Bilinear biorthogonal expansions and the Dunkl kernel on the real line

BILINEAR BIOR THOGONAL EXP ANSIONS AND THE DUNKL KERNEL ON THE REAL LINE LU ´ IS DANIEL ABREU, ´ OSCAR CIAURRI, AND JUAN LUIS V ARONA Abstra ct. W e study an ext ension of the classical P aley-Wiener space stru cture, whic h is based on bilinear expansions of integ ral kernels in to biorthogonal sequ ences of functions. The structure includes b oth sampling expansions and F ourier-Neumann type series as sp ecial cases, and it also provides a bilinear expansion for the D unkl kernel (in the rank 1 case) which is a D unkl analogue of Gegenbauer’s expansion of the plane wa v e and the corresponding sampling expansions. In fact, w e sho w how to derive sampling and F ourier-Neumann type exp an sions from the results related to the bilinear ex pansion for the Dunk l kernel. 1. Introduction The function K ( x, t ) = e ixt has sev eral w ell-kno w n bilinear expansion form ulas, for example: the F ourier series expansion, (1) e ixt = ∞ X n = −∞ sin( x − π n ) x − π n e iπ nt , t ∈ [ − 1 , 1]; the expansion in terms of the prolate spher oidal wa v efunctions ϕ n , (2) e − ixt = √ 2 π ∞ X n =0 i n λ n ϕ n ( x ) ϕ n ( t ) , where λ n are the square ro ots of the eigen v alues arising f r om the time-band limiting inte gral equation (see the recen t pap er [28]); and Gege n bauer’s expansion of the plane w a ve in Gegen b auer p olynomials and Bessel functions (see [16, § 4.8, formula (4.8.3), p . 116]) (3) e ixt = Γ ( β )  x 2  − β ∞ X n =0 i n ( β + n ) J β + n ( x ) C β n ( t ) , t ∈ [ − 1 , 1] , (in the p articular case β = 0, this formula is the so-called Jacobi-Anger iden tit y). Here and in what follo ws in this pap er we are using C β n to denote the Gegen bauer p olynomials of order β and J ν to denote the Bessel functions of order ν . Eac h of the ab o ve expans ions is asso ciated w ith imp ortan t deve lopmen ts in mathemat- ical analysis. The fir st one is equiv alen t to the Whittak er-Shannon-Kotel’nik o v sampling theorem (see [27, Ch. 2]), the second one is the protot yp e of a Mercer kernel [28], and the third one has b een the main to ol in the diagonalization of certain integ ral op er ators [17]. Our main in terest is in expansions of the t yp e (3). T o see wh y b iorthogonalit y is requir ed, recall the defin ition of the Pa ley-Wiener space P W , P W =  f ∈ L 2 ( R ) : f ( z ) = (2 π ) − 1 / 2 Z 1 − 1 u ( t ) e iz t dt, u ∈ L 2 ( − 1 , 1)  . THIS P APER HAS BEEN PUBLISHED IN Exp o. Math. 30 (2012), 32–48. 2000 M athematics Subje ct Classific ation. Primary 94A20; Secondary 42A38, 42C10, 33D45. Key wor ds and phr ases. Bilinear ex pansion, biorthogonal expansion, plane w a v e exp ansion, sampling theorem, F ourier-Neumann expansion, Dunk l transform, sp ecial functions. Researc h of the first author supp orted by CMUC/F CT and FCT p ost-do ctoral gran t SFRH/BPD/26078/20 05, POCI 2010 and FS E. Researc h of the second and third authors supp orted by grant MTM2009-12740-C0 3-03 of th e DGI. 1 2 L. D. ABREU, ´ O. CIAURRI, AN D J. L. V ARONA W e immediately obtain orthogonal expansions for f ∈ P W b y simp ly integrati ng equa- tions (1) and (2), by using th e orthogonalit y of the exp onen tials and the pr olate sph eroidal functions. How eve r, if we try to do the same th ing in (3), we must restrict ourselves to the case β = 1 / 2, w hen the wei gh t function of the Gegen bauer’s p olynomials (actually Le- gendre) is 1. T herefore, even in this simple case it is not clear ho w to expand P aley-W iener functions into Gegen bauer p olynomials or Bessel fun ctions w ith general parameter β , and the biorthogonal formulatio n serve to introduce the parameter β of (3) in a natural wa y . T o organize the presentati on of our ideas, w e first construct a structure in v olving biorthogonal expansions, fr om which the results are obtained, after explicit ev aluation of some inte grals. In particular (but not exclusiv ely), we use this structure to analyze the solution of the ab ov e mentio ned expansion problem and its extension to the Dun kl k ernel E α ( ix ) = 2 α Γ( α + 1)  J α ( x ) x α + J α +1 ( x ) x α +1 xi  (as we will see in subsection 4.1, the Dunkl k ernel is used to define the Dunkl transform on the real line similarly to ho w the kernel e ixt is us ed to define th e F our ier transf orm), and so we expan d functions in P W and its generalizatio n stu died in [6] in terms of F ourier- Neumann series. F rom the follo w in g extension of (3) to the Dunkl ke rnel E α ( ixt ) = Γ( α + β + 1)  x 2  − α − β − 1 ∞ X n =0 i n ( α + β + n + 1) J α + β + n +1 ( x ) C ( β +1 / 2 ,α +1 / 2) n ( t ) , where C ( β +1 / 2 ,α +1 / 2) n are the so-calle d generalized Gegen bauer p olynomials, we obtain uniformly con v ergen t F our ier-Neumann t yp e expansions f ( x ) = ∞ X n =0 a n ( f )( α + β + n + 1) x − α − β − 1 J α + β + n +1 ( x ) , v alid for f ∈ P W α (the natural generalizatio n of the P aley-Wie ner space as in [6]), and where a n ( f ) = 2 α + β +1 Γ( α + β + 1) Z R f ( t ) J α + β + n +1 ( t ) t α + β +1 dµ α + β ( t ) . Moreo ver, in some cases, th e co efficients a n ( f ) are identi fied as F ourier co efficients. The pap er is organized as follo ws. In the s econd section we describ e our problem in abstract terms. Fi rst w e build the general setup for bilinear orthogonal exp ansions and , later, we mo d ify it to consider biorthogonal s equences in the exp an s ions (see Theorem 1). In the third section w e describ e the results which are obtained in the case of the F our ier k ernel. The fourth section stu d ies the expansion asso ciated with the Dun kl ke rnel (see Theorem 2), that we think is a new and in teresting result; moreo v er, w e also sh o w its consequences for the Hankel transform . In the last sectio n w e collect the ev aluation of some integrals inv olving sp ecial f u nctions wh ic h we re essen tial for the pap er but could not b e found in the literature. 2. Structure 2.1. Orthogonal expansions. W e b egin with K ( x, t ), a f unction of t w o v ariables defined on Ω × Ω ⊂ R × R satisfying K ( x, t ) = K ( t, x ) almost ev erywhere for ( x, t ) ∈ Ω × Ω , and an in terv al I ⊂ Ω. Using this function as a k ernel, define on L 2 (Ω , dµ ), with dµ a non-negativ e real measure, an in tegral transformation b y (4) ( K f )( t ) = Z Ω f ( x ) K ( x, t ) dµ ( x ) . BILINEAR BIOR THOGONAL EXP ANSIONS 3 Moreo ver, we su p p ose that K is inv ertible and that the in v erse is (5) ( e K g )( x ) = Z Ω g ( t ) K ( x, t ) dµ ( t ) . Then, from F ubin i’s th eorem w e get th e m ultiplicatio n f orm ula (6) Z Ω ( K f ) g dµ = Z Ω ( K g ) f dµ and also Z Ω ( e K f ) g dµ = Z Ω ( e K g ) f dµ. Moreo ver, if in the multiplic ation formula w e tak e g = K ( f ) and use K ( h ) = e K ( h ), we get kK f k L 2 (Ω ,dµ ) = k f k L 2 (Ω ,dµ ) . As usual, it is enough to supp ose that the op erators K and e K , d efined by (4) and (5), are defined on a suitable den se sub set of L 2 (Ω , dµ ), and later extend ed to th e w hole L 2 (Ω , dµ ) in the stand ard wa y . Moreo ver, let us also assu me that, as a f unction of t , K ( x, t ) χ I ( t ) b elongs to L 2 (Ω , dµ ) (or, in other words, K ( x, · ) ∈ L 2 ( I , dµ )). Here, χ I stands for the c haracte ristic fun ction of I . No w, let N b e a subset of the integ ers, { φ n } n ∈ N b e an orth on orm al basis of the space L 2 ( I , dµ ) and { S n } n ∈ N b e a sequence of functions in L 2 (Ω , dµ ) such that, for ev ery n ∈ N , (7) ( K S n )( t ) = χ I ( t ) φ n ( t ) (notice the small abuse of n otation in the use of χ I φ n ; h ere, φ n is a fu n ction that is defined only on I , and by χ I φ n w e mean that w e extend this function to Ω by ha ving it b e the n ull f unction on Ω \ I ; we will use this kind of notation often in this pap er). Consider th e subspace P of L 2 (Ω , dµ ) constituted by those fun ctions f su c h that K f v anishes outside of I . This can also b e written as P = n f ∈ L 2 (Ω) : f ( x ) = Z I u ( t ) K ( x, t ) dµ ( t ) , u ∈ L 2 ( I , dµ ) o . On the one hand, b y using th at K is an isometry , it f ollo ws that S n is a c omplete or- thonorm al s equence in P . This implies that eve ry function f in P has an expansion of the form (8) f ( x ) = X n ∈ N c n S n ( x ) . On the other hand, from (7) w e ha v e e K ( χ I φ n ) = e K ( K S n ) = S n . Consequent ly , the F our ier co efficien ts of K ( x, t ), as a fun ction of t , in th e b asis { φ n } n ∈ N on L 2 ( I , dµ ) are S n ( x ). As a result, K ( x, t ) has the follo wing bilinear expansion form ula: (9) K ( x, t ) = X n ∈ N S n ( x ) φ n ( t ) , that m ust b e understo od as in L 2 ( I , dµ ( t )) for ev ery x ∈ Ω. Remark 1. The reader familiar with sampling theory , in particular with the generaliza- tion du e to Kramer (see [19] or, also, [27 , Theorem 3.5]), h as pr obably noticed strong resem blances. Indeed, Kramer’s lemma corresp onds to a particular case of the ab ov e situ- ation when there exists a sequen ce of p oints x k suc h that S n ( x k ) = δ n,k . This imp lies that { K ( x n , · ) } = { φ n } is an orthogonal basis of L 2 ( I , dµ ) and that P has an orthogonal basis giv en by { e K ( χ I K ( x n , · )) } = { e K ( χ I φ n ( · )) } = { S n } . The orthogonal exp an s ion in the basis { S n } n ∈ N is th e sampling theorem. In [14] there is giv en a detailed exp osition of a similar structure, which, although restricted to sampling theory , is in its essence equiv al en t to the one that w e ha v e describ ed. 4 L. D. ABREU, ´ O. CIAURRI, AN D J. L. V ARONA The ob jects that w e are inte rested in here are mainly those exp an s ions that fi t in the ab o v e setup, bu t that are not sampling expansions. As w e will see, there exist quite a few of these. W e will see in this w ork a wea lth of situations where explicit computations of certain in tegrals yield new expansion formulas of the t yp e (9), bu t in general they are sp ecial cases of the more general setting that we will pr o vide in the next section. Remark 2. Perhaps the most remark able feature that this setup inherits from samp ling theory is the fact that, in many situations, un iform con v ergence can b e guaran teed, once w e kn o w th at the expansion con v erge s in norm. This h app ens b ecause P is a Hilb ert sp ace with a repro d u cing ke rnel giv en by k ( x, y ) = X n ∈ N S n ( x ) S n ( y ) = Z I K ( x, t ) K ( y , t ) dµ ( t ) . This fact can b e prov ed using Saitoh’s theory of linear transformation in Hilb ert space [22, 23] in a w a y similar to what has b een done in [1] and also by the same arguments in [14 ]. The uniform con v erge nce of the expansions (8) is no w a consequence of the w ell- kno wn fact that if the sequence f n con v erges to f in the norm of a Hilb ert space with repro du cing k ernel k ( · , · ), then the conv ergence is p oint wise to f and u niform in ev ery set where k K ( x, · ) k L 2 ( I ,dµ ) is b oun ded. 2.2. Biorthogonal expansions. W e no w consider the same s etup as in su bsection 2.1 (in particular, the notation for the op erators K f and its inv erse e K g in terms of a kernel K ( x, t ) that satisfies the multiplica tion formula (6)), b ut instead of the orthonormal basis { φ n } n ∈ N of the space L 2 ( I , dµ ), w e assume th at we ha v e a pair of complete biorthonormal sequences of fun ctions in L 2 ( I , dµ ), { P n } n ∈ N and { Q n } n ∈ N , that is, Z I P n ( x ) Q m ( x ) dµ ( x ) = δ n,m and ev ery g ∈ L 2 ( I , dµ ) can b e written, in a uniqu e wa y , as g ( t ) = X n ∈ N c n ( g ) P n ( t ) , c n ( g ) = Z I g ( t ) Q n ( t ) dµ ( t ) . Let us also defin e, in L 2 (Ω , dµ ), the sequences of functions { S n } n ∈ N and { T n } n ∈ N giv en b y (10) S n ( x ) = e K ( χ I Q n )( x ) , x ∈ Ω , and T n ( x ) = K ( χ I P n )( x ) , x ∈ Ω (note that if P n = Q n then S n = T n ). Our p u rp ose is to prov e the follo w ing theorem, w hic h sa ys that it is s till p ossib le to fi nd a bilinear expansion in this con text. Theorem 1. F or e ach x ∈ Ω , the fol lowing exp ansion 1 holds, with r esp e ct to t , in L 2 ( I , dµ ) : (11) K ( x, t ) = X n ∈ N P n ( t ) S n ( x ) . Mor e over, { S n } n ∈ N and { T n } n ∈ N ar e a p air of c omplete biortho gonal se q uenc es in P such that every f ∈ P c an b e written as (12) f ( x ) = X n ∈ N c n ( f ) S n ( x ) , x ∈ Ω , 1 The condition t ∈ I in the identit y (11) is not a mistake. A lthough K ( x, t ) is defined on Ω × Ω, the functions P n ( t ) are d efined, in general, only on I . BILINEAR BIOR THOGONAL EXP ANSIONS 5 with c n ( f ) = Z Ω f ( t ) T n ( t ) dµ ( t ) . The c onver genc e is u niform in every set wher e k K ( x, · ) k L 2 ( I ,dµ ) is b ounde d. Pr o of. Let us start by pr oving (11). Since K ( x, · ) ∈ L 2 ( I , dµ ) for ev ery x ∈ Ω, as { P n } n ∈ N and { Q n } n ∈ N are a complete biorthogonal system on L 2 ( I , dµ ), we can write K ( x, t ) = X n ∈ N b n ( x ) P n ( t ) with (by (10 )) b n ( x ) = Z I K ( x, t ) Q n ( t ) dµ ( t ) = e K ( χ I Q n )( x ) = S n ( x ) . No w, let u s pro v e the biorthogonalit y of { S n } n ∈ N and { T n } n ∈ N . By definition and the m ultiplicativ e f orm ula, w e hav e Z Ω S n T m dµ = Z Ω S n K ( χ I P m ) dµ = Z Ω K ( S n ) χ I P m dµ = Z I Q n P m dµ = δ n,m . Finally , for f ∈ P , b y applying (11 ), interc hanging th e sum and the integ ral, and using the m ultiplicativ e formula, we h a v e f ( x ) = Z I u ( t ) K ( x, t ) dµ ( t ) = X n ∈ N  Z I u ( t ) P n ( t ) dµ ( t )  S n ( x ) = X n ∈ N  Z Ω ( K f )( t ) χ I ( t ) P n ( t ) dµ ( t )  S n ( x ) = X n ∈ N  Z Ω f ( t ) K ( χ I P n )( t ) dµ ( t )  S n ( x ) = X n ∈ N  Z Ω f ( t ) T n ( t ) dµ ( t )  S n ( x ) and the pro of is finish ed.  3. The F ourier kerne l As an example to clarify the tec hnique, and to sho w how usefu l the us e of the biortho gonal sequences is, let us lo ok at (1) and (3) in the ligh t of the ab ov e scheme. 3.1. The classical sampling formula. With the notation of the ab o v e section, take dµ ( x ) = dx , Ω = R , I = [ − 1 , 1] and the k ernel K ( x, t ) = 1 √ 2 π e ixt , so the op erator K is the F ourier transform . The sp ace P b ecomes th e classical P aley-Wie ner s pace P W . No w, tak e N = Z and the functions P n ( t ) = Q n ( t ) = φ n ( t ) = 1 √ 2 e iπ nt , n ∈ Z . Then, S n ( x ) is S n ( x ) = e K ( χ I Q n )( x ) = Z 1 − 1 e ixt √ 2 π e − iπ nt √ 2 dt = sin( x − π n ) √ π ( x − π n ) . F rom this exp r ession, b y us ing (11) we obtain (1). Moreo ver, usin g th e identit y 1 π Z R sin( x − π n ) x − π n f ( x ) dx = f ( π n ) , f ∈ P W, 6 L. D. ABREU, ´ O. CIAURRI, AN D J. L. V ARONA and (12), w e deduce the classical Whittak er -S hannon-Kotel’nik o v s ampling theorem f ( x ) = ∞ X n =0 sin( x − π n ) x − π n f ( πn ) . 3.2. Gegen bauer’s plane wa ve expa nsion. As in the p r evious case, take dµ ( x ) = dx , Ω = R , I = [ − 1 , 1], the ke rnel K ( x, t ) = 1 √ 2 π e ixt , K the F our ier transform, and P = P W . But, this time, consider N = N ∪ { 0 } an d , using C β n ( t ) to den ote th e Gegenbauer p oly- nomial of order β > − 1 / 2 (with th e usu al tric k of emp lo ying the Chebyshev p olynomials if β = 0), tak e the b iorthonormal system P n ( t ) = C β n ( t ) , Q n ( t ) = (1 − t 2 ) β − 1 / 2 C β n ( t ) /h n with h n = Z 1 − 1 C β n ( t ) 2 (1 − t 2 ) β − 1 / 2 dt = π 1 / 2 Γ( β + 1 / 2) Γ(2 β + n ) Γ( β )Γ(2 β )( n + β ) n ! . Using the inte gral Z R e − ixt x − β J β + n ( x ) dx = 2 − β +1 π 1 / 2 Γ(2 β )( − 1) n i n n ! Γ( β + 1 / 2) Γ(2 β + n ) (1 − t 2 ) β − 1 / 2 C β n ( t ) χ [ − 1 , 1] ( t ) (see [11, Ch. 3.3, (9), p. 123]) w e deduce that S n ( x ) = 2 β − 1 / 2 π − 1 / 2 i n Γ( β )( β + n ) x − β J β + n ( x ) . Then, (11) b ecomes (3). Moreo ver, every f unction f ∈ P W admits an expan s ion in a un iformly con v er gent F ourier-Neumann ser ies of the form f ( x ) = 2 β − 1 / 2 π − 1 / 2 Γ( β ) ∞ X n =0 c n ( f ) i n ( β + n ) x − β J β + n ( x ) , with c n ( f ) = Z R f ( t ) K ( χ [ − 1 , 1] C β n )( t ) dt, corresp ondin g to the expansion (12 ). A more explicit expression (see (32)) f or the co effi- cien ts c n ( f ) will b e giv en in the next section for some v alues of β . Finally , since k K ( x, · ) k L 2 ( I ,dx ) = 1 √ 2 π   e ix ·   L 2 ([ − 1 , 1] ,dx ) = 1 √ π , Remark 2 automaticall y ensure that the ab o v e mentio ned expansions conv erge uniformly on the real line. 4. The Dunkl kern el on the real line 4.1. The Dunkl t ransform. F or α > − 1, let J α denote th e Bessel function of ord er α and, for complex v alues of the v ariable z , let I α ( z ) = 2 α Γ( α + 1) J α ( iz ) ( iz ) α = Γ ( α + 1) ∞ X n =0 ( z / 2) 2 n n ! Γ( n + α + 1) ( I α is a small v ariation of the so-called mo d ified Bessel function of the fi rst kind and order α , usually denoted by I α ; see [26]). Moreo v er, let us tak e E α ( z ) = I α ( z ) + z 2( α + 1) I α +1 ( z ) , z ∈ C . BILINEAR BIOR THOGONAL EXP ANSIONS 7 The Dunkl op erators on R n are different ial-difference op erators asso ciated with some finite r eflection groups (see [7]). W e consider the Dun kl op erator Λ α , α ≥ − 1 / 2, asso ciated with the reflection group Z 2 on R giv en b y (13) Λ α f ( x ) = d dx f ( x ) + 2 α + 1 x  f ( x ) − f ( − x ) 2  . F or α ≥ − 1 / 2 and λ ∈ C , the in itial v alue pr oblem (14) ( Λ α f ( x ) = λf ( x ) , x ∈ R , f (0) = 1 has E α ( λx ) as its u nique solution (see [8] and [18]); this function is called the Dunkl k ernel. F or α = − 1 / 2, it is clear that Λ − 1 / 2 = d/dx , and E − 1 / 2 ( λx ) = e λx . Let dµ α ( x ) = (2 α +1 Γ( α + 1)) − 1 | x | 2 α +1 dx and w rite (15) E α ( ix ) = 2 α Γ( α + 1)  J α ( x ) x α + J α +1 ( x ) x α +1 xi  . In a similar wa y to th e F ourier transform (whic h is the particular case α = − 1 / 2), the Dunkl transform of order α ≥ − 1 / 2 is giv en by (16) F α f ( y ) = Z R f ( x ) E α ( − iy x ) dµ α ( x ) , y ∈ R , for f ∈ L 1 ( R , dµ α ). By means of the S c h w artz class S ( R ), the d efinition is extended to L 2 ( R , dµ α ) in the usu al wa y . In [18], it is sho wn that F α is an isometric isomorp hism on L 2 ( R , dµ α ) and th at F − 1 α f ( y ) = F α f ( − y ) for fun ctions such that f , F α f ∈ L 1 ( R , dµ α ). F rom F ubini’s theorem, it follo w s that the Dunkl tran s form satisfies the multiplicat ion form ula (17) Z R u ( y ) F α v ( y ) dµ α ( y ) = Z R F α u ( y ) v ( y ) dµ α ( y ) . Finally , let us take into account th at th e Dunkl trans form F α can also b e defin ed in L 2 ( R , dµ α ) for − 1 < α < − 1 / 2, although the expression (16) is no longer v alid for f ∈ L 1 ( R , dµ α ) in general. How eve r, it preserve s the same pr op erties in L 2 ( R , dµ α ); see [20] for details. This allo ws us to extend our study to the case α > − 1. 4.2. The sampling theorem related to the Dunkl transform. In our general s etup dev eloped in sub section 2.2, let us s tart by taking Ω = R , I = [ − 1 , 1], dµ = dµ α , with α > − 1, and L 2 ( I , dµ ) = L 2 ([ − 1 , 1] , dµ α ). On this space, we consider the k ernel K ( x, t ) = E α ( ixt ), so the corresp ond ing op erator is K = F α , i.e., the ab ov ementio ned Dunkl transform. No w, as usual in sampling theory , we tak e th e space of P aley-W iener type that, in our setting, is defin ed as (18) P W α =  f ∈ L 2 ( R , dµ α ) : f ( x ) = Z 1 − 1 u ( t ) E α ( ixt ) dµ α ( t ) , u ∈ L 2 ([ − 1 , 1] , dµ α )  endo w ed with the norm of L 2 ( R , dµ α ). (This sp ace is c haracte rized in [2, Th eorem 5.1] as b eing the sp ace of en tire f unctions of exp onen tia l t yp e 1 that b elong to L 2 ( R , dµ α ) wh en restricted to the real line.) Th en, tak e, of course P = P W α . It is w ell-kno wn that the Bessel f u nction J α +1 ( x ) h as an increasing sequence of p ositiv e zeros { s n } n ≥ 1 . Cons equ en tly , the r eal function Im( E α ( ix )) = x 2( α +1) I α +1 ( ix ) is o dd and it has an infin ite sequence of zeros { s n } n ∈ Z (with s − n = − s n and s 0 = 0). Then, follo wing [6 ] (or [5]), let us define the functions (19) e α,n ( t ) = d n E α ( is n t ) , n ∈ Z , t ∈ [ − 1 , 1] , 8 L. D. ABREU, ´ O. CIAURRI, AN D J. L. V ARONA where d n = 2 α/ 2 (Γ( α + 1)) 1 / 2 |I α ( is n ) | , n 6 = 0 , d 0 = 2 ( α +1) / 2 (Γ( α + 2)) 1 / 2 . With this notation, the sequence of fun ctions { e α,n } n ∈ Z is a complete orthon orm al sys tem in L 2 ([ − 1 , 1] , dµ α ), for α > − 1. Thus, let us take N = Z and P n ( t ) = Q n ( t ) = e α,n ( t ). On the other hand, let us use that, for x, y ∈ R , x 6 = y and α > − 1, w e ha v e (20) Z 1 − 1 E α ( ixt ) E α ( iy t ) dµ α ( t ) = 1 2 α +1 Γ( α + 2) x I α +1 ( ix ) I α ( iy ) − y I α +1 ( iy ) I α ( ix ) x − y (the pro of can b e found in [3] or [6]). Then, S n ( x ) = e K ( χ [ − 1 , 1] Q n )( x ) = Z 1 − 1 E α ( ixt ) e α,n ( t ) dµ α ( t ) = d n 2 α +1 Γ( α + 2) x I α +1 ( ix ) I α ( is n ) − s n I α +1 ( is n ) I α ( ix ) x − s n = d n 2 α +1 Γ( α + 2) x I α +1 ( ix ) I α ( is n ) x − s n b ecause I α +1 ( is n ) = 0. C on s equen tly , E α ( ixt ) = X n ∈ Z e α,n ( t ) d n 2 α +1 Γ( α + 2) x I α +1 ( ix ) I α ( is n ) x − s n = I α +1 ( ix ) + X n ∈ Z \{ 0 } E α ( is n t ) 2( α + 1) I α ( is n ) x I α +1 ( ix ) x − s n , whic h corresp ond s to the formula (11) in Th eorem 1. Finally , the form ula (12) in Theorem 1 says that, if f ∈ P W α , then f has the represen- tation (21) f ( x ) = f ( s 0 ) I α +1 ( ix ) + X n ∈ Z \{ 0 } f ( s n ) x I α +1 ( ix ) 2( α + 1) I α ( is n )( x − s n ) , that conv erges in the norm of L 2 ( R , dµ α ). This is so b ecause th e co efficien ts c n ( f ) in (12) are c n ( f ) = d n f ( s n ), as w e can see in w h at follo ws: c n ( f ) = Z R f ( t ) F α ( χ [ − 1 , 1] e α,n )( t ) dµ α ( t ) = Z R f ( t ) Z 1 − 1 e α,n ( x ) E α ( − ixt ) dµ α ( x ) dµ α ( t ) = Z 1 − 1 e α,n ( x ) Z R f ( t ) E α ( − ixt ) dµ α ( t ) dµ α ( x ) = d n Z 1 − 1 E α ( is n x ) Z R f ( t ) E α ( − ixt ) dµ α ( t ) dµ α ( x ) = d n f ( s n ) , where in the last step w e ha v e used that f ∈ P W α . Moreo v er , by using L’Hopital rule in (20), it is not difficult to c hec k that     E α ( x · ) ( x · ) α     2 L 2 ([ − 1 , 1] ,dµ α ) = Z 1 − 1 | E α ( ixr ) | 2 dµ α ( r ) = 1 2 α +1 Γ( α + 2)  x 2 2( α + 1) I 2 α +1 ( ix ) − (2 α + 1) I α +1 ( ix ) I α ( ix ) + 2( α + 1) I 2 α ( ix )  , BILINEAR BIOR THOGONAL EXP ANSIONS 9 and this norm is b ounded on ev ery compact set on the r eal line. So, by Remark 2, the series (21) conv erges un iformly in compact su bsets of R . (21) is the sampling th eorem related to the Dunkl transform that has b een established in [6]. 4.3. F ourier-Neumann t yp e expansion. F ollo wing [9, Definition 1.5.5, p. 27], let us in tro duce th e generalized Gegen bauer p olynomials C ( λ,ν ) n ( t ) for λ > − 1 / 2, ν ≥ 0 and n ≥ 0 (the case ν = 0 corresp onding to the ordin ary Gegen bauer p olynomials); actually , for con v enience with the notation of this pap er, we are going to use C ( β +1 / 2 ,α +1 / 2) n ( x ). In this wa y , for β > − 1 and α ≥ − 1 / 2, the generalized Gegen b auer p olynomials are defin ed b y C ( β +1 / 2 ,α +1 / 2) 2 n ( t ) = ( − 1) n ( α + β + 1) n ( α + 1) n P ( α,β ) n (1 − 2 t 2 ) , (22) C ( β +1 / 2 ,α +1 / 2) 2 n +1 ( t ) = ( − 1) n ( α + β + 1) n +1 ( α + 1) n +1 tP ( α +1 ,β ) n (1 − 2 t 2 ) , (23) where in the co efficien ts we are us ing the P oc hhammer symbol ( a ) n = a ( a + 1) · · · ( a + n − 1) = Γ( a + n ) / Γ( a ). Note that there is n o problem in extending the d efinition of the generalized Gegen bauer p olynomials taking α > − 1, so we w ill assume this situation. F rom th e L 2 -norm of the Jacobi p olynomials (see [12, Ch . 16.4, (5), p. 285]), it is easy to find h ( β ,α ) 2 n = Z 1 − 1 h C ( β +1 / 2 ,α +1 / 2) 2 n ( t ) i 2 (1 − t 2 ) β dµ α ( t ) (24) = 1 2 α +1 Γ( α + 1)Γ( β + n + 1)Γ( α + β + n + 1) ( α + β + 2 n + 1)Γ( α + β + 1) 2 Γ( α + n + 1) n ! , h ( β ,α ) 2 n +1 = Z 1 − 1 h C ( β +1 / 2 ,α +1 / 2) 2 n +1 ( t ) i 2 (1 − t 2 ) β dµ α ( t ) (25) = 1 2 α +1 Γ( α + 1)Γ( β + n + 1)Γ( α + β + n + 2) ( α + β + 2 n + 2)Γ( α + β + 1) 2 Γ( α + n + 2) n ! . Finally , giv en α > − 1, we define the functions J α,n ( x ) b y J α,n ( x ) = J α + n +1 ( x ) x α +1 , x ∈ R , n = 0 , 1 , 2 , . . . ; as these functions arise in F ourier-Neumann series, we will allud e to J α,n ( x ) using the name of Neumann functions. 2 F rom the ident ities Z ∞ 0 J a ( x ) J b ( x ) x dx = 2 π sin(( b − a ) π / 2) b 2 − a 2 , a > 0 , b > 0 , a 6 = b, Z ∞ 0 J a ( x ) 2 x dx = 1 2 a , a > 0 , and taking in to account that J α,n ( x ) is ev en or od d according to n b eing ev en or od d, resp ectiv ely , it follo w s that {J α,n } n ≥ 0 is an orthogonal system on L 2 ( R , dµ α ), namely , Z R J α,n ( x ) J α,m ( x ) dµ α ( x ) = δ n,m 2 α +1 Γ( α + 1)( α + n + 1) . Generalized Gegen bauer p olynomials and Neumann functions are th e main in gredien ts for obtaining the Dunkl analogue of Gegen bauer’s expansion of the plane wa v e. T o establish this result we need a relation b etw een them. By using the notation P ( α,β ) n ( t ) = C ( β +1 / 2 ,α +1 / 2) n ( t ) , (26) Q ( α,β ) n ( t ) =  h ( β ,α ) n  − 1 (1 − t 2 ) β C ( β +1 / 2 ,α +1 / 2) n ( t ) (27) 2 In the literature, the name “Neumann functions” is sometimes used for the Bessel functions of the second kind Y α ( x ), but these functions will not arise in this pap er. 10 L. D. ABREU, ´ O. CIAURRI, AN D J. L. V ARONA (where h ( β ,α ) n is giv en in (24) and (25)), this relation is giv en in th e follo wing lemma that, moreo v er, can hav e indep end en t in terest: Lemma 1. L et α, β > − 1 , α + β > − 1 , and k = 0 , 1 , 2 , . . . . The Dunkl tr ansform of or der α of J α + β ,k ( x ) is (28) F α ( J α + β ,k )( t ) = ( − i ) k 2 α + β +1 Γ( α + β + 1)( α + β + k + 1) Q ( α,β ) k ( t ) χ [ − 1 , 1] ( t ) . Mor e over, if β < 1 , we have 3 (29) F α ( | · | 2 β J α + β ,k )( t ) = 2 β Γ( α + β + 1) Γ( α + 1) ( − i ) k P ( α,β ) k ( t ) , t ∈ [ − 1 , 1] . Remark 3. There is a delicacy with th e f orm ulas in Lemma 1. Actually , F α w as defined, as a first step, as a Leb esgue inte gral for su itably integrable functions. Then, F α is extended to L p spaces where the inte gral represen tation is n o longer v alid f or some fu nctions. No w , the in tegrals Z ∞ 0 x − λ J µ ( ax ) J ν ( bx ) dx from [12, Ch. 8.11] or [26, Ch. XI I I] that we will use in the pro of are improp er Riemann in tegrals. Hence, the prop er un d erstanding of those integ rals sh ould b e as lim N → ∞ Z N 0 x − λ J µ ( ax ) J ν ( bx ) dx. Since J α + β ,k χ [ − N ,N ] and | · | 2 β J α + β ,k χ [ − N ,N ] are integrable fun ctions, the in tegral form of F α is v alid here and (28) and (29 ) can b e under s to o d as lim N → ∞ F α ( J α + β ,k χ [ − N ,N ] )( t ) , lim N → ∞ F α ( | · | 2 β J α + β ,k χ [ − N ,N ] )( t ) , and the identiti es in the lemma h old in the almost everywhere sense. Finally , the L 2 b ound edness of F α allo ws us to und erstand these identit ies in the L 2 sense. F rom no w on , w e will no longer men tion these details. W e p ostp one the pro of of Lemma 1 to s ubsection 5.2. With Lemma 1 , we already h a v e all the to ols for pr oving: Theorem 2. L e t α, β > − 1 and α + β > − 1 . Then for e ach x ∈ R the fol lowing exp ansion holds in L 2 ([ − 1 , 1] , dµ α ) : (30) E α ( ixt ) = 2 α + β +1 Γ( α + β + 1) ∞ X n =0 i n ( α + β + n + 1) J α + β ,n ( x ) C ( β +1 / 2 ,α +1 / 2) n ( t ) . Mor e over, for β < 1 and f ∈ P W α , we have the ortho gonal exp ansion (31) f ( x ) = ∞ X n =0 a n ( f )( α + β + n + 1) J α + β ,n ( x ) with (32) a n ( f ) = 2 α + β +1 Γ( α + β + 1) Z R f ( t ) J α + β ,n ( t ) dµ α + β ( t ) . F urthermor e, the series c onver ges uniformly in c omp act subsets of R . 3 Observe that nothing is said ab out outside the interv al [ − 1 , 1]; this do es not mean that this expression v anishes for | t | > 1, that is n ot true when β 6 = 0. BILINEAR BIOR THOGONAL EXP ANSIONS 11 Pr o of. In the biorthogonal setup give n in subsection 2.2, let Ω = R , I = [ − 1 , 1], the space L 2 ( I , dµ ) = L 2 ([ − 1 , 1] , dµ α ), and the k ernel K ( x, t ) = E α ( ixt ), from which the op erator K b ecomes the Dunkl tr an s form F α (and e K = F − 1 α ). Also, consider the P aley-Wiener space P = P W α (see (18)). Finally , for N = N ∪ { 0 } , take the biorthonormal system giv en by P n ( t ) = P ( α,β ) n ( t ) and Q n ( t ) = Q ( α,β ) n ( t ) as in (26) and (27). F rom (28), we ha v e S n ( x ) = 2 α + β +1 Γ( α + β + 1) i n ( α + β + n + 1) J α + β ,n ( x ) . In this situation, the form ula (11) in Theorem 1 giv es (30). No w, let us consid er T n ( x ) = K ( χ [ − 1 , 1] P n )( x ) = F α ( χ [ − 1 , 1] P ( α,β ) n )( x ) . Then, the identit y giv en by (12) b ecomes f ( x ) = 2 α + β +1 Γ( α + β + 1) ∞ X n =0 c n ( f ) i n ( α + β + n + 1) J α + β ,n ( x ) , f ∈ P W α , with c n ( f ) = Z R f ( t ) F α ( χ [ − 1 , 1] P ( α,β ) n )( t ) dµ α ( t ) . Let us see that, when β < 1, th e coefficient c n ( f ) can b e written as c n ( f ) = ( − i ) n Z R f ( t ) J α + β ,n ( t ) dµ α + β ( t ) , whic h implies (32) with a n ( f ) = 2 α + β +1 Γ( α + β + 1) i n c n ( f ). Indeed, if we consid er u su c h that f = F − 1 α ( uχ [ − 1 , 1] ) and u se the m ultiplica tion for- m ula (17), w e can write c n ( f ) = Z 1 − 1 u ( x ) P ( α,β ) n ( x ) dµ α ( x ) . No w, b y (29) and the multiplica tion formula again, we hav e c n ( f ) = Z 1 − 1 u ( x ) i n Γ( α + 1) 2 β Γ( α + β + 1) F α ( | · | 2 β J α + β ,n )( x ) dµ α ( x ) = i n Z R F α ( uχ [ − 1 , 1] )( t ) J α + β ,n ( t ) dµ α + β ( t ) . It is clear that F α ( uχ [ − 1 , 1] )( t ) = f ( − t ), so the c hange of v ariable from t to − t giv es c n ( f ) = ( − i ) n Z R f ( t ) J α + β ,n ( t ) dµ α + β ( t ) b ecause J α + β ,n ( − t ) = ( − 1) n J α + β ,n ( t ).  Remark 4. Actually , formulas (3) and (30) are equiv alen t for α ≥ − 1 / 2. The pr o of in one d irection is clear, ju st by sp ecializing the parameters. T o obtain (30) from (3), we can use the in tert wining op erator V α g ( t ) = Γ(2 α + 2) 2 2 α +1 Γ( α + 1 / 2)Γ( α + 3 / 2) Z 1 − 1 g ( st )(1 − s ) α − 1 / 2 (1 + s ) α +1 / 2 ds (see [9, Definition 1.5.1, p. 24], w e c hange the parameter µ in the definition give n in [9] by α + 1 / 2), defined for α ≥ − 1 / 2. With this notation w e ha v e (33) V α C α + β +1 n ( t ) = C ( β +1 / 2 ,α +1 / 2) n ( t ) and V α e i · ( t ) = E α ( it ) . In this w a y , app lying V α to (3) (with α + β + 1 instead of β ) we get (30). This idea has b een used for the higher r ank in [21], wh ere the author assumes that (3) w as already kno wn, 12 L. D. ABREU, ´ O. CIAURRI, AN D J. L. V ARONA and th en (30) is established for α ≥ − 1 / 2 by using the in tert wining op erator. Th is giv es a considerably shorter pro of. In stead, w ith the metho d follo w ed in the p ro of of Theorem 2, the identit y (30) not only can b e found for α > − 1, but also is pr ov ed d irectly and then, as a particular case, (3) holds. Remark 5. Another wa y of obtaining (30) is as follo ws. Some results from [13] were generalized in [25] to ∞ X m =0 a m b m ( z w ) m m ! = ∞ X n =0 ( − z ) n n ! ( γ + n ) n ∞ X r =0 b n + r z r r ! ( γ + 2 n + 1) r ! n X s =0 ( − n ) s ( n + γ ) s s ! a s w s ! . When z and w are r eplaced by z γ and w/γ , resp ectiv ely , and we let γ → ∞ , we get the companion formula ∞ X m =0 a m b m ( z w ) m = ∞ X n =0 ( − z ) n n ! ∞ X j =0 b n + j j ! z j ! n X k =0 ( − n ) k a k w k ! (these formulas are also stated in [16, Ch. 9]). These expansions conta in sev eral expan- sions in terms of J acobi p olynomials of argum ent 1 − 2 t 2 (i.e., generalized Gege n bauer p olynomials). In particular, (30 ) f ollo ws in th is wa y . 4.4. Consequences for the Hank el transform. F or α > − 1, consid er the so-ca lled mo dified Hankel transform H α , that is (34) H α f ( y ) = Z ∞ 0 J α ( xy ) ( xy ) α f ( y ) x 2 α +1 dx, x > 0 . The ke rnel E α ( ixt ) of the Dunkl transform (16) can b e written in terms of the Bessel functions of order α and α + 1, and this clearly allo ws us to stud y th e Hank el transform as a s im p le consequence of the Dunk l tr an s form. In particular, if we hav e a fun ction f ∈ L 2 ((0 , ∞ ) , x 2 α +1 dx ), we can tak e the ev en extension f ( | · | ) ∈ L 2 ( R , dµ α ). Then, usin g that J α ( x ) /x α is ev en and J α +1 ( x ) /x α is o dd, w e write (34) as H α f ( y ) = F α ( f ( | · | ))( y ) . The P aley-W iener space for the Hank el transform is giv en b y P W ′ α = ( f ∈ L 2 ((0 , ∞ ) , x 2 α +1 dx ) : f ( t ) = Z 1 0 u ( x ) J α ( xt ) ( xt ) α x 2 α +1 dx, u ∈ L 2 ((0 , 1) , x 2 α +1 dx ) ) ; also, n ote that if f ∈ P W ′ α , then b oth the even extension f ( | · | ) and the o dd extension sgn( · ) f ( | · | ) b elong to P W α . So, let us adap t the samp ling formula of subsection 4.2 and the Theorem 2 of su b sec- tion 4.3 to the con text of the Hank el tr an s form. 4.4.1. The sampling formula for the Hankel tr ansfo rm. F or f ∈ P W ′ α , taking its ev en extension f ( | · | ), usin g that s − n = − s n and grouping the summ an d s corresp onding to 1 / ( x − s n ) and 1 / ( x + s n ) in (21 ), w e get f ( x ) = f ( s 0 ) I α +1 ( ix ) + ∞ X n =1 f ( s n ) I α +1 ( ix ) ( α + 1) I α ( is n ) x 2 x 2 − s 2 n . Similarly , w ith the o dd extension of f , (21) b ecomes f ( x ) = ∞ X n =1 f ( s n ) I α +1 ( ix ) ( α + 1) I α ( is n ) s 2 n x 2 − s 2 n . BILINEAR BIOR THOGONAL EXP ANSIONS 13 The latter iden tit y corresp onds to the w ell-kno wn Higgins sampling theorem for th e Hank el transform [15]. 4.4.2. A version of the The or em 2 for the H ankel tr ansform. Let us observe that J α ( xt ) ( xt ) α = 1 2 α +1 Γ( α + 1)  E α ( ixt ) + E α ( ixt )  . F rom this, it is v ery easy to adapt (30 ) to the new con text, and to write it in terms of Jacobi p olynomials by u sing (22). Giv en f ∈ P W ′ α , let us consider its ev en extension f ( | · | ) ∈ P W α . Applying (31) to this ev en fu nction, it b ecomes an expansion that only con tains J α + β , 2 n ( x ) = J α + β +2 n +1 ( x ) /x α + β +1 (i.e., only with ev en indexes). Th us, the results corresp onding to the Hanke l transf orms can b e summarized in this w a y: Corollary 3. L et α, β > − 1 and α + β > − 1 . Then for e ach x ∈ (0 , ∞ ) the fol lowing exp ansion holds in L 2 ((0 , 1) , x 2 α +1 dx ) : (35) J α ( xt ) ( xt ) α = ∞ X n =0 2 β +1 ( α + β + 2 n + 1)Γ( α + β + n + 1) Γ( α + n + 1) J α + β , 2 n ( x ) P ( α,β ) n (1 − 2 t 2 ) . Mor e over, for β < 1 and f ∈ P W ′ α , we have the ortho gonal exp ansion f ( x ) = ∞ X n =0 a n ( f )( α + β + 2 n + 1) J α + β , 2 n ( x ) with a n ( f ) = 2 Z ∞ 0 f ( t ) J α + β , 2 n ( t ) t 2 α +2 β +1 dt. F urthermor e, the series c onver ges uniformly in c omp act subsets of (0 , ∞ ) . Remark 6. In the particular case α = − 1 / 2, on usin g J − 1 / 2 ( z ) = 2 1 / 2 π − 1 / 2 z − 1 / 2 cos( z ) and (22), the formula (35) b ecomes (36) x β +1 / 2 2 β +1 / 2 Γ( β + 1 / 2) cos( xt ) = ∞ X n =0 ( − 1) n (2 n + β + 1 / 2) J β +2 n +1 / 2 ( x ) C β +1 / 2 2 n ( t ) , whic h is already known (see [26, § 11. 5, formula (5), p. 369]). On the other hand , follo wing the pro cedure d escrib ed in Remark 4, fr om this expression w e can obtain another p ro of of (35), v alid for α > − 1 / 2. Let us assume that (36) is already known, and w e wr ite it w ith α + β + 1 / 2 instead of β ; th en, by applying the inte rt wining op erator V α and using (33 ), (22) and V α cos( t ) = 2 α Γ( α + 1) J α ( t ) t α , w e get the iden tit y (35 ), as desired. Let us conclud e by observin g that the L p con v ergence of the orth ogonal series that app ear in the previous corollary has b een stud ied in the p ap ers [24, 4] for functions in an appropriate L p extension of the P aley-W iener sp ace. 5. Technical lemmas The main goal of this section is to pr o v e Lemma 1 , wh ic h is key in our s tu dy of the Dunkl transform on the real line. Th e p ro of is con tained in sub section 5.2. With this target, we need to previously establish some formulas. T hey are giv en in su bsection 5.1. 14 L. D. ABREU, ´ O. CIAURRI, AN D J. L. V ARONA 5.1. Some integrals inv olving Bessel functions. F or th e sak e of completeness, let us start by p ro ving t w o identi ties that exp ress some in tegrals in v olving the pro d uct of t w o Bessel fu nctions in terms of Jacobi p olynomials. Such integral s are usu ally wr itten in terms of hyp ergeometric fun ctions; ho w ev er their expr essions as Jacobi p olynomials are not easily f ound in the literature. F or instance, they do not app ear in the standard references [10, 26, 12]. In wh at follo ws, w e can tak e in to accoun t the commen ts in Remark 3, but w e will not rep eat them. Lemma 2. F or α, β > − 1 with α + β > − 1 , and n = 0 , 1 , 2 , . . . , let us define I − ( α, β , n )( t ) = t − α Z ∞ 0 x − β J α + β +2 n +1 ( x ) J α ( xt ) dx, I + ( α, β , n )( t ) = t − α Z ∞ 0 x β J α + β +2 n +1 ( x ) J α ( xt ) dx. Then, we have (37) I − ( α, β , n )( t ) = 2 − β Γ( n +1) Γ( β + n +1) (1 − t 2 ) β P ( α,β ) n (1 − 2 t 2 ) χ [0 , 1] ( t ) , t ∈ (0 , ∞ ) . Assume further that β < 1 ; then, (38) I + ( α, β , n )( t ) = 2 β Γ( α + β + n +1) Γ( α + n +1) P ( α,β ) n (1 − 2 t 2 ) , t ∈ (0 , 1) . Pr o of. W e use the formula (39) Z ∞ 0 x − λ J µ ( ax ) J ν ( bx ) dx = b ν a λ − ν − 1 Γ( µ + ν − λ +1 2 ) 2 λ Γ( ν + 1)Γ( λ + µ − ν +1 2 ) 2 F 1  µ + ν − λ + 1 2 , ν − λ − µ + 1 2 ; ν + 1; b 2 a 2  , v alid when 0 < b < a and − 1 < λ < µ + ν + 1; h ere, 2 F 1 denotes the hyp ergeometric function (see [12, Ch. 8.11, (9), p. 48] or [26, Ch. XI I I, 13.4 (2), p. 401]). Then, let u s start with (37). T aking a = 1 and t = b in (39), and making the corr e- sp ond ing c hanges of v ariable and parameters ( ν = α , µ = α + β + 2 n + 1, λ = β ) we get I − ( α, β , n )( t ) = Γ( α + n + 1) 2 β Γ( α + 1)Γ( β + n + 1) 2 F 1 ( α + n + 1 , − n − β ; α + 1; t 2 ) , whic h is v alid for α > − 1 and β > − 1 in the interv al 0 < t < 1. Moreo v er, w e ha v e 2 F 1 ( α + n + 1 , − n − β ; α + 1; t ) = (1 − t ) β 2 F 1 ( − n, α + β + n + 1; α + 1; t ) , where α, β > − 1, n = 0 , 1 , 2 , . . . , and (40) P ( α,β ) n ( y ) = Γ( n + α +1) Γ( α +1)Γ( n +1) 2 F 1 ( − n, α + β + n + 1; α + 1; 1 − y 2 ) , whenev er α, β > − 1 and − 1 < y < 1. Therefore, I − ( α, β , n )( t ) = 2 − β Γ( n +1) Γ( β + n +1) (1 − t 2 ) β P ( α,β ) n (1 − 2 t 2 ) , t ∈ (0 , 1) . No w, we are going to ev al uate I − ( α, β , n )( t ) for t > 1. T o do that, let us take t = a , b = 1, ν = α + β + 2 n + 1, µ = α , and λ = β in (39). In this wa y , 1 2 ( λ + µ − ν + 1) = 0 , − 1 , − 2 , . . . , so the co efficien t 1 / Γ( 1 2 ( λ + µ − ν + 1)) v anishes and we get I − ( α, β , n )( t ) = 0. Finally , let us pr o v e th e second part of the lemma. T o this end, we take , in (39 ), a = 1 and t = b , with parameters λ = − β , µ = α + β + 2 n + 1 and ν = α . Then, for β < 1, α + β > − 1, and 0 < t < 1 we get I + ( α, β , n )( t ) = 2 β Γ( α + β + n + 1) Γ( α + 1)Γ( n + 1) 2 F 1 ( α + β + n + 1 , − n ; α + 1; t 2 ) . Then, by using (40 ), (38) follo w s .  BILINEAR BIOR THOGONAL EXP ANSIONS 15 5.2. Pro of of Lemma 1 . W e s tart by ev aluating F α ( J α + β ,k )( t ) for α > − 1 and α + β > − 1. By definition, F α ( J α + β ,k )( t ) = 1 2 Z R J α + β + k +1 ( x ) x α + β +1  J α ( xt ) ( xt ) α − J α +1 ( xt ) ( xt ) α +1 xti  | x | 2 α +1 dx. F or th e case k = 2 n , by decomp osing into even and o dd fu nctions, w e can wr ite (41) F α ( J α + β , 2 n )( t ) = Z ∞ 0 J α + β +2 n +1 ( x ) x α + β +1 J α ( xt ) ( xt ) α x 2 α +1 dx. Then, for t > 0, by using (37) in Lemma 2, (22) and (24), it follo ws that F α ( J α + β , 2 n )( t ) = t − α Z ∞ 0 x − β J α + β +2 n +1 ( x ) J α ( xt ) dx = Γ( n + 1) 2 β Γ( β + n + 1) (1 − t 2 ) β P ( α,β ) n (1 − 2 t 2 ) χ [0 , 1] ( t ) = ( − 1) n Γ( α + β + 1)Γ( n + 1)Γ( α + n + 1) 2 β Γ( α + 1)Γ( β + n + 1)Γ( α + β + n + 1) (1 − t 2 ) β C ( β +1 / 2 ,α +1 / 2) 2 n ( t ) χ [0 , 1] ( t ) = i k 2 α + β +1 Γ( α + β + 1)( α + β + k + 1) Q ( α,β ) k ( t ) χ [0 , 1] ( t ) . F or t < 0, let us make, in (41), the change t 1 = − t , u se the ev enness of the f unction J α ( z ) /z α , p ro ceed as in the case t > 0, and u ndo the c hange. T hen, w e get F α ( J α + β , 2 n )( t ) = i k 2 α + β +1 Γ( α + β + 1)( α + β + k + 1) Q ( α,β ) k ( t ) χ [ − 1 , 0] ( t ) . Th us, (28) for ev en k is pro v ed. The case k = 2 n + 1 is completely similar. Pro ceeding in the same w a y , the formula (29) follo ws from (38). Ac kno wledgement. 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