 2(2 a + n − 2)(4 a + 5 n ) cos 2 φ + (2 a + n − 3) 2 cos 4 φ  . The p olynomial I ( z ) (3.21) is I ( z ) = − 3840 sin 3 φ sin 2 φ  (38 + 24 cos 2 φ + cos 4 φ ) z 5 + 10  8(7 a − 8) + 4(1 0 a − 13) cos 2 φ + (2 a − 3) cos 4 φ  sin φ · z 4 + 20  2(78 a 2 − 173 a + 78) + 4(32 a 2 − 82 a + 47) cos 2 φ + (8 a 2 − 24 a + 17) cos 4 φ  sin 2 φ · z 3 + 40  4(2 a − 1)(25 a 2 − 67 a + 35) + 4(48 a 3 − 180 a 2 + 200 a − 65) cos 2 φ + (2 a − 3)(8 a 2 − 24 a + 15) cos 4 φ  sin 3 φ · z 2 + 32  280 a 4 − 1100 a 3 + 1310 a 2 − 645 a + 131 + 2(160 a 4 − 760 a 3 + 1180 a 2 − 690 a + 119 ) cos 2 φ + (40 a 4 − 240 a 3 + 510 a 2 − 450 a + 137) cos 4 φ  sin 4 φ · z + (2 a − 5)(2 a − 1)  744 a 3 − 828 a 2 + 406 a − 297 + 4(2 a − 3)(2 a + 1)(62 a − 57) cos 2 φ + (2 a − 3)(124 a 2 − 352 a + 193 ) cos 4 φ  sin 5 φ  . W e ha v e also obtained 9- term recurrence relations fo r D = { 1 , 2 } with non- minimal X ( η ) 16 ( Y ( η ) = η ), and 11-term recurrence relations for D = { 1 , 4 } , { 2 , 3 } , { 1 , 2 , 4 } , { 1 , 2 , 3 , 4 } with X ( η ) = X min ( η ) and D = { 2 } , { 1 , 2 } with non- minimal X ( η ) ( Y ( η ) = η 2 ). Since the explicit forms of r X, D n,k are somewhat lengthy , we do not write dow n them here. 3.3.2 m ulti-indexed con tin uous Hahn p olynomials W e set σ 1 = a 1 + a ∗ 1 , σ 2 = a 1 a ∗ 1 , σ ′ 1 = a 2 + a ∗ 2 and σ ′ 2 = a 2 a ∗ 2 . Ex.1 D = { 2 I } ( M I = 1 , M II = 0), Y ( η ) = 1 ( ⇒ ℓ D = 2 , X ( η ) = X min ( η ) , L = 3): 7- term recurrence relations X ( η ) = η 24  4( σ 1 − σ ′ 1 − 4) 2 η 2 + 6 i ( σ 1 − σ ′ 1 − 3)  a 1 − a ∗ 1 + 2( a 1 a ∗ 2 − a ∗ 1 a 2 ) + 3( a 2 − a ∗ 2 )  η + 12 +  36 a 2 ( a 2 + 1) − 2 σ ′ 1 − 24 σ ′ 2 − 7  σ 1 +  1 − 12 a 2 ( a 2 + 1)  σ 2 1 + 24 a 2 ( a 2 + 1)( a 2 1 − 3 a 1 + σ 2 ) − σ ′ 1 (23 σ ′ 1 + 17) − 12 a 1 ( a 1 − 3) σ ′ 1 ( σ ′ 1 + 1) + 24( a 2 1 − 3 a 1 + 3 + σ 2 ) σ ′ 2  , r X, D n, 3 = ( σ 1 − σ ′ 1 − 4) 2 ( n + 1) 3 ( n + σ 1 − 3)( n + b 1 − 1) 3 6( n + σ 1 )(2 n + b 1 − 1) 6 , r X, D n, 2 = i ( a 1 − a ∗ 1 − a 2 + a ∗ 2 )( σ 1 − σ ′ 1 − 3)( b 1 − 2)( n + 1) 2 × ( n + σ 1 − 3)( n + σ ′ 1 + 2)( n + b 1 − 1) 2 2(2 n + b 1 − 2) 5 (2 n + b 1 + 4) , r X, D n, 1 = ( n + σ 1 − 3)( n + σ ′ 1 + 2)( n + b 1 − 1) 2(2 n + b 1 − 3) 4 (2 n + b 1 + 2) 2 × A, (3.22) where A is a p olynomial of degree 5 in n and w e presen t it in App endix D.3 , (D.1), b ecause it is a bit long . The p olynomial I ( z ) of degree 4 (3.21 ) is o mitted b ecause it has a length y expression. The example with D = { 2 II } ( M I = 0 , M II = 1) and Y ( η ) = 1 can b e obtained b y exc hanging a 1 and a 2 . W e ha ve also obtained 11-term recurrence relations fo r D = { 4 I } , { 4 II } with X ( η ) = X min ( η ). Since the explicit forms of r X, D n,k are somewhat lengthy , w e do not write do wn them here. 4 Generalize d Clos ure Relatio n s and C reation/An ni- hilation Op erators In this section we discuss the generalized closure relations and the creation/annihilation op er- ators o f the m ulti-indexed Meixner-P ollaczek and contin uous Hahn idQM systems described 17 b y H D (A.19). First let us recapitulate the essence o f the ( g eneralized) closure relation [34]. The closure relation of order K is an alg ebraic relatio n b etw een a Hamiltonian H and some op era t o r X (= X ( η ( x )) = ˇ X ( x )) [34]: (ad H ) K X = K − 1 X i =0 (ad H ) i X · R i ( H ) + R − 1 ( H ) , (4.1) where (ad H ) X = [ H , X ], (a d H ) 0 X = X and R i ( z ) = R X i ( z ) is a p olynomial in z . The original closure relation [43, 6] corresp onds to K = 2. Since the closure relatio n of order K implies that of or der K ′ > K , w e are in terested in the smallest integer K satisfying (4.1). W e assume that the matrix A = ( a ij ) 1 ≤ i,j ≤ K ( a i +1 ,i = 1 ( 1 ≤ i ≤ K − 1 ) , a i +1 ,K = R i ( z ) (0 ≤ i ≤ K − 1 ) , a ij = 0 ( o thers)) has K distinct real non-v anishing eigen v alues α i = α i ( z ) for z ≥ 0, whic h are indexed in decreasing order α 1 ( z ) > α 2 ( z ) > · · · > α K ( z ). Then w e obtain the exact Heisen b erg solution of X , X H ( t ) def = e i H t X e − i H t = ∞ X n =0 ( it ) n n ! (ad H ) n X = K X j =1 a ( j ) e iα j ( H ) t − R − 1 ( H ) R 0 ( H ) − 1 . (4.2) Here a ( j ) = a ( j ) ( H , X ) (1 ≤ j ≤ K ) are creation or a nnihilation op erators, a ( j ) =  K X i =1 (ad H ) i − 1 X · p ij ( H ) + R − 1 ( H ) α j ( H ) − 1  K Y k =1 k 6 = j ( α j ( H ) − α k ( H )) − 1 , (4.3) where p ij ( z ) (1 ≤ i, j ≤ K ) are p ij ( z ) = α j ( z ) K − i − K − i X k =1 R K − k ( z ) α j ( z ) K − i − k . (4.4) Let us consider the idQM systems describ ed b y the m ulti- indexed Meixner-P ollaczek and con tinuous Hahn p olynomials. The Hamiltonian is H D (A.19) a nd a candidate of the op erator X is a p olynomial X ( η ( x )) = ˇ X ( x ) discuss ed in § 3.2. The closure relation (4.1) is no w (ad H D ) K X = K − 1 X i =0 (ad H D ) i X · R i ( H D ) + R − 1 ( H D ) . (4.5) F rom the for m of H D , the p olynomials R i ( z ) = R X i ( z ) hav e at most the followin g degrees, R i ( z ) = K − i X j =0 r ( j ) i z j (0 ≤ i ≤ K − 1) , R − 1 ( z ) = K X j =0 r ( j ) − 1 z j , (4.6) 18 where r ( j ) i = r X ( j ) i are co efficien t s. Let us define α j ( z ) (1 ≤ j ≤ 2 L ) as follows: MP : α j ( z ) = ( 2( L + 1 − j ) sin φ (1 ≤ j ≤ L ) − 2( j − L ) sin φ ( L + 1 ≤ j ≤ 2 L ) , (4.7) cH : α j ( z ) = ( ( L + 1 − j ) 2 + ( L + 1 − j ) p 4 z + ( b 1 − 1) 2 (1 ≤ j ≤ L ) ( j − L ) 2 − ( j − L ) p 4 z + ( b 1 − 1) 2 ( L + 1 ≤ j ≤ 2 L ) . (4.8) F or MP , α j ( z )’s are constan t functions. The pair of α j ( z ) and α 2 L +1 − j ( z ) (1 ≤ j ≤ L ) satisfies α j ( z ) + α 2 L +1 − j ( z ) =  0 : MP 2( L + 1 − j ) 2 : cH , (4.9) α j ( z ) α 2 L +1 − j ( z ) =  − 4( L + 1 − j ) 2 sin 2 φ : MP ( L + 1 − j ) 2  ( L + 1 − j ) 2 − 4 z − ( b 1 − 1) 2  : cH . (4.10) These α j ( z ) satisfy α 1 ( z ) > α 2 ( z ) > · · · > α L ( z ) > 0 > α L +1 ( z ) > α L +2 ( z ) > · · · > α 2 L ( z ) ( z ≥ 0 ) , (4.11) for 0 < φ < π (MP) and b 1 > 2 L (cH). W e remark that α j ( E n ) for cH is square ro ot free, p 4 E n + ( b 1 − 1) 2 = 2 n + b 1 − 1. It is easy to sho w the follo wing: α j ( E n ) =  E n + L +1 − j − E n > 0 (1 ≤ j ≤ L ) E n − ( j − L ) − E n < 0 ( L + 1 ≤ j ≤ 2 L ) . (4.12) Lik e the Wilson and Ask ey-Wilson cases [34], w e conjecture the follow ing. Conjecture 3 T ake X ( η ) as The or e m 2 an d take R i ( z ) ( − 1 ≤ i ≤ 2 L − 1) as fol lows: R i ( z ) = ( − 1) i +1 X 1 ≤ j 1 2 L fo r cH), we obtain (4.12) and a ( j ) φ D n ( x ) = ( r X, D n,L +1 − j φ D n + L +1 − j ( x ) (1 ≤ j ≤ L ) r X, D n, − ( j − L ) φ D n − ( j − L ) ( x ) ( L + 1 ≤ j ≤ 2 L ) , (4.15) − R − 1 ( E n ) R 0 ( E n ) − 1 = r X, D n, 0 , (4.16) where r X, D n,k = 0 for n + k < 0 . Note that (4 .16) is consisten t with Conjecture 2. Therefore a ( j ) (1 ≤ j ≤ L ) and a ( j ) ( L + 1 ≤ j ≤ 2 L ) are creation and annihilatio n op erators, resp ectiv ely . Among them, a ( L ) and a ( L +1) are fundamen ta l, a ( L ) φ D ,n ( x ) ∝ φ D n +1 ( x ) a nd a ( L +1) φ D ,n ( x ) ∝ φ D n − 1 ( x ). F urthermore, X = X min case is the most basic. By the similarity transformation (see (A.35)), the closure relat io n (4.5) b ecomes (ad e H D ) K X = K − 1 X i =0 (ad e H D ) i X · R i ( e H D ) + R − 1 ( e H D ) , (4.17) and the creation/annihilation op erato r s for eigenp olynomials can b e obta ined, ˜ a ( j ) def = ψ D ( x ) − 1 ◦ a ( j ) ( H D , X ) ◦ ψ D ( x ) = a ( j ) ( e H D , X ) , (4.18) 20 ˜ a ( j ) ˇ P D ,n ( x ) = ( r X, D n,L +1 − j ˇ P D ,n + L +1 − j ( x ) (1 ≤ j ≤ L ) r X, D n, − ( j − L ) ˇ P D ,n − ( j − L ) ( x ) ( L + 1 ≤ j ≤ 2 L ) . (4.19) Remark Since the closure relations (4.5) (or (4.17)) is an algebraic relation b et w een H D (or e H D ) and X , it is expected to hold ev en when H D (or e H D ) is singular. Esp ecially w e conjecture that ˜ a ( j ) (4.18) are the creation/annihilation op erators for eigenp olynomials ˇ P D ,n ( x ) with any D , (4.19), whic h can b e v erified b y direct calculation for small d j , n a nd deg Y . F or an illustratio n, w e presen t an example. Let us consider Ex.1 in § 3.3.1. The denomi- nator p olynomial Ξ D ( η ) is Ξ D ( η ) = (1 − cos 2 φ ) η 2 − (2 a − 3) sin 2 φ · η + ( a − 1)  a − 1 + ( a − 2) cos 2 φ  , and R i ( z ) ar e R 5 ( z ) = R 3 ( z ) = R 1 ( z ) = 0 , R 4 ( z ) = 56 sin 2 φ, R 2 ( z ) = − 784 sin 4 φ, R 0 ( z ) = 2304 sin 6 φ, R − 1 ( z ) = 4 8 sin φ sin 2 φ  (4 + cos 2 φ ) z 3 + 6  6 a − 5 + 2( a − 1) cos 2 φ  sin φ · z 2 + 4  4(6 a 2 − 9 a + 2) + (12 a 2 − 24 a + 11) cos 2 φ  sin 2 φ · z + (2 a − 3)(2 a − 1)  14 a + 7 + (14 a − 11 ) cos 2 φ  sin 3 φ  . The creation/annihilation op erat ors for eigenp olynomials ˜ a ( j ) are ˜ a (1) = 1 7680  384 X + 64 sin φ (ad e H ) X − 120 sin 2 φ (ad e H ) 2 X − 20 sin 3 φ (ad e H ) 3 X + 6 sin 4 φ (ad e H ) 4 X + 1 sin 5 φ (ad e H ) 5 X + 1 6 sin 6 φ R − 1 ( e H )  , ˜ a (2) = 1 1920  − 576 X − 144 sin φ (ad e H ) X + 160 sin 2 φ (ad e H ) 2 X + 40 sin 3 φ (ad e H ) 3 X − 4 sin 4 φ (ad e H ) 4 X − 1 sin 5 φ (ad e H ) 5 X − 1 4 sin 6 φ R − 1 ( e H )  , ˜ a (3) = 1 1536  1152 X + 576 sin φ (ad e H ) X − 104 sin 2 φ (ad e H ) 2 X − 52 sin 3 φ (ad e H ) 3 X + 2 sin 4 φ (ad e H ) 4 X + 1 sin 5 φ (ad e H ) 5 X + 1 2 sin 6 φ R − 1 ( e H )  , ˜ a (4) = 1 1536  1152 X − 576 sin φ (ad e H ) X − 104 sin 2 φ (ad e H ) 2 X + 52 sin 3 φ (ad e H ) 3 X + 2 sin 4 φ (ad e H ) 4 X − 1 sin 5 φ (ad e H ) 5 X + 1 2 sin 6 φ R − 1 ( e H )  , 21 ˜ a (5) = 1 1920  − 576 X + 144 sin φ (ad e H ) X + 160 sin 2 φ (ad e H ) 2 X − 40 sin 3 φ (ad e H ) 3 X − 4 sin 4 φ (ad e H ) 4 X + 1 sin 5 φ (ad e H ) 5 X − 1 4 sin 6 φ R − 1 ( e H )  , ˜ a (6) = 1 7680  384 X − 64 sin φ (ad e H ) X − 120 sin 2 φ (ad e H ) 2 X + 20 sin 3 φ (ad e H ) 3 X + 6 sin 4 φ (ad e H ) 4 X − 1 sin 5 φ (ad e H ) 5 X + 1 6 sin 6 φ R − 1 ( e H )  . Here ( a d e H ) i X are (ad e H ) X = V ′ D ( x )  X ( x − iγ ) − X ( x )  e γ p + V ′ ∗ D ( x )  X ( x + iγ ) − X ( x )  e − γ p , (ad e H ) 2 X = V ′ D ( x ) V ′ D ( x − iγ )  X ( x ) − 2 X ( x − iγ ) + X ( x − 2 iγ )  e 2 γ p + V ′ D ( x )  V D ( x ) + V ∗ D ( x ) − V D ( x − iγ ) − V ∗ D ( x − iγ )  X ( x ) − X ( x − iγ )  e γ p + 2 V ′ D ( x ) V ′ ∗ D ( x − iγ )  X ( x ) − X ( x − iγ )  + 2 V ′ ∗ D ( x ) V ′ D ( x + iγ )  X ( x ) − X ( x + iγ )  + V ′ ∗ D ( x )  V D ( x ) + V ∗ D ( x ) − V D ( x + iγ ) − V ∗ D ( x + iγ )  X ( x ) − X ( x + iγ )  e − γ p + V ′ ∗ D ( x ) V ′ ∗ D ( x + iγ )  X ( x ) − 2 X ( x + iγ ) + X ( x + 2 iγ )  e − 2 γ p , and so on (we omit them b ecause they are somewhat lengthy ). By direct calculation, the relations (4.19) can b e v erified for small n . 5 Summary and Comments F ollo wing t he preceding pap ers on the case-(1) m ulti- indexed o r thogonal po lynomials (La - guerre a nd Jacobi cases in oQM [28, 31, 33], Wilson and Ask ey-Wilson cases in idQM [28, 31, 35] and Racah and q - Racah cases in rdQM [35]), w e ha ve discussed t he recurrence relations for the case-(1) m ulti- indexed Meixner-P o llaczek a nd con tin uous Hahn orthogo nal p olynomials in idQM, whose ph ysical range of the co ordinate is the whole r eal line. The 3 + 2 M term recurrence relations with v aria ble dep enden t co efficien ts (3 .6) (Theorem 1) pro- vide an efficien t metho d to calculate the multi-indexed MP and cH p olynomials. The 1 + 2 L term ( L ≥ M + 1) recurrence relatio ns with constan t co efficien ts (3.15) (Theorem 2), and their examples are presen ted. Since Y ( η ) is arbitrary , w e obtain infinitely many recurrence relations. Not all of them are indep endent, but the relatio ns among them are unclear. T o clarify their relations is an imp ortant problem. Corresp onding to the recurrence relatio ns with constan t co efficien ts, the idQM systems describ ed by the multi-index ed MP and cH 22 orthogonal p olynomials satisfy the generalized closure relatio ns (4.5) ( Conjecture 3 ), from whic h the creation and annihilation op erators are obta ined. There are man y creation and annihilation op erator s and it is an in t eresting problem to study their relations. The Hamiltonian of the deformed system is determined b y the denominator p olynomial Ξ D ( η ), whose degree is ℓ D (A.26). There is no restriction on ℓ D for L, J, W and A W cases, whereas the degree ℓ D m ust b e ev en for MP a nd cH cases, in order that the deformed Hamil- tonian is hermitian. This is b ecause the ph ysical rang e of the co ordinate of the deformed MP and cH systems is the whole real line, see (A.27). The range of the co o rdinate of the harmonic oscillator, whose eigenstates a re described b y the Hermite p olynomial, is also the whole real line, but the case-(1) multi-indexe d Hermite orthogonal p olynomials do not exist. In § 2.2.1 w e ha v e defined the multi-indexed MP and cH p olynomials for any index set D , namely ℓ D ma y b e o dd and they ma y not b e o r thogonal p olynomials. The recurrence relations with v ariable dep enden t co efficien ts (3.6 ) for the m ulti-indexed MP and cH p olynomials hold ev en for non-orthogonal case. W e conjecture that the recurrence relations with constan t co effi- cien ts (3.15), the generalized closure relations ( 4.17) and the creation/annihilat io n op erators (4.19) also hold eve n for non- orthogonal case. Ac kn o w ledgmen ts This w ork w as supp orted b y JSPS KAKENHI Grant Num b ers JP19K036 67. A Discrete Quan tum Me chanics With Pure Imaginary Shifts and D eformed Systems In t his App endix we recapitulate the discrete quan tum mec hanics with pure imaginary shifts (idQM) and deformed systems [2, 5, 17, 36]. The dynamical v ariables of idQM are the real co ordinate x ( x 1 < x < x 2 ) and the conjugate momen tum p = − i∂ x , which are gov erned by the follow ing factorized p o sitive semi-definite Hamiltonian: H def = p V ( x ) e γ p p V ∗ ( x ) + p V ∗ ( x ) e − γ p p V ( x ) − V ( x ) − V ∗ ( x ) = A † A , (A.1) A def = i  e γ 2 p p V ∗ ( x ) − e − γ 2 p p V ( x )  , A † def = − i  p V ( x ) e γ 2 p − p V ∗ ( x ) e − γ 2 p  . (A.2) 23 Here the p ot ential function V ( x ) is an analytic function of x and γ is a real constan t. The ∗ - op eration o n an analytic function f ( x ) = P n a n x n ( a n ∈ C ) is defined by f ∗ ( x ) = P n a ∗ n x n , in whic h a ∗ n is the complex conjuga t ion of a n . Since the momen tum op erator app ears in exp o nen t iated forms, the Sc hr¨ odinger equation H φ n ( x ) = E n φ n ( x ) ( n = 0 , 1 , 2 , . . . ) , (A.3) is an a nalytic difference equation with pure imaginary shifts instead of a differen tial equation. W e consider those systems whic h ha v e a square-inte grable groundstate to gether with an infinite n um b er of discrete energy lev els: 0 = E 0 < E 1 < E 2 < · · · . The o rthogonality relation reads ( φ n , φ m ) def = Z x 2 x 1 dx φ ∗ n ( x ) φ m ( x ) = h n δ nm ( n, m = 0 , 1 , 2 , . . . ) , 0 < h n < ∞ . (A.4) The eigenfunctions φ n ( x ) can b e c ho sen ‘real’, φ ∗ n ( x ) = φ n ( x ), a nd the g roundstate w av e- function φ 0 ( x ) is determined as the zero mo de of the op erator A , A φ 0 ( x ) = 0. The norm o f a f unction f ( x ) is | | f | | def = ( f , f ) 1 2 . The Hamiltonian H should b e hermitian. F r o m its form H = A † A , it is formally her- mitian, H † = ( A † A ) † = ( A ) † ( A † ) † = A † A = H . Ho w ev er, the true hermiticit y is defined in terms of the inner pro duct, ( f 1 , H f 2 ) = ( H f 1 , f 2 ) [2, 41 , 1 7]. T o show the hermiticit y o f H , singularit ies of some functions in the rectangular domain D γ are imp orta n t. Here D γ is defined b y [17] D γ def =  x ∈ C   x 1 ≤ Re x ≤ x 2 , | Im x | ≤ 1 2 | γ |  . (A.5) The Meixner-P olla czek (MP), con tin uous Hahn (cH), Wilson (W), Ask ey-Wilson (A W) p olynomials etc. are mem b ers of the Ask ey-sc heme of the (basic) h yp ergeometric orthogonal p olynomials and satisfy the second order a nalytic difference equation with pure imaginary shifts [1]. These orthogona l p o lynomials can b e studied in the framew ork of idQM, in whic h they a pp ear as part of the eigenfunction as f o llo ws: φ n ( x ) = φ 0 ( x ) ˇ P n ( x ) , ˇ P n ( x ) def = P n  η ( x )  ( n = 0 , 1 , 2 , . . . ) . (A.6) Here η ( x ) is a sin usoidal co ordinate [43, 41] a nd P n ( η ) is a orthog onal p olynomial of degree n in η . The orthog o nalit y relation (A.4) giv es that of ˇ P n ( x ), Z x 2 x 1 dx φ 0 ( x ) 2 ˇ P n ( x ) ˇ P m ( x ) = h n δ nm ( n, m = 0 , 1 , 2 , . . . ) . (A.7) 24 W e call this idQM system b y the name of the orthogonal p olynomial: MP system, cH system, W system, A W system etc. These idQM syste ms ha v e the prop ert y of shap e inv ariance, whic h is a sufficien t condition for exact solv ability . Concrete idQM systems hav e a set of parameters λ = ( λ 1 , λ 2 , . . . ). V arious quan tities dep end on them and their dep endence is expresse d lik e, f = f ( λ ), f ( x ) = f ( x ; λ ). (W e sometimes omit writing λ -dep endence, when it do es not cause confusion.) The shap e in v ariant condition is the follow ing [2, 41, 5]: A ( λ ) A ( λ ) † = κ A ( λ + δ ) † A ( λ + δ ) + E 1 ( λ ) , (A.8) where κ is a real p ositiv e constant and δ is the shift o f the parameters. This condition com bined with the Crum’s theorem allow s the wa v efunction φ n ( x ) and energy eigen v alue E n of the excited states to b e expres sed in terms of the ground state wa v efunction φ 0 ( x ) and the first excited state energy eigen v alue E 1 with shifted parameters. As a consequence of the shap e in v ariance, we hav e A ( λ ) φ n ( x ; λ ) = f n ( λ ) φ n − 1 ( x ; λ + δ ) , A ( λ ) † φ n − 1 ( x ; λ + δ ) = b n − 1 ( λ ) φ n ( x ; λ ) , (A.9) where f n ( λ ) and b n − 1 ( λ ) are some constants satisfying f n ( λ ) b n − 1 ( λ ) = E n ( λ ). These rela- tions can b e rewritten a s the fo rw ard and bac kward shift relations: F ( λ ) ˇ P n ( x ; λ ) = f n ( λ ) ˇ P n − 1 ( x ; λ + δ ) , B ( λ ) ˇ P n − 1 ( x ; λ + δ ) = b n − 1 ( λ ) ˇ P n ( x ; λ ) . (A.10) Here t he fo r ward and backw ard shift op erators F ( λ ) and B ( λ ) a re defined b y F ( λ ) def = φ 0 ( x ; λ + δ ) − 1 ◦ A ( λ ) ◦ φ 0 ( x ; λ ) = iϕ ( x ) − 1 ( e γ 2 p − e − γ 2 p ) , (A.11) B ( λ ) def = φ 0 ( x ; λ ) − 1 ◦ A ( λ ) † ◦ φ 0 ( x ; λ + δ ) = − i  V ( x ; λ ) e γ 2 p − V ∗ ( x ; λ ) e − γ 2 p  ϕ ( x ) , (A.12) where ϕ ( x ) is an auxiliary function ( ϕ ( x ) ∝ η ( x − i γ 2 ) − η ( x + i γ 2 )). The difference op erator e H acting on the p olynomial eigenfunctions is square ro ot free: e H def = φ 0 ( x ) − 1 ◦ H ◦ φ 0 ( x ) = B F = V ( x )( e γ p − 1) + V ∗ ( x )( e − γ p − 1) , (A.13) e H ˇ P n ( x ) = E n ˇ P n ( x ) ( n = 0 , 1 , 2 , . . . ) . (A.14) By the Darb o ux transformation, we can deform idQM systems kee ping their exact solv- abilit y . The m ulti-step Darb o ux transformations with virtual state wa v efunctions as seed 25 solutions giv e iso-sp ectral deformations and the case-(1) multi-indexe d orthogonal po lyno- mials are obtained [1 7, 36]. The virtual state wa v efunctions are obtained b y using the twis t op eration. The t wist o p eration t is a map for parameters λ and giv es a linear relation b et w een tw o Hamiltonia ns: H ( λ ) = α ( λ ) H  t ( λ )  + α ′ ( λ ) , (A.15) where α and α ′ are constan ts. The constant ˜ δ is introduced as t ( λ + β δ ) = t ( λ ) + β ˜ δ ( ∀ β ∈ R ). There are tw o types o f t wist op erations (t yp e I and I I) for cH, W and A W systems , and one ty p e of t wist op eration for MP system. The virtual state w av efunctions ˜ φ v ( x ) are obtained from the eigenfunction φ n ( x ) (A.6) as fo llo ws: ˜ φ v ( x ; λ ) def = φ v  x ; t ( λ )  = φ 0  x ; t ( λ )  ˇ P v  x ; t ( λ )  , ˜ ξ v ( x ; λ ) def = ξ v  η ( x ); λ  def = ˇ P v  x ; t ( λ )  = P v  η ( x ); t ( λ )  , (A.16) whic h satisfies the Sc hr¨ odinger equation H ˜ φ v ( x ) = ˜ E v ˜ φ v ( x ) with the virtual state energy ˜ E v , ˜ E v ( λ ) = α ( λ ) E v  t ( λ )  + α ′ ( λ ) . (A.17) The Hamiltonian is deformed as H → H d 1 → H d 1 d 2 → · · · → H d 1 ...d s → · · · → H d 1 ...d M = H D b y M - step D a rb oux tr a nsformations with virtual state w av efunctions as seed solutions. Here the index set D = { d 1 , . . . , d M } ( d j : m utually distinct) lab els the virtual state w av efunctions used in the transfor ma t io ns. Exactly sp eaking, D is an or dered set. F or cH, W and A W systems , there a re t w o types of virtual states (t yp e I and I I) and D is D = { d 1 , . . . , d M } = { d I 1 , . . . , d I M I , d II 1 , . . . , d II M II } ( M = M I + M II , d I j : mu tually distinct, d II j : m utually distinct). V arious quan tit ies of the defor med systems are denoted as H D , φ D n , A D , etc. The Sc hr¨ odinger equation of the deformed system is H D φ D n ( x ) = E n φ D n ( x ) ( n = 0 , 1 , 2 , . . . ) . (A.18) The deformed Hamiltonian H D and eigenfunctions φ D n ( x ) are giv en b y H D def = p V D ( x ) e γ p q V ∗ D ( x ) + q V ∗ D ( x ) e − γ p p V D ( x ) − V D ( x ) − V ∗ D ( x ) = A † D A D , (A.19) A D def = i  e γ 2 p q V ∗ D ( x ) − e − γ 2 p p V D ( x )  , A † D def = − i  p V D ( x ) e γ 2 p − q V ∗ D ( x ) e − γ 2 p  , (A.20) V D ( x ; λ ) def = V ( x ; λ ′ ) ˇ Ξ D ( x + i γ 2 ; λ ) ˇ Ξ D ( x − i γ 2 ; λ ) ˇ Ξ D ( x − iγ ; λ + δ ) ˇ Ξ D ( x ; λ + δ ) , (A.21) 26 φ D n ( x ) def = Aψ D ( x ) ˇ P D ,n ( x ) , ψ D ( x ; λ ) def = φ 0 ( x ; λ ′ ) q ˇ Ξ D ( x − i γ 2 ; λ ) ˇ Ξ D ( x + i γ 2 ; λ ) , (A.22) where A and λ ′ are A = ( κ − 1 4 M ( M +1) α ( λ ′ ) 1 2 M : MP κ − 1 4 M I ( M I +1) − 1 4 M II ( M II +1)+ 5 2 M I M II α I ( λ ′ ) 1 2 M I α II ( λ ′ ) 1 2 M II : cH,W,A W , (A.23) λ ′ = ( λ + M ˜ δ : MP λ [ M I ,M II ] def = λ + M I ˜ δ I + M II ˜ δ II : cH,W,A W . (A.24) Note that A = 1 for MP , cH and W. Here ˇ Ξ D ( x ) and ˇ P D ,n ( x ) are p olynomials in η ( x ), ˇ Ξ D ( x ) def = Ξ D  η ( x )  , ˇ P D ,n ( x ) def = P D ,n  η ( x )  , (A.25) and their explicit fo rms are giv en in [17, 36]. The denominato r p o lynomial Ξ D ( η ) and the m ulti-indexed p olynomial P D ,n ( η ) are p olynomials in η and t heir degrees are ℓ D and ℓ D + n , resp ectiv ely . Here ℓ D is ℓ D def = M X j =1 d j − 1 2 M ( M − 1) +  0 : MP 2 M I M II : cH,W,A W . (A.26) The deformed Hamiltonian H D is hermitian, if the f o llo wing condition is satisfied [17, 36]: The denominator p olynomial ˇ Ξ D ( x ) has no zero in D γ (A.5). (A.27) The eigenfunctions φ D n ( x ) are orthogonal eac h other, whic h giv es the orthog onalit y relation of ˇ P D ,n ( x ) : Z x 2 x 1 dx ψ D ( x ) 2 ˇ P D ,n ( x ) ˇ P D ,m ( x ) = h D ,n δ nm ( n, m = 0 , 1 , 2 , . . . ) , (A.28) h D ,n = A − 2 h n ×    Q M j =1 ( E n − ˜ E d j ) : MP Q M I j =1 ( E n − ˜ E I d I j ) · Q M II j =1 ( E n − ˜ E II d II j ) : cH,W,A W , (A.29) where A is giv en by ( A.2 3). The m ulti- indexed orthogo nal p olynomial P D ,n ( η ) has n zeros in the phys ical region ( η ( x 1 ) < η < η ( x 2 ) for MP , cH, W, η ( x 2 ) < η < η ( x 1 ) for A W), whic h in terlace the n + 1 zeros of P D ,n +1 ( η ) in the phy sical region, and ℓ D zeros in the unphys ical region ( η ∈ C \{ phys ical regio n of η } ). The shap e in v ariance of the original system is inherited b y the deformed systems. By the argumen t of [17], the Hamiltonian H D ( λ ) is shap e in v ariant: A D ( λ ) A D ( λ ) † = κ A D ( λ + δ ) † A D ( λ + δ ) + E 1 ( λ ) . (A.30) 27 As a conseq uence of t he shap e in v aria nce, the actions of A D ( λ ) a nd A D ( λ ) † on the eigen- functions φ D n ( x ; λ ) are A D ( λ ) φ D n ( x ; λ ) = κ M 2 f n ( λ ) φ D n − 1 ( x ; λ + δ ) , A D ( λ ) † φ D n − 1 ( x ; λ + δ ) = κ − M 2 b n − 1 ( λ ) φ D n ( x ; λ ) . (A.31) The forw ard and bac kw ard shift op era t o rs are defined by F D ( λ ) def = ψ D ( x ; λ + δ ) − 1 ◦ A D ( λ ) ◦ ψ D ( x ; λ ) = i ϕ ( x ) ˇ Ξ D ( x ; λ )  ˇ Ξ D ( x + i γ 2 ; λ + δ ) e γ 2 p − ˇ Ξ D ( x − i γ 2 ; λ + δ ) e − γ 2 p  , (A.32) B D ( λ ) def = ψ D ( x ; λ ) − 1 ◦ A D ( λ ) † ◦ ψ D ( x ; λ + δ ) (A.33) = − i ˇ Ξ D ( x ; λ + δ )  V ( x ; λ ′ ) ˇ Ξ D ( x + i γ 2 ; λ ) e γ 2 p − V ∗ ( x ; λ ′ ) ˇ Ξ D ( x − i γ 2 ; λ ) e − γ 2 p  ϕ ( x ) , ( λ ′ is giv en by (A.24)) and their actions on ˇ P D ,n ( x ; λ ) are F D ( λ ) ˇ P D ,n ( x ; λ ) = f n ( λ ) ˇ P D ,n − 1 ( x ; λ + δ ) , B D ( λ ) ˇ P D ,n − 1 ( x ; λ + δ ) = b n − 1 ( λ ) ˇ P D ,n ( x ; λ ) . (A.34) The similarit y transformed Hamiltonian is square ro ot free: e H D ( λ ) def = ψ D ( x ; λ ) − 1 ◦ H D ( λ ) ◦ ψ D ( x ; λ ) = B D ( λ ) F D ( λ ) = V ( x ; λ ′ ) ˇ Ξ D ( x + i γ 2 ; λ ) ˇ Ξ D ( x − i γ 2 ; λ )  e γ p − ˇ Ξ D ( x − iγ ; λ + δ ) ˇ Ξ D ( x ; λ + δ )  + V ∗ ( x ; λ ′ ) ˇ Ξ D ( x − i γ 2 ; λ ) ˇ Ξ D ( x + i γ 2 ; λ )  e − γ p − ˇ Ξ D ( x + iγ ; λ + δ ) ˇ Ξ D ( x ; λ + δ )  . (A.35) By defining V ′ D ( x ) as V ′ D ( x ; λ ) def = V ( x ; λ ′ ) ˇ Ξ D ( x + i γ 2 ; λ ) ˇ Ξ D ( x − i γ 2 ; λ ) , (A.36) it is written a s e H D = V ′ D ( x ) e γ p + V ′ ∗ D ( x ) e − γ p − V D ( x ) − V ∗ D ( x ) . (A.37) The m ulti-indexed orthogonal p olynomials ˇ P D ,n ( x ) are its eigenp olynomials: e H D ˇ P D ,n ( x ) = E n ˇ P D ,n ( x ) ( n = 0 , 1 , 2 , . . . ) . (A.38) 28 B Some Prop ert i e s of the Mu lti-inde xed Meixner-P ol- laczek P o l ynomials W e presen t some prop erties of t he multi-indexe d Meixner-P olla czek p olynomials [36]. • co efficien ts of t he highest degree terms : Ξ D ( η ; λ ) = c Ξ D ( λ ) η ℓ D + (lo w er order terms) , c Ξ D ( λ ) = M Y j =1 c d j  t ( λ )  · Y 1 ≤ j < k ≤ M ( d k − d j ) , (B.1) P D ( η ; λ ) = c P D ,n ( λ ) η ℓ D + n + (lo w er order terms) , c P D ,n ( λ ) = c Ξ D ( λ ) c n ( λ ) M Y j =1 ( − 2 a − n + d j + 1) . (B.2) • ˇ P D , 0 ( x ; λ ) vs ˇ Ξ D ( x ; λ ) : ˇ P D , 0 ( x ; λ ) = A ˇ Ξ D ( x ; λ + δ ) , A = M Y j =1 ( − 2 a + d j + 1) . (B.3) • d j = 0 case : ˇ P D ,n ( x ; λ )    d M =0 = A ˇ P D ′ ,n ( x ; λ + ˜ δ ) , D ′ = { d 1 − 1 , . . . , d M − 1 − 1 } , A = ( − 1) M (2 a + n − 1)(2 sin φ ) M − 1 . (B.4) • v a lues at special p oin ts : Let x 0 and η 0 b e x 0 def = − iγ ( a − 1 2 M ) , η 0 def = η ( x 0 ) . (B.5) Note tha t, as co ordinates x and η , these v alues x 0 and η 0 are unph ysical (they a re imag ina ry). The m ulti-indexed p olynomials P D ,n ( η ) tak e ‘simple’ v alues at these ‘unph ysical’ v a lues η 0 : P D ,n ( η 0 ; λ ) = c P D ,n ( λ ) e iφ ( ℓ D − n ) (2 a ) n (2 sin φ ) ℓ D + n M Y j =1 (1 − 2 a ) d j (1 − 2 a ) j − 1 · M Y j =1 d j + 1 − 2 a d j + 1 − n − 2 a , (B.6) where w e hav e assumed 0 ≤ d 1 < · · · < d M . C Some Prop ertie s of th e Multi-ind e xed Con tin uous Hahn P olynomials W e presen t some prop erties of t he multi-indexe d con tin uous Hahn p olynomials [36]. 29 • co efficien ts of t he highest degree terms : Ξ D ( η ; λ ) = c Ξ D ( λ ) η ℓ D + (lo w er order terms) , c Ξ D ( λ ) = M I Y j =1 c d I j  t I ( λ )  · M II Y j =1 c d II j  t II ( λ )  · Y 1 ≤ j < k ≤ M I ( d I k − d I j ) · Y 1 ≤ j < k ≤ M II ( d II k − d II j ) × M I Y j =1 M II Y k =1 ( − a 2 − a ∗ 2 − d I j + a 1 + a ∗ 1 + d II k ) , (C.1) P D ( η ; λ ) = c P D ,n ( λ ) η ℓ D + n + (lo w er or der terms) , c P D ,n ( λ ) = c Ξ D ( λ ) c n ( λ ) M I Y j =1 ( − a 1 − a ∗ 1 − n + d I j + 1) · M II Y j =1 ( − a 2 − a ∗ 2 − n + d II j + 1) . (C.2) • ˇ P D , 0 ( x ; λ ) vs ˇ Ξ D ( x ; λ ) : ˇ P D , 0 ( x ; λ ) = A ˇ Ξ D ( x ; λ + δ ) , A = M I Y j =1 ( − a 1 − a ∗ 1 + d I j + 1) · M II Y j =1 ( − a 2 − a ∗ 2 + d II j + 1) . (C.3) • d j = 0 case : ˇ P D ,n ( x ; λ )    d I M I =0 = A ˇ P D ′ ,n ( x ; λ + ˜ δ I ) , D ′ = { d I 1 − 1 , . . . , d I M I − 1 − 1 , d II 1 + 1 , . . . , d II M II + 1 } , A = ( − 1) M I ( a 1 + a ∗ 1 + n − 1) M I − 1 Y j =1 ( − a 1 − a ∗ 1 + a 2 + a ∗ 2 + d I j + 1) · M II Y j =1 ( d II j + 1) , (C.4) ˇ P D ,n ( x ; λ )    d II M II =0 = B ˇ P D ′ ,n ( x ; λ + ˜ δ II ) , D ′ = { d I 1 + 1 , . . . , d I M I + 1 , d II 1 − 1 , . . . , d II M II − 1 − 1 } , B = ( − 1) M ( a 2 + a ∗ 2 + n − 1) M II − 1 Y j =1 ( − a 2 − a ∗ 2 + a 1 + a ∗ 1 + d II j + 1) · M I Y j =1 ( d I j + 1) . (C.5) • v a lues at special p oin ts : Let x 0 and η 0 b e x 0 def = − iγ  a ∗ 2 + 1 2 ( M I − M II )  , η 0 def = η ( x 0 ) . (C.6) Note tha t, as co ordinates x and η , these v alues x 0 and η 0 are unph ysical (they a re imag ina ry). The m ulti-indexed p olynomials P D ,n ( η ) tak e ‘simple’ v alues at these ‘unph ysical’ v a lues η 0 : P D ,n ( η 0 ; λ ) 30 = ( − i ) ℓ D + n c P D ,n ( λ )( − 1) P M II j =1 d II j − 1 2 M II ( M II − 1) × M I Y j =1 ( a ∗ 2 − a ∗ 1 + 1 , a 2 + a ∗ 2 ) d I j ( a 2 + a ∗ 2 − a 1 − a ∗ 1 + d I j + 1) d I j · Q 1 ≤ j < k ≤ M I ( a 2 + a ∗ 2 − a 1 − a ∗ 1 + d I j + d I k + 1) M I Q j =1 ( a ∗ 2 − a ∗ 1 + 1 , a 2 + a ∗ 2 ) j − 1 × M II Y j =1 ( a ∗ 1 − a ∗ 2 + 1 , 1 − a 2 − a ∗ 2 ) d II j ( a 1 + a ∗ 1 − a 2 − a ∗ 2 + d II j + 1) d II j · Q 1 ≤ j < k ≤ M II ( a 1 + a ∗ 1 − a 2 − a ∗ 2 + d II j + d II k + 1) M II Q j =1 ( a ∗ 1 − a ∗ 2 + 1 , 1 − a 2 − a ∗ 2 ) j − 1 × M I Y j =1 M II Y k =1 ( a ∗ 2 − a ∗ 1 + j − k )( a 2 + a ∗ 2 + j − k ) a 2 + a ∗ 2 − a 1 − a ∗ 1 + d I j − d II k × ( a 1 + a ∗ 2 , a 2 + a ∗ 2 ) n ( a 1 + a ∗ 1 + a 2 + a ∗ 2 + n − 1) n M I Y j =1 a 2 + a ∗ 2 + d I j + n a 2 + a ∗ 2 + j − 1 · M II Y j =1 d II j + 1 − a 2 − a ∗ 2 d II j + 1 − n − a 2 − a ∗ 2 , (C.7) where w e hav e assumed 0 ≤ d I 1 < · · · < d I M I and 0 ≤ d II 1 < · · · < d II M II . W e remark that a similar form ula can b e obtained by in terc hanging a 1 ↔ a 2 and I ↔ I I. D More Examples for § 3 W e presen t more examples for § 3. Unlik e in § 3.3 , the examples presen t ed here do not satisfy the condition (A.27 ). Namely , the multi-indexe d p o lynomials P D ,n ( η ) are not orthogonal p olynomials. Except for Ex.1 in § D.1, w e prese nt only r X, D n,k (1 ≤ k ≤ L ), b ecause r X, D n,k ( − L ≤ k ≤ 0) are obtained b y (3.16)–(3.19). An explicit form of A in (3.22 ) is also presen ted. D.1 Examples for § 3.3.1 Ex.1 D = { 1 } , Y ( η ) = 1 ( ⇒ ℓ D = 1 , X ( η ) = X min ( η ) , L = 2) : 5-term recurrence relations X ( η ) = η  sin φ · η + 2(1 − a ) cos φ  , r X, D n, 2 = ( n + 1) 2 4 sin φ 2 a + n − 2 2 a + n , r X, D n, − 2 = (2 a + n − 2) 2 4 sin φ , r X, D n, 1 = − ( n + 1)(2 a + n − 2) cot φ, r X, D n, − 1 = − (2 a + n − 2) 2 cot φ, r X, D n, 0 = a (6 a + 10 n − 7) + 3 n 2 − 7 n + 1 2 sin φ − 1 4 (2 a + 2 n − 1)(6 a + 2 n − 5) sin φ. The p olynomial I ( z ) (3.21) is I ( z ) = − 8 sin φ  (2 + cos 2 φ ) z 2 + 2  6 a − 4 + (4 a − 3) cos 2 φ  sin φ · z 31 +  12 a ( a − 1) − 1 + (2 a − 1) ( 6 a − 5) cos 2 φ  sin 2 φ  . Ex.2 D = { 1 } , Y ( η ) = η ( ⇒ ℓ D = 1 , L = 3): 7-term recurrence relations X ( η ) = η 6  4 sin φ · η 2 + 6(1 − a ) cos φ · η + sin φ  , r X, D n, 3 = ( n + 1) 3 12 sin 2 φ 2 a + n − 2 2 a + n + 1 , r X, D n, 2 = − ( n + 1) 2 (3 a + 2 n ) cos φ 4 sin 2 φ 2 a + n − 2 2 a + n , r X, D n, 1 = ( n + 1)(2 a + n − 2) 4 sin 2 φ  2 a + 3 n + 1 + 2( a + n ) cos 2 φ  . The p olynomial I ( z ) (3.21) is I ( z ) = − 48 sin 2 φ  (4 + cos 2 φ ) z 3 + 3  10 a − 6 + (3 a − 2) cos 2 φ  sin φ · z 2 + 2  30 a ( a − 1) + 4 + (12 a 2 − 15 a + 4) cos 2 φ  sin 2 φ · z + ( a − 1)  20 a ( a + 1) − 3 + (2 a − 1)(1 0 a − 3) cos 2 φ  sin 3 φ  . Ex.3 D = { 1 } , Y ( η ) = η 2 ( ⇒ ℓ D = 1 , L = 4 ): 9- term recurrence r elat io ns X ( η ) = η 12  6 sin φ · η 3 + 8(1 − a ) cos φ · η 2 + 3 sin φ · η + 2(1 − a ) cos φ  , r X, D n, 4 = ( n + 1) 4 32 sin 3 φ 2 a + n − 2 2 a + n + 2 , r X, D n, 3 = − ( n + 1) 3 (4 a + 3 n + 2) cos φ 12 sin 3 φ 2 a + n − 2 2 a + n + 1 , r X, D n, 2 = ( n + 1) 2 8 sin 3 φ 2 a + n − 2 2 a + n  5 a ( a + 1) + 2(5 a + 1) n + 4 n 2 + (5 a 2 + 2 a + 8 an + 3 n 2 + n ) cos 2 φ  , r X, D n, 1 = − ( n + 1)(2 a + n − 2) cos φ 4 sin 3 φ  2 a 2 + 4 a + 8 an + 5 n 2 + 2 n + 1 + 2( a + n ) 2 cos 2 φ  . The p olynomial I ( z ) (3.21) is I ( z ) = − 192 sin φ  3(18 + 16 cos 2 φ + cos 4 φ ) z 4 + 4  18(7 a − 4) + 4(29 a − 17) cos 2 φ + (8 a − 5 ) cos 4 φ  sin φ · z 3 + 12  126 a ( a − 1) + 30 + 4(31 a 2 − 33 a + 8) cos 2 φ + (10 a 2 − 12 a + 3) cos 4 φ  sin 2 φ · z 2 + 16  3(32 a 3 − 33 a 2 + 13 a − 6) + 2(5 4 a 3 − 69 a 2 + 29 a − 8) cos 2 φ + ( a − 1)(12 a 2 − 9 a + 1) cos 4 φ  sin 3 φ · z +  3(2 a + 1)(56 a 3 + 4 a 2 − 58 a − 3) + 4(112 a 4 − 16 a 3 − 112 a 2 + 16 a + 3) cos 2 φ + (2 a − 1)(56 a 3 − 100 a 2 + 38 a + 3) cos 4 φ  sin 4 φ  . Ex.4 D = { 3 } , Y ( η ) = 1 ( ⇒ ℓ D = 3 , X ( η ) = X min ( η ) , L = 4) : 9-term recurrence relations X ( η ) = η 12  4 sin 3 φ · η 3 − 16( a − 2) cos φ sin 2 φ · η 2 32 +  12 a ( a − 3) + 29 + (12 a ( a − 4 ) + 43) cos 2 φ  sin φ · η − 2( a − 2)  (2 a − 1) 2 + (4 a ( a − 4) + 11) cos 2 φ  cos φ  , r X, D n, 4 = ( n + 1) 4 48 sin φ 2 a + n − 4 2 a + n , r X, D n, 3 = − 1 6 ( n + 1) 3 (2 a + n − 4) cot φ, r X, D n, 2 = ( n + 1) 2 12 sin φ (2 a + n − 4)  6 a + 4 n + 3(2 a + n − 1) cos 2 φ  , r X, D n, 1 = − 1 6 ( n + 1)(2 a + n − 1)(2 a + n − 4 ) cot φ  4 a + 5 n + 2 + 2(2 a + n − 2) cos 2 φ  . The p olynomial I ( z ) (3.21) is I ( z ) = − 384 sin 3 φ  (18 + 16 cos 2 φ + cos 4 φ ) z 4 + 4  54( a − 1) + 4(13 a − 14) cos 2 φ + (4 a − 5) cos 4 φ  sin φ · z 3 + 4  6(2 a − 1)(19 a − 27) + 8(30 a 2 − 63 a + 28) cos 2 φ + (24 a 2 − 60 a + 35) cos 4 φ  sin 2 φ · z 2 + 16  3(2 a − 1)(16 a 2 − 35 a + 12) + 8(14 a 3 − 42 a 2 + 35 a − 8) cos 2 φ + (4 a − 5)(4 a 2 − 10 a + 5) cos 4 φ  sin 3 φ · z + 3(2 a − 1)  (2 a − 1)(60 a 2 − 116 a − 9) + 4(40 a 3 − 116 a 2 + 70 a + 5) cos 2 φ + (2 a − 3)(20 a 2 − 56 a + 31) cos 4 φ  sin 4 φ  . W e hav e also obtained 9 -term recurrence relations for D = { 1 , 3 } , { 1 , 2 , 3 } with X ( η ) = X min ( η ). Since the explicit forms of r X, D n,k are somewhat lengthy , w e do not write do wn them here. D.2 Examples for § 3.3.2 W e set σ 1 = a 1 + a ∗ 1 , σ 2 = a 1 a ∗ 1 , σ ′ 1 = a 2 + a ∗ 2 and σ ′ 2 = a 2 a ∗ 2 . Ex.1 D = { 1 I } ( M I = 1 , M II = 0), Y ( η ) = 1 ( ⇒ ℓ D = 1 , X ( η ) = X min ( η ) , L = 2): 5- term recurrence relations X ( η ) = η 2  (2 − σ 1 + σ ′ 1 ) η − 2 i ( a 2 − a ∗ 2 + a 1 a ∗ 2 − a ∗ 1 a 2 )  , r X, D n, 2 = (2 − σ 1 + σ ′ 1 )( n + 1) 2 ( n + σ 1 − 2)( n + b 1 − 1) 2 2( n + σ 1 )(2 n + b 1 − 1) 4 , r X, D n, 1 = − i ( a 1 − a ∗ 1 − a 2 + a ∗ 2 )( b 1 − 2)( n + 1)( n + σ 1 − 2)( n + σ ′ 1 + 1)( n + b 1 − 1) (2 n + b 1 − 2) 3 (2 n + b 1 + 2) . The p olynomial I ( z ) (3.21) is I ( z ) 33 = − 4(2 − σ 1 + σ ′ 1 ) z 3 + 2  16 − σ 3 1 − 12 σ 2 + 12 a 2 σ 2 + 6 a 2 1 (2 a 2 − σ ′ 1 ) + 14 σ ′ 1 + 4 σ 2 σ ′ 1 − 3 σ ′ 2 1 + σ ′ 3 1 + σ 2 1 (3 − 6 a 2 + 3 σ ′ 1 ) + σ 1 ( − 10 + 12 a 2 + 6 a 2 2 + 6 σ 2 − 16 σ ′ 1 − 3 σ ′ 2 1 − 4 σ ′ 2 ) + 20 σ ′ 2 − 6 σ ′ 1 σ ′ 2 − 6 a 1 (4 a 2 + 2 a 2 2 − 2 σ ′ 1 − σ ′ 2 1 + 2 σ ′ 2 )  z 2 − 2  6 − 24 σ 2 + 48 a 2 σ 2 + 16 a 2 2 σ 2 + 4 a 3 2 σ 2 + 4 a 2 2 a ∗ 2 σ 2 + 27 σ ′ 1 − 20 σ 2 σ ′ 1 + 4 σ ′ 2 1 + 10 σ 2 σ ′ 2 1 − 11 σ ′ 3 1 − 10 σ 2 σ ′ 3 1 + 3 σ ′ 4 1 + σ 4 1 (1 + 2 σ ′ 1 ) + 16 σ 2 σ ′ 2 + 24 σ ′ 1 σ ′ 2 + 4 σ 2 σ ′ 1 σ ′ 2 − 14 σ ′ 2 1 σ ′ 2 − 2 a 3 1 (6 a 2 + 2 a 2 2 − 3 σ ′ 1 − σ ′ 2 1 + 2 σ ′ 2 ) + σ 3 1 ( − 1 + 6 a 2 + 2 a 2 2 − 8 σ ′ 1 + 10 σ ′ 2 ) − σ 2 1  8 + 24 a 2 + 2 a 3 2 + 2 a 2 2 (4 + a ∗ 2 ) − 19 σ ′ 1 − 12 σ ′ 2 1 + 2 σ 2 (3 + 5 σ ′ 1 ) + 46 σ ′ 2 − 12 σ ′ 1 σ ′ 2  + 2 a 1  − 24 a 2 + 2 a 3 2 + 2 a 2 2 ( − 4 + a ∗ 2 ) + 4 σ ′ 2 1 − σ ′ 3 1 − 8 σ ′ 2 + 2 σ ′ 1 (6 + σ ′ 2 )  + 2 a 2 1  − 6( − 4 + a ∗ 1 ) a 2 + 2 a 3 2 + a 2 2 (8 − 2 a ∗ 1 + 2 a ∗ 2 ) + ( − 4 + a ∗ 1 ) σ ′ 2 1 − σ ′ 3 1 − 2( − 4 + a ∗ 1 ) σ ′ 2 + σ ′ 1 ( − 12 + 3 a ∗ 1 + 2 σ ′ 2 )  − σ 1  − 9 + 2 a 3 2 + 12 a 2 ( − 2 + σ 2 ) + 2 a 2 2 ( − 4 + a ∗ 2 + 2 σ 2 ) + 40 σ ′ 1 + 23 σ ′ 2 1 − 12 σ ′ 3 1 + 2 σ ′ 4 1 − 52 σ ′ 2 + 42 σ ′ 1 σ ′ 2 − 10 σ ′ 2 1 σ ′ 2 + 2 σ 2 ( − 12 − 15 σ ′ 1 + 6 σ ′ 2 1 + 2 σ ′ 2 )   z − 1 2 ( b 1 − 3) 2  − 32 a 2 2 σ 2 − 8 a 3 2 σ 2 − 8 a 2 2 a ∗ 2 σ 2 − 2 σ ′ 1 + 16 σ 2 σ ′ 1 − 7 σ ′ 2 1 + 20 σ 2 σ ′ 2 1 − 5 σ ′ 3 1 + σ 3 1  − 4 a 2 2 + (1 + 2 σ ′ 1 ) 2  − 4 a 2 1  2 a 3 2 + a 2 2 (8 − 2 a ∗ 1 + 2 a ∗ 2 ) + ( − 4 + a ∗ 1 − σ ′ 1 )( σ ′ 2 1 − 2 σ ′ 2 )  + 8 a 1  2 a 3 2 + 2 a 2 2 (2 + a ∗ 2 ) − (2 + σ ′ 1 )( σ ′ 2 1 − 2 σ ′ 2 )  − 32 σ 2 σ ′ 2 + 8 σ ′ 1 σ ′ 2 − 8 σ 2 σ ′ 1 σ ′ 2 + a 3 1 (8 a 2 2 − 4 σ ′ 2 1 + 8 σ ′ 2 ) + σ 2 1  − 1 + 4 a 3 2 + 4 a 2 2 (4 + a ∗ 2 ) + σ ′ 1 − 8 σ ′ 2 1 − 4 σ ′ 3 1 + 12 σ ′ 2 + 16 σ ′ 1 σ ′ 2  − σ 1  2 + 8 a 3 2 + 8 a 2 2 (2 + a ∗ 2 − σ 2 ) + ( − 11 + 16 σ 2 ) σ ′ 2 1 − 12 σ ′ 3 1 + 24 σ ′ 2 − 8 σ 2 σ ′ 2 + 4 σ ′ 1 (2 + 5 σ 2 + 9 σ ′ 2 )   . The example with D = { 1 II } ( M I = 0 , M II = 1) and Y ( η ) = 1 can b e obtained b y exc hanging a 1 and a 2 . W e ha v e also obtained 9-term recurrence relations for D = { 3 I } , { 3 II } , { 1 I , 1 II } with X ( η ) = X min ( η ). Since the explicit forms of r X, D n,k are somewhat lengthy , w e do not write dow n them here. D.3 Explicit form of A in (3. 2 2) A in (3.2 2) = ( σ 1 − σ ′ 1 − 4) 2 n 5 + ( σ 1 − σ ′ 1 − 4) 2 (1 + 2 b 1 ) n 4 34 +  − 36 + 12 σ 2 − 38 a 2 σ 2 + 12 a 2 2 σ 2 + 5 a 3 1 (2 a 2 − σ ′ 1 ) − 5 σ ′ 1 + 6 σ 2 σ ′ 1 + 29 σ ′ 2 1 − σ 2 σ ′ 2 1 + 16 σ ′ 3 1 + σ 3 1 ( − 2 − 5 a 2 + σ ′ 1 ) +  12 + 12 σ 2 + σ ′ 1 ( − 13 + 5 σ ′ 1 )  σ ′ 2 − σ 2 1  5 + a 2 ( − 19 + 6 a 2 ) − 5 σ 2 + σ ′ 1 (19 + 2 σ ′ 1 ) + σ ′ 2  + a 2 1  2 a 2 ( − 19 + 5 a ∗ 1 + 6 a 2 ) + 19 σ ′ 1 − 5 a ∗ 1 σ ′ 1 − 6 σ ′ 2 1 + 12 σ ′ 2  + σ 1  45 − 5 a 3 2 + a 2 2 (13 − 5 a ∗ 2 ) + 2 a 2 ( − 6 + 5 σ 2 ) + σ 2 ( − 19 + σ ′ 1 ) − 6 σ ′ 2 + σ ′ 1 (52 + σ ′ 1 (5 + σ ′ 1 ) + σ ′ 2 )  + a 1  2 a 2 (12 + a 2 ( − 13 + 5 σ ′ 1 )) − 26 σ ′ 2 + σ ′ 1 ( − 12 + 13 σ ′ 1 − 5 σ ′ 2 1 + 10 σ ′ 2 )   n 3 +  − 36 + 5 a 4 2 (2 a 1 − σ 1 ) + 10 a 3 2 a ∗ 2 (2 a 1 − σ 1 ) − 15 σ 1 + 30 σ 2 1 − 7 σ 3 1 + 3 σ 4 1 − σ 5 1 + 12 σ 2 − 7 σ 1 σ 2 − 14 σ 2 1 σ 2 + 5 σ 3 1 σ 2 + a 3 2  2 a 1 ( − 8 + 11 a 1 ) + (8 − 11 σ 1 ) σ 1 + 22 σ 2  + a 2 2 a ∗ 2  2 a 1 ( − 8 + 11 a 1 ) + (8 − 11 σ 1 ) σ 1 + 22 σ 2  + a 2  2 a 1 (12 + a 1 ( − 7 − 14 a 1 + 5 a 2 1 + 2( − 7 + 5 a 1 ) a ∗ 1 )) − ( − 3 + σ 1 ) σ 1 (1 + σ 1 )( − 4 + 5 σ 1 ) + 2( − 7 + σ 1 ( − 14 + 5 σ 1 )) σ 2 + 10 σ 2 2  + a 2 2  2 a 1 ( − 1 + a 1 ( − 26 + 11 σ 1 )) − 52 σ 2 + σ 1 (1 + (26 − 11 σ 1 ) σ 1 + 22 σ 2 )  +  − 65 + a 1 ( − 12 + a 1 (7 + a 1 (14 − 5 a 1 − 10 a ∗ 1 ) + 14 a ∗ 1 )) + 44 σ 1 + 8 σ 2 1 − 6 σ 3 1 + 6( − 1 + σ 1 ) 2 σ 2 − 5 σ 2 2  σ ′ 1 +  a 1 (1 + a 1 (26 − 11 σ 1 )) + σ 1 (38 + ( − 12 + σ 1 ) σ 1 ) + 2( − 7 + 9 σ 2 )  σ ′ 2 1 +  17 + (8 − 11 a 1 ) a 1 + σ 1 (8 + σ 1 ) − 6 σ 2  σ ′ 3 1 + (7 − 5 a 1 ) σ ′ 4 1 − σ ′ 5 1 − 2  − 6 + a 1 + a 2 1 (26 − 11 σ 1 ) + 3( − 1 + σ 1 ) 2 σ 1 + 26 σ 2 − 11 σ 1 σ 2  σ ′ 2 +  − 1 + 2 a 1 ( − 8 + 11 a 1 ) − 18 σ 1 + 22 σ 2  σ ′ 1 σ ′ 2 + 2( − 4 + 5 a 1 + 3 σ 1 ) σ ′ 2 1 σ ′ 2 + 5 σ ′ 3 1 σ ′ 2 + 5(2 a 1 − σ 1 ) σ ′ 2 2  n 2 +  24 − 50 σ 1 + 20 σ 2 1 − 8 σ 3 1 + 8 σ 4 1 − 2 σ 5 1 − 12 σ 2 + 46 σ 1 σ 2 − 41 σ 2 1 σ 2 + 9 σ 3 1 σ 2 + a 4 2  6 a 1 + 8 a 2 1 − σ 1 (3 + 4 σ 1 ) + 8 σ 2  + 2 a 3 2 a ∗ 2  6 a 1 + 8 a 2 1 − σ 1 (3 + 4 σ 1 ) + 8 σ 2  + a 2  2 a 1 ( − 12 + a 1 (46 − 41 a ∗ 1 + a 1 ( − 41 + 9 a 1 + 18 a ∗ 1 ))) − ( − 3 + σ 1 ) σ 1 (4 + σ 1 ( − 14 + 9 σ 1 )) + 92 σ 2 + 2 σ 1 ( − 41 + 9 σ 1 ) σ 2 + 18 σ 2 2  + a 2 2  2 a 1 (10 + a 1 ( − 22 − 3 a ∗ 1 + a 1 ( − 3 + 4 a 1 + 8 a ∗ 1 ))) + 3 σ 3 1 − 4 σ 4 1 + 4 σ 2 ( − 11 + 2 σ 2 ) − 2 σ 1 (5 + 3 σ 2 ) + σ 2 1 (22 + 8 σ 2 )  + a 2 2 σ ′ 1  2 a 1 ( − 5 + a 1 ( − 9 + 8 σ 1 )) − 18 σ 2 + σ 1 (5 + (9 − 8 σ 1 ) σ 1 + 16 σ 2 )  −  14 + a 1 ( − 12 + a 1 (46 − 41 a ∗ 1 + a 1 ( − 41 + 9 a 1 + 18 a ∗ 1 ))) + σ 1 (40 + σ 1 ( − 46 + σ 1 (18 + ( − 5 + σ 1 ) σ 1 ))) + 36 σ 2 + σ 1 ( − 19 − 4( − 3 + σ 1 ) σ 1 ) σ 2 + 9 σ 2 2  σ ′ 1 +  − 36 + a 1 ( − 10 + a 1 (22 + a 1 (3 − 4 a 1 − 8 a ∗ 1 ) + 3 a ∗ 1 )) − 3 σ 3 1 + σ 1 (14 − 6 σ 2 ) + (17 − 4 σ 2 ) σ 2 + σ 2 1 (6 + 4 σ 2 )  σ ′ 2 1 +  − 4 + a 1 (5 + a 1 (9 − 8 σ 1 )) + σ 1 (14 + σ 1 ( − 5 + 2 σ 1 ) − 4 σ 2 ) + 12 σ 2  σ ′ 3 1 +  6 − a 1 (3 + 4 a 1 ) + 5 σ 1 − 4 σ 2  σ ′ 4 1 − σ 1 σ ′ 5 1 +  − 12 + 2 a 1 (10 + a 1 ( − 22 35 − 3 a ∗ 1 + a 1 ( − 3 + 4 a 1 + 8 a ∗ 1 ))) + 12 σ 3 1 − 4 σ 4 1 − 6 σ 1 ( − 6 + σ 2 ) − 44 σ 2 + 8 σ 2 2 + σ 2 1 ( − 19 + 8 σ 2 )  σ ′ 2 +  10 + 2 a 1 ( − 5 + a 1 ( − 9 + 8 σ 1 )) − 18 σ 2 + σ 1 ( − 17 + 6 σ 1 − 4 σ 2 1 + 16 σ 2 )  σ ′ 1 σ ′ 2 +  − 5 + 6 a 1 + 8 a 2 1 + 4( − 3 + σ 1 ) σ 1 + 8 σ 2  σ ′ 2 1 σ ′ 2 + (3 + 4 σ 1 ) σ ′ 3 1 σ ′ 2 +  6 a 1 + 8 a 2 1 − σ 1 (3 + 4 σ 1 ) + 8 σ 2  σ ′ 2 2  n + (1 + σ ′ 1 )( b 1 − 3) 2  4 + σ 1 − 2 σ 2 + 2 a 2 ( σ 1 − 2 σ 2 1 + 4 σ 2 ) + a 2 1 (8 a 2 − 4 σ ′ 1 ) + 3 σ ′ 1 + 2 a 1 ( − 2 a 2 + σ ′ 1 ) − ( σ 1 − σ ′ 1 )( σ 2 1 − 4 σ 2 + σ ′ 1 − σ 1 σ ′ 1 ) − 2 σ ′ 2 + 4 σ 1 σ ′ 2  . 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