Using a computer algebra system to simplify expressions for Titchmarsh-Weyl m-functions associated with the Hydrogen Atom on the half line

Using a computer algebra system to simplify expressions for Titchma rsh-W eyl m-functions asso ciated with the Hydrogen Atom on the ha lf line b y Cecilia Knoll and Charles F ulton TECHNICAL REPOR T FLORID A INSTITUTE OF TECHNOLOGY Septem b er 2007 1 Using a computer algebra system to simplify expressio ns for Titc hmarsh-W eyl m-functions asso ciated with the Hydrogen A tom on the half line ∗ CECILIA KNOLL Departmen t of Applied Mathematics Florida Institute of T ec hnology Melb ourne, Florida 32901-6 975 CHARLES FUL TON Departmen t of Mathematic al Sciences Florida Institute of T ec hnology Melb ourne, Florida 32901-6 975 Abstract Abstract: In this pap er we give simplified for mulas for certain p olynomials which aris e in s ome new Titch marsh-W eyl m-functions for the radial part of the separ ated Hydro g en atom o n the half line (0 , ∞ ) and t wo indep endent progra ms for g enerating them using the symbolic manipulator Ma thematica. 1 In tro d uction Recen tly F ulton[1] and F ulton and Langer [2] c onsidered the follo wing Sturm-Liouville problem for the h yd rogen atom on the half line, x ∈ (0 , ∞ ): − y ′′ +  − a x + ℓ ( ℓ + 1) x 2  y = λy , 0 < x < ∞ (1.1) lim x → 0 W x  y , x 1 2 J 2 ℓ +1 ( √ 4 ax )  = 0 . (1.2) In [1] a fundamenta l system of F rob en ius solutions defin ed at x = 0 w as in tro duced having th e forms: φ ( x, λ ) = x ℓ +1 " 1 + ∞ X n =1 a n ( λ ) x n # = 1 ( − 2 √ λ ) ℓ +1 M β ,ℓ + 1 2 ( − 2 ix √ λ ) , β := ia 2 √ λ (1.3) ∗ This research partially supp orted by National Science F oundation Gran t DMS-0109022 to Florida Institut e of T echnology . 2 and θ ( x, λ ) = − 1 2 ℓ + 1 " K ℓ ( λ ) φ ( x, λ ) ln x + x − 1 + ∞ X n =1 d n ( λ ) x − ℓ + n # (1.4) Where a n ( λ ) and d n ( λ ) are p olynomials in λ , M β ,ℓ + 1 2 is the Whittak er function of first kind and K ℓ ( λ ) = − a (2 ℓ + 1)!( 2 ℓ )! ℓ Y j =1  4 λj 2 + a 2  = ( − 2 i √ λ ) 2 ℓ +1 ( − ℓ − β ) 2 ℓ +1 (2 ℓ )!(2 ℓ + 1)! . (1.5) Then a Titc hmarsh -W eyl m -function w as in tro du ced in [1] b y the r equiremen t θ ( x, λ ) − m ℓ ( λ ) φ ( x, λ ) ∈ L 2 (0 , ∞ ) , (1.6) whic h gives m ℓ as m ℓ ( λ ) = − ak ℓ ( λ )  log( − 2 i √ λ ) + Ψ  1 − ia 2 √ λ  − H 2 ℓ + 2 γ  − ak ℓ ( λ )    ℓ X j =1 1  j − ia 2 √ λ     + ( i √ λ ) 2 ℓ +1 (2 ℓ + 1)!   2 ℓ X k =0 2 k  − ℓ − ia 2 √ λ  k k !(2 ℓ + 1 − k )   (1.7) where Ψ( z ) = Γ ′ ( z ) Γ( z ) is the psi or digamma fun ction, H 2 ℓ = P 2 ℓ j =1 1 j , γ = E u ler’s constan t, and the Poc h ham- mer symb ol is d efined for an y complex z as ( z ) k = z ( z + 1) ... ( z + k − 1). Here k ℓ ( λ ) is the p olynomial of degree ℓ defined b y k ℓ ( λ ) = − 1 a (2 ℓ + 1) K ℓ ( λ ) = − 1 (2 ℓ + 1)! ℓ Y j =1  4 λj 2 + a 2  . (1.8) F or ℓ = 0, w e define k 0 ( λ ) = 1. The function m ℓ ( λ ) is an analytic function in the h alf p lanes Im λ < 0 and Im λ > 0, has p oles on the negativ e x -axis at the eigen v alues of the pr oblem (1.1)-(1.2), and a branc h cut, corresp onding to th e cont in u ous s p ectrum, on the p ositiv e real λ -axis. In [2] a Pick-N ev alinna represent ation of m 0 ( λ ) was obtained for ℓ = 0 and for ℓ ≥ 1 it wa s s ho wn that m ℓ ( λ ) has a Q-fun ction represent ation whic h puts it in the class N κ of generalize d Nev alinna functions with κ =  ℓ +1 2  . Our purp ose in th is pap er is to s ho w that the last tw o terms in m ℓ ( λ ) can b e d ecomp osed in to real and imaginary parts as − ak ℓ ( λ )    ℓ X j =1 1 j − ia 2 √ λ    + ( i √ λ ) 2 ℓ +1 (2 ℓ + 1)!   2 ℓ X k =0 2 k  − ℓ − ia 2 √ λ  k k !(2 ℓ + 1 − k )   = i √ λk ℓ ( λ ) + r ℓ ( λ ) 2 ℓ + 1 , (1.9) where r ℓ ( λ ) is a p olynomial of degree ℓ in λ . Effectiv ely , the decomp osition (1.9) into real and imaginary parts b ecomes a defining equation for r ℓ ( λ ). Our first Mathematic a pr ogram enables the decomp osition to b e v er ifi ed. Next, in Section 3 w e separate the left hand side of (1.9) into real and imaginary parts by introd ucing p olynomial rep r esen tations of ( − ℓ − t ) k in t and ℓ Y j =1 j 6 = m  λ + a 2 4 j 2  in λ, so that th e real p art can b e represen ted as a p olynomial of degree ℓ . This yields a r eal, somewh at explicit, represent ation for r ℓ / (2 ℓ +1), and using it a second Mathemat ic a program sho ws that this real represen tation yields the same result as the firs t Mathematic a p rogram. 3 2 The First Ma thematic a program A simp le program in Mathematic a can b e used to v erify that the p olynomial r ℓ ( λ ) in (1.9) is real v alued . This mak es use of the built-in function P o c hhammer [ a, k ] wh ic h execute s the m ultiplications in the P o chhammer sym b ol ( a ) k . Using the second expression in (1.5) for K ℓ ( λ ), w e h a ve using (1.8) that b := − ak ℓ ( λ ) = K ℓ ( λ ) 2 ℓ + 1 = ( − 2 i √ λ ) 2 ℓ +1 ( − ℓ − β ) 2 ℓ +1 [(2 ℓ + 1)! ] 2 . Accordingly , solving (1.9) for r ℓ ( λ ) 2 ℓ +1 , w e ha ve r ℓ ( λ ) 2 ℓ + 1 = b    ℓ X j =1 1 j − β    + c + bi √ λ a , w here c := ( i √ λ ) 2 ℓ +1 (2 ℓ + 1)!   2 ℓ X k =0 2 k  − ℓ − ia 2 √ λ  k k !(2 ℓ + 1 − k )   . (2.1) F ollo wing is the Mathematic a program wh ic h implements equation (2.1). The output for r ℓ ( λ ) 2 ℓ +1 for ℓ = 1 , 2 , 3 , and 4 is sh o wn. The pr ogram w as executed up to ℓ = 30 showing that r ℓ ( λ ) remained real- v alued. O bserv e that the constan t a multiplies all terms of r ℓ ( λ ) 2 ℓ +1 whic h in the Mathematic a outpu t is ans1. This is also p ov ed in (3.7) b elo w . First Program Program input : ℓ ℓ = 4; While[ ℓ < 6 , β = I a 2 √ λ ; b = Expand[( − 2 I √ λ ) 2 l +1 Poc hhammer [ − ℓ − β , 2 ℓ +1] (2 ℓ +1)! 2 ; (*This generates K ℓ / (2 ℓ + 1). S ee (1.5).*) c = Apart[Simplify[ ( I √ λ ) 2 ℓ +1 (2 ℓ +1)! P 2 ℓ k =0 Poc hhammer [ − ℓ − β ,k ]2 k k !(2 ℓ +1 − k ) ]]; g = b  − P 2 ℓ j =1 1 j  ; e = b P ℓ j =1 1 j − β ; f = bI √ λ a ; Print [“F or ℓ =”, ℓ ]; ans 1 = Simplify[ c + e + f ]; Print [“ ans 1 = ” ans 1]; (*This generates the RHS of (2.1 ).*) ans 2 = Series[ ans 1 , { λ, 0 , ℓ } ]; Print[“ = ” , ans 2]; ans 3 = Simplify[ g + c + e + f ]; Print [“ ans 3 = ” , an s 3]; (*This generates the RHS of (2.6)*) ans 4 = Series[ ans 3 , { λ, 0 , ℓ } ]; Print[“ = ” , ans 4]; ℓ = ℓ + 1] Program out put: F or ℓ = 1 ans 1 = − aλ 36 ans 3 = 1 72 (3 a 3 + 10 aλ ) = a 3 24 + 5 aλ 36 (2.2) 4 F or ℓ = 2 ans 1 = − aλ ( a 2 + 13 λ ) 7200 = − a 3 λ 7200 − 13 aλ 2 7200 ans 3 = 25 a 5 + 476 a 3 λ + 1288 aλ 2 17280 0 = a 5 6912 + 119 a 3 λ 43200 + 161 aλ 2 21600 (2.3) F or ℓ = 3 ans 1 = − aλ ( a 4 + 46 a 2 λ + 400 λ 2 ) 84672 00 = − a 5 λ 84672 00 − 23 a 3 λ 2 42336 00 − aλ 3 21168 ans 3 = 49 a 7 + 2684 a 5 λ + 35656 a 3 λ 2 + 8889 6 aλ 3 50803 2000 = a 7 10368 000 + 671 a 3 λ 12700 8000 + 4457 a 3 λ 2 63504 000 + 463 aλ 3 26460 00 (2.4) F or ℓ = 4 ans 1 = − aλ ( a 6 + 107 a 4 λ + 3124 a 2 λ 2 + 2254 8 λ 3 ) 32920 473600 = − a 7 λ 32920 473600 − 107 a 5 λ 2 32920 473600 − 781 a 3 λ 3 82301 18400 − 1879 aλ 4 27433 72800 ans 3 = 761 a 9 + 9020 0 a 7 λ + 3204208 a 5 λ 2 + 3643 8400 a 3 λ 3 + 86960256 aλ 4 36870 930432000 = 761 a 9 36870 930432000 + 451 a 7 λ 18435 4652160 + 4087 a 3 λ 3 11522 165760 + 22645 9 aλ 4 96018 048000 . (2.5) Output for the p olynomials p ℓ ( λ ) := r ℓ ( λ ) 2 ℓ + 1 + ak ℓ ( λ ) H 2 ℓ , (2.6) whic h arise in the representa tion of the m ℓ function from [1, Equation(8.15)], m ℓ ( λ ) = k ℓ ( λ )  − a log( − 2 i √ λ ) − a Ψ  1 − ia 2 √ λ  − 2 γ a + i √ λ  + p ℓ ( λ ) is also giv en b elo w for ℓ = 1 , 2 , 3 , 4 . p 1 = a 3 24 + 5 aλ 36 , p 2 = a 5 6912 + 119 a 3 43200 λ + 161 a 21600 λ 2 , p 3 = a 7 10368 000 + 671 a 5 12700 8000 λ + 4457 a 3 63504 000 λ 2 + 463 a 26460 00 λ 3 p 4 = 761 a 9 36870 930432000 + 451 a 7 18435 4652160 λ + 4087 a 5 47029 248000 λ 2 + 11387 a 3 11522 165760 λ 3 + 22645 9 a 96018 048000 λ 4 . Here p ℓ is print ed as ans 3 in the ab o ve program. 5 3 Explicit Represen tation for r ℓ In this section we giv e a metho d f or separating the expression in equation (1.9 ) into real and imaginary parts, yielding a real representa tion for the p olynomial r ℓ ( λ ). The difficult y arises from the complicated pro du ct in the Poc hh ammer symb ol ( ℓ − β ) k where β = ia 2 √ λ . Replacing β by a r eal v ariable t , we let the co efficien ts of the p olynomial ( − ℓ − t ) k b e defined by g k ( t ) := ( − ℓ − t ) k = k − 1 Y j =0 ( − ℓ + j − t ) = k X n =0 α ( k , n ) t n = k 1 X j =0 α ( k , 2 j ) t 2 j + k 2 X j =0 α ( k , 2 j + 1) t 2 j +1 , (3.1) where k 1 =  k 2  and k 2 =  k − 1 2  . He re α ( k , n ) = α ℓ ( k , n ), and w e are in terested for fixed ℓ to ha ve α ℓ ( k , n ) a v ailable for all 0 ≤ k ≤ 2 ℓ , and all 0 ≤ n ≤ k . F or n = 0, th e constan t term is α ℓ ( k , 0) = Q k − 1 j =0 ( − ℓ + j ) = ( − ℓ ) k , k = 0 , 1 , · · · 2 ℓ . F ormula s for larger v alues of n b ecome increasingly more complicated and are not kno wn in closed form. W e can, ho wev er, represent the real and the im aginary parts of (1.9) in terms of α ( k , n ). Putting t = β = ia 2 √ λ in (3.1) giv es g k ( β ) = k X n =0 α ( k , n )  a 2  n i n ( λ − 1 2 ) n = k 1 X j =0 ( − 1) j α ( k , 2 j )  a 2  2 j i n ( λ − 1 2 ) 2 j + i k 2 X j =0 ( − 1) j α ( k , 2 j + 1)  a 2  2 j +1 ( λ − 1 2 ) 2 j +1 . (3.2) No w f or th e ( i √ λ ) 2 ℓ +1 term in (1.9) w e ha ve ( i ) 2 ℓ +1 = ( − 1) ℓ i , ℓ = 0 , 1 , 2 , ... T h u s the second su m in (1.9) ma y b e w ritten as i ( − 1) ℓ 2 ℓ X k =0 2 k (2 ℓ + 1)! k !(2 ℓ + 1 − k ) g k ( β )( λ 1 2 ) 2 ℓ +1 = − ( − 1) ℓ 2 ℓ X k =1 2 k (2 ℓ + 1)! k !(2 ℓ + 1 − k )   k 2 X j =0 ( − 1) j α ( k , 2 j + 1)  a 2  2 j +1 ( λ 1 2 ) 2 ℓ +1 − (2 j +1)   + i ( − 1) ℓ 2 ℓ X k =0 2 k (2 ℓ + 1)! k !(2 ℓ + 1 − k )   k 1 X j =0 ( − 1) j α ( k , 2 j )  a 2  2 j +1 ( λ 1 2 ) 2 ℓ +1 − 2 j   = 2 ℓ X k =0 2 k A k (2 ℓ + 1)! k !(2 ℓ + 1 − k ) + i 2 ℓ X k =0 2 k B k (2 ℓ + 1)! k !(2 ℓ + 1 − k ) (3.3) where A k = ( − 1) ℓ +1 k 2 X j =0 ( − 1) j α ( k , 2 j + 1)  a 2  2 j λ ℓ − j (3.4) and B k = ( − 1) ℓ k 1 X j =0 ( − 1) j α ( k , 2 j )  a 2  2 j λ ℓ − j λ 1 2 (3.5) 6 Similarly , we ma y d ecomp ose the first term in (1.9) as ℓ X m =1 1 m − ia 2 √ λ = ℓ X m =1 4 λm 4 λm 2 + a 2 + i ℓ X m =1 2 aλ 1 2 4 λm 2 + a 2 . (3.6) Using (1.8) and cancelli ng one 4 λj 2 + a 2 factor for eac h term in the ab o ve sums, w e fi nd that − ak ℓ ( λ ) " ℓ X m =1 1 m − ia 2 √ λ # = − a [(2 ℓ + 1)!] 2     ℓ X m =1 4 λm ℓ Y j =1 j 6 = m (4 λj 2 + a 2 )     + i   − a [(2 ℓ + 1)! ] 2   ℓ X m =1 2 aλ 1 2 ℓ Y j =1 (4 λj 2 + a 2 )     . (3.7) Com bin ing (3.3) and (3.7) and taking real and imaginary parts of (1.9), w e th us obtain: √ λk ℓ ( λ ) = − 2 a 2 √ λ [(2 ℓ + 1)!] 2     ℓ X m =1 ℓ Y j =1 j 6 = m (4 λj 2 + a 2 )     + 2 ℓ X k =0 2 k B k (2 ℓ + 1)! k !(2 ℓ + 1 − k ) (3.8) and r ℓ ( λ ) 2 ℓ + 1 = − 4 λa [(2 ℓ + 1)!] 2     ℓ X m =1 m ℓ Y j =1 j 6 = m (4 λj 2 + a 2 )     + 2 ℓ X k =1 2 k A k (2 ℓ + 1)! k !(2 ℓ + 1 − k ) . (3.9) Since every A k in volv es a as a factor, it follo ws th at a factor of a o ccur s in b oth terms on th e right hand side of (3.9). Hence a = 0 will giv e r ℓ ( λ ) = 0. T o brin g the expression in (3.9) in to a simpler form, w e no w isolate th e coefficients of the p ow ers of λ . F or the second term in (3.9) w e inser t the form ula (3.4) for A k in to the sum, change the summation ind ex ( m = ℓ − j ), and then interc hange the order of su mmation: 2 ℓ X k =1 2 k A k (2 ℓ + 1)! k !(2 ℓ + 1 − k ) = 2 ℓ X k =1 k 2 X j =0 2 k ( − 1) ℓ +1+ j α ( k , 2 j + 1)  a 2  2 j +1 λ ℓ − j (2 ℓ + 1)! k !(2 ℓ + 1 − k ) = 2 ℓ X k =1 ℓ X m = ℓ − k 2 " 2 k ( − 1) m +1 α ( k , 2( ℓ − m ) + 1)  a 2  2( ℓ − m )+1 (2 ℓ + 1)! k !(2 ℓ + 1 − k ) # λ m = ℓ X m =1 2 ℓ X k =2( ℓ − m )+1 " 2 k ( − 1) m +1 α ( k , 2( ℓ − m ) + 1)  a 2  2( ℓ − m )+1 (2 ℓ + 1)! k !(2 ℓ + 1 − k ) # λ m = ℓ X j =1 d j λ j (3.10) where 7 d j := a 2 2 ℓ X k =2( ℓ − j )+1 " 2 k ( − 1) m +1  a 2  2( ℓ − j ) α ( k , 2( ℓ − j ) + 1) (2 ℓ + 1)! k !(2 ℓ + 1 − k ) # . (3.11) Similarly , to isolate the p o w ers of λ in the fir s t term in (3.7) we first define the co efficien ts γ ( m, n ) of the ℓ − 1 degree p olynomial, ℓ Y j =1 j 6 = m  λ + a 2 4 j 2  = ℓ − 1 X n =0 γ ( m, n ) λ n . (3.12) Insertion of this in to the first term in (3.9), c hanging the su mmation ind ex ( j = n + 1), and inte rc hanging the order of summation then yields: − 4 λa [(2 ℓ + 1)! ] 2     ℓ X m =1 m ℓ Y j =1 j 6 = m (4 λj 2 + a 2 )     = − a 4 ℓ ( ℓ !) 2 [(2 ℓ + 1)! ] 2 " ℓ X m =1 ℓ − 1 X n =0  γ ( m, n ) m  λ n +1 # = − a 4 ℓ ( ℓ !) 2 [(2 ℓ + 1)! ] 2   ℓ X m =1 ℓ X j =1  γ ( m, j − 1) m  λ j   = − a 4 ℓ ( ℓ !) 2 [(2 ℓ + 1)! ] 2   ℓ X j =1 ℓ X m =1  γ ( m, j − 1) m  λ j   = ℓ X j =1 c j λ j (3.13) where c j := − a 4 ℓ ( ℓ !) 2 [(2 ℓ + 1)! ] 2 ℓ X m =1  γ ( m, j − 1) m  . (3.14) Putting (3.10) and (3.13) in (3.9) w e thus ha ve for the p olynomial r ℓ ( λ ) / (2 ℓ + 1) th e representa tion, r ℓ ( λ ) 2 ℓ + 1 = ℓ X j =1 ( c j + d j ) λ j . (3.15) Here it is clear that a is a common factor in c j and d j for all j , and hence for the case a = 0, r ℓ ≡ 0. The expression (3.15) is somewhat more explicit than the expr ession (2.1) and (3.9) since the co efficien ts of the p o w ers of λ are isolated, and there are no complex terms pr esen t. On the other hand , further simp lifications are certainly desirable; h o we v er, this requires closed form formulas for α ( k , n ) and γ ( m, n ) which remain elusiv e. Ho w ever, the formulas for c j and d j are easily implemen ted using a sym b olic manip ulator. 4 The Second Mathematic a Program As an indep enden t chec k on the fi r st Mathematic a program w e implemen ted th e form ulas (3.11) and (3.14) for d j and c j and computed th e p olynomial of degree ℓ in (3.15). T h is required computing and storing the 8 co efficien t α ( k , n ) and γ ( m, n ) in (3.1 ) and (3.12) . F ollo wing is the M athematic a program that do es th is to compute r ℓ ( λ ) / (2 ℓ + 1). The outpu t for ℓ = 1 , 2 , 3 and 4 is sh o wn. The program wa s executed up to ℓ = 30 and ga v e exact agreemen t with the first Mathematic a program. Second Program Program input : ℓ ℓ = 4; k = 0; While[ k ≤ 2 ℓ, p [ t ] = Expand[P o c h hammer[ − ℓ − t, k ]]; n = k ; i = 0; While[ i ≤ n, α [ k , i ] = p [0]; p [ t ] = Simplify[Expand[( p [ t ] − α [ k , i ]) /t ]]; i + +] k + +] Clear[ k ]; j = 1; While[ j ≤ ℓ , d [ j ] = P 2 ℓ k =2( ℓ − j )+1 2 k ( − 1) j − 1 α [ k , 2 ℓ − 2 j +1] ( a 2 ) 2 ℓ − 2 j + 1 (2 ℓ +1)! k !(2 ℓ +1 − k ) ; j + +] Clear[ j ]; m = 1; While[ m ≤ ℓ, q [ t ] =  Q ℓ j =1  t + a 2 4 j 2  /  t + a 2 4 m 2  ; n = ℓ − 1; i = 0; While[ i ≤ n, γ [ m, i ] = q [0]; q [ t ] = Simplify[Expand[( q [ t ] − γ [ m, i ]) /t ]]; i + +] m + + ] Clear[ i ]; j = 1; While[ j ≤ ℓ , c [ j ] = − a 4 ℓ ( ℓ !) 2 ((2 ℓ +1)!) 2 P ℓ m =1 γ [ m,j − 1] m ; j + + ; Print[ P ℓ j =1 ( c [ j ] + d [ j ]) t j ] Output of second program F or ℓ = 1 : r ℓ (2 ℓ + 1) = − at 36 F or ℓ = 2 : r ℓ (2 ℓ + 1) = − a 3 t 7200 − 13 at 2 7200 F or ℓ = 3 : r ℓ (2 ℓ + 1) = − a 5 t 84672 00 − 23 a 3 t 2 42336 00 − at 3 21168 F or ℓ = 4 : r ℓ (2 ℓ + 1) = − a 7 t 32920 473600 − 107 a 5 t 2 32920 473600 − 781 a 3 t 3 82301 18400 − 1879 at 4 27433 72800 This outp u t is in agreemen t with the output of the first p rogram, and was also c hec ked up to ℓ = 30. The t wo Mathematic a pr ograms verify that th e imaginary part of the expression (2.1) f or r ℓ ( λ ) / (2 ℓ + 1) is zero 9 for all v alues of ℓ for whic h the programs we r e executed. A general p ro of that th is is true for all ℓ , that is, a p ro of of (3.8) (or, equiv alen tly , a pro of of (3 .9)) requ ires that the α ( k , n ) and γ ( m, n ) co efficien ts b e obtained in a simplified form, and th is app ears to b e a formid able task. I t is quite difficu lt to p erform induction on ℓ to prov e (3.8) or (3.9) b ecause of th e complicated manner in whic h α ( k , n ) = α ℓ ( k , n ) and γ ( m, n ) = γ ℓ ( m, n ) change w ith ℓ . Nev ertheless, it ma y b e p ossible to construct rigorous pro ofs b y making use of some com binatorial analysis. F or example there are general form s for the coefficien ts of a p olynomial ha ving k kn o wn r o ot s x i , i = 1 , 2 , ..., k . Sen and Krishnamurth y [3], for example, show that q k ( x ) := k Y j =1 ( x − x j ) = x k + k X m =1 a m x k − m (4.1) with a m := ( − 1) m S m := ( − 1) m X x 1 x 2 ...x m , (4.2) where the su m is tak en ov er all the p ro ducts of x i tak en m at a time. Thus, for the p olynomial g k ( t ) of (3.1) w e ha v e g k ( t ) = ( − 1) k k − 1 Y j =0 [ t − ( j − ℓ )] = ( − 1) k k Y j =1 [ t − ( j − 1 − ℓ )] = ( − 1) k t k + k X m =1 a m t k − m ! , (4.3) with a m = a ℓ ( k , m ) = ( − 1) m X t 1 t 2 ...t m (4.4) where the sum is tak en o ve r all the pro du cts of t j := j − 1 − ℓ tak en m at a time. But, a general solution for a m as a fun ction of k and ℓ , v alid for all ℓ and all 0 ≤ k ≤ 2 ℓ , remains elusiv e. An other idea wo uld b e to establish (3.8) or (3.9 ) by induction on ℓ , but it unfortu nately app ears difficult to employ the indu ction h yp othesis. So, an analytic pro of of (3.8 ) or (3.9) v alid for all ℓ ≥ 1 remains an op en problem. 5 Conclusion The simp lification of a ℓ ( k , n ) and a rigorous p ro of of (3.8) and /or (3.9) remain as op en p roblems. W e ha ve, ho wev er, give n in this note tw o Mathematic a programs wh ic h implicitly establish b oth of these r esults. References [1] C. F ulton, Titc hmarsh -W eyl m-functions for Second-order Sturm Liouville Problems with tw o singular endp oints, Math. Nac h r. 281 (10) (2008), 1418-1475. [2] C. F u lton and H. Langer, Sturm-Liouville op erators with singularities and Ge n eralized Nev anlinna functions, Complex Anal. and Op er. Theory , to app ea r . [3] E.V. Krishnamurth y and S.K. Sen, Numerical Algorithms: Computations in Science and Engineering, Affiliated East W est Press, New Delhi, 2001. 10