arXiv:math-ph/0011021v2 24 May 2002 as o eti constants certain for h ieidpnetSh¨dne qainfroeelectr is one potential Coulomb for Schr¨odinger equation time-independent The Introduction 1 M 908 n M 0100539. DMS and 9970389 DMS ∗ uigtepeaaino hspprteato a partiall was author the paper this of preparation the During aurePlnma rhgnlt n the and Polynomial Laguerre A ihdffrn nrylvl.I sas hw htti sthe analogous is polynomials. an Meixner basis this for polynomial that obtained e Laguerre is shown usual By orthogonality the also polynomials. of is terms Laguerre fu in It for wave orthogonality of natural levels. orthogonality other energy the of differing degree. proof with expo the elementary on an depending and an factor polynomial presents a by Laguerre scaled a variable the of with product the is potential h ailpr ftewv ucino neeto naCoulom a in electron an of function wave the of part radial The − ,c m, 2 ∆ ~ eateto of Department m 2 ψ ∆ ( nvriyo Virginia of University yrgnAtom Hydrogen seRbnt 7,C.1) ert h equation the Rewrite 18). Ch. [7], Robinett (see x ψ + ) hre .Dunkl F. Charles ( x a 7 2002 17, May ) || − Abstract x k || || ψ x c 1 || ( x ψ = ) ( x = ) − λψ ∗ Eψ ( x ( ) x , upre yNFgrants NSF by supported y ) , nsbett a to subject on hsnote This xpanding strange nctions nential only b where k is a positive parameter and λ denotes the rescaled energy level; let 1/2 RN N 2 x ∈ (usually N = 3, of course) and ||x|| = j=1 xj . We separate variables by considering solutions of the form ψ (xP)= φ (||x||) pl (x) where pl is a harmonic homogeneous polynomial of degree l = 0, 1, 2,... (the angular momentum quantum number). Then φ must satisfy the equation

d2 N + 2l − 1 d k φ (r)+ φ (r)+ + λ φ (r) = 0. (1) dr2 r dr r   Using spherical polar coordinates for RN we see that φ must be square- integrable for the measure r2l+N−1dr on {r : r > 0} (we restrict our at- tention to the point spectrum). Furthermore solutions for the same value of l but different energy levels must be orthogonal for this measure, by the self-adjointness of the equation. The topic of this note is a direct elementary proof of this orthogonality. It will also be shown that these functions and the standard set form the only possible (in a sense to be specified) orthogo- nal sets of scaled Laguerre polynomials. Then by expanding the functions in terms of the usual Laguerre basis an analogous orthogonality is obtained for Meixner polynomials. This orthogonal set incorporates a parameter depen- dent on the degree and is a set of eigenfunctions of a self-adjoint second-order difference operator resembling equation (1). In equation (1) set λ = −µ2 (the bound energy levels are negative) and substitute φ (r)= e−µrg (2µr), then g (s) satisfies

d2 d 1 k g (s) + (N + 2l − 1 − s) g (s)+ 1+ − N − 2l g (s) = 0. ds2 ds 2 µ   1 k The integrable solutions are obtained by setting 2 1+ µ − N − 2l = n α for some n = 0, 1, 2, 3,... then φ (r) = Ln (2µnr) exp (−µnr) whereµn = k 2(α+2n+1) and α = N + 2l − 2 ; see (3) below (φ is not normalized). The 2 2 k energy levels are −µn (specifically − 4(n+l+1) for N = 3). These radial Laguerre-type functions continue to be useful in the re- search literature: Beckers and Debergh [1] used them for supersymmetric properties of the parastatistical hydrogen atom, Blaive and Cadilhac [2] derived an expansion in powers of µn (for fixed l), Dehesa et al [3] found approximations for entropies which involve of the logarithms of the Laguerre polynomials, Xu and Kong [9] constructed shift operators for the radial Hamiltonian.

2 The Pochhammer symbol is defined by

n R (a)n = (a + i − 1) , a ∈ , Yi=1 and the hypergeometric ( 2F1) series is ∞ (a)j (b)j j 2F1 (a, b; c; x)= x . (c)j j! Xj=0 The basic facts about the Laguerre polynomial of index α and degree n that will be needed are: n (α + 1) (−n) xj Lα (x)= n j , n n! (α + 1) j! j=0 j ∞ X − (α + 1) Lα (x) Lα (x) xαe xdx = δ Γ (α + 1) n , (2) m n mn n! Z0 d2 d x + (α + 1 − x) + n Lα (x) = 0, (3) dx2 dx n   α α α (n + 1) Ln+1 (x) − (α +1+2n − x) Ln (x) + (n + α) Ln−1 (x) = 0. (4)

Their proofs and further information are in Szeg¨o[8], Ch.5. The author gratefully acknowledges a useful discussion with R. Y´a˜nez.

2 Orthogonality relation

Fix a parameter α > −1. The scaling factor k appearing in equation (1) is set equal to 1.

Definition 1 For n = 0, 1, 2,... and x ∈ R let

x x φ (x)= Lα exp − . n n α + 2n + 1 2 (α + 2n + 1)    

Theorem 2 For m,n = 0, 1, 2,... the following orthogonality relation holds: ∞ δ φ (x) φ (x) xα+1 dx = mn (α + 2n + 1)α+3 Γ (α +1+ n) . m n n! Z0

3 The proof is composed of two parts. First, the case m = n follows from the orthogonality relation (2) and the three-term recurrence (4) in the form

α α α α xLn (x)= − (n + 1) Ln+1 (x) + (α + 2n + 1) Ln (x) + (α + n) Ln−1 (x) ,

∞ α 2 α −x and thus 0 xLn (x) x e dx = (α + 2n + 1) Γ (α +1+ n) /n!. In the in- ∞ 2 α+1 x tegral φn (x) x dx change the variable to s = . 0 R ∞ α+2n+1 Second, fix m 6= n and in the φ (x) φ (x) xα+1 dx change R 0 m n the variable to R x 1 1 s = + , 2 α + 2m + 1 α + 2n + 1   and let α + 2n + 1 β = ; α + m + n + 1

(α + 2m + 1) (α + 2n + 1) α+2 then the desired integral equals times α + m + n + 1   ∞ α α α+1 −s Lm (βs) Ln ((2 − β) s) s e ds. (5) Z0 We will show this equals zero by expressing the two polynomials in terms of α+1 Lj (s), which simplifies evaluation of the integral. Recall the identities

α α+1 α+1 Ln (x)= Ln (x) − Ln−1 (x) , (6) n (δ +1+ j) − − Lδ (bx)= n j bj (1 − b)n j Lδ (x) , (7) n (n − j)! j Xj=0 which hold for arbitrary x, b ∈ R, δ > −1 and n = 0, 1, 2,... . Then (with δ = α + 1 in (7))

α α+1 α+1 Lm (βs)= Lm (βs) − Lm−1 (βs) m (α +2+ j) − − − − = m 1 j βj (1 − β)m 1 j Lα+1 (s) (m − j)! j Xj=0 × ((1 − β) (α +1+ m) − (m − j)) m (α +2+ j) − − − − = m 1 j βj (1 − β)m 1 j (j − n (2 − β)) Lα+1 (s) (m − j)! j Xj=0

4 and α α+1 α+1 Ln ((2 − β) s)= Ln ((2 − β) s) − Ln−1 ((2 − β) s) n (α +2+ j) − − − − = n 1 j (2 − β)j (β − 1)n 1 j Lα+1 (s) (n − j)! j Xj=0 × ((β − 1) (α +1+ n) − (n − j)) n (α +2+ j) − − − − = n 1 j (2 − β)j (β − 1)n 1 j (j − mβ) Lα+1 (s) . (n − j)! j Xj=0 Now multiply the two expressions together and integrate with respect to the measure sα+1e−sds on {s : s> 0}. By the orthogonality (2) with parameter α + 1 integral (5) equals Γ (α + 2) times

(α + 2) − (α + 2) − − − m 1 n 1 (1 − β)m 1 (β − 1)n 1 (8) m! n! min(m,n) j (−m)j (−n)j β (β − 2) × 2 (j − n (2 − β)) (j − mβ) . (α + 2)j j! (1 − β) Xj=0   1 The formula holds when one of m,n = 0 by the convention (α + 2)−1 = α+1 . Let β (β − 2) 1 γ = = 1 − (1 − β)2 (1 − β)2 then γ (j − n (2 − β)) (j − mβ)= j (α +1+ j)+ (mn − j (α + m + n + 1)) , γ − 1 γ since β (2 − β)= γ−1 . The second line of the sum (8) can be written as γmn mnγ F (1 − m, 1 − n; α + 2; γ)+ F (−m, −n; α + 2; γ) 2 1 γ − 1 2 1 γ2mn (α + m + n + 1) − F (1 − m, 1 − n; α + 3; γ) ; (γ − 1) (α + 2) 2 1 γ which equals mn γ−1 times

(γ − 1) 2F1 (1 − m, 1 − n; α + 2; γ)+ 2F1 (−m, −n; α + 2; γ) γ (α + m + n + 1) − F (1 − m, 1 − n; α + 3; γ) ; α + 2 2 1 and this expression is zero by the contiguity relations for the 2F1-series (by the straightforward calculation of the coefficient of γj for each j). This completes the proof of the theorem.

5 3 Uniqueness ∞ α 1 We show that the only possible orthogonal sets Ln (γnx)exp − 2 γnx n=0 α+κ for the measure x dx on {x : x> 0} are the two cases (1) κ = 0, γn = γ0 α+1   for all n; (2) κ = 1, γn = γ0 α+2n+1 . By rescaling x we may assume γ0 = 1. The choice of the basis functions is motivated by their general utility and α 1 precise asymptotic formulae. Let φn (x)= Ln (γnx)exp − 2 γnx (with each γ > 0) and define the inner product n  ∞ − hf, gi = Γ (α + κ + 1) 1 f (x) g (x) xα+κdx, Z0 for suitable functions f, g. By using the same method as in Section 2 (for- mula (5)) the inner product hφm, φni is expressed as a multiple of ∞ α α −s α+κ Lm (βs) Ln ((2 − β) s) e s ds, Z0 2γm where β = γm+γn . The evaluations are straightforward algebraic calcula- tions and were carried out in the computer algebra system MapleTM , which was also used to find the polynomials q2,q3 and their resultant.

3.1 The case κ 6=0, 1

We show that the equations hφ0, φ1i = hφ0, φ2i = hφ1, φ2i = 0 are incompat- α+1 ible unless κ = 0 or 1. The equation hφ0, φ1i = 0 implies γ1 = α+2κ+1 . The condition hφ0, φ2i = 0 is equivalent to a certain quadratic equation q2 (γ2)= 0 (not displayed here). The discriminant of q2 equals 16κ (α + κ + 1) (α + 2). Integrability requires α + κ + 1 > 0 hence there are real solutions for γ2 only α+1 if κ ≥ 0. The conditions hφ1, φ2i = 0 and γ1 = α+2κ+1 are equivalent to a certain cubic equation q3 (γ2) = 0. The resultant of q2,q3 (a polynomial in the coefficients which is zero if and only if there exists a simultaneous root) is found to be the product of κ (κ − 1) by a polynomial in (α + 1) and κ with all coefficients nonnegative. This shows that q2 (γ2)=0= q3 (γ2) is impossible for κ ≥ 0,α> −1 unless κ = 0 or 1.

3.2 The case κ =0 or 1

For the case κ = 0 the condition hφ0, φni = 0 (for n > 0) and the formula n 2 (7) with δ = α show that (β − 1) = 0 where β = 1+γn , hence γn =1= γ0 for all n (the variable β is not identical to the one used in Section 2, because the scaling parameters γn are initially undetermined).

6 α+1 For the case κ = 1 , hφ0, φ1i = 0 implies γ1 = α+3 ; further for n ≥ 2 the condition hφ0, φni = 0 and formulas (6), (7) imply − (β − 1)n 1 ((α +1+ n) (β − 1) − n) = 0,

2 α+2n+1 α+1 (where β = 1+γn ) with the solutions β = 1 and α+n+1 . Thus γn = α+2n+1 α+1 or 1. But γ1 = α+3 , and the computation of hφ1, φni with γn = 1 yields

n−1 (α + 3) (α + 3)n−2 hφ1, φni = (−1) n ((α + 3) n − 1) − (α + 1)n 2 (α + 2) which is nonzero for n ≥ 2. This rules out the possibility that γn = 1 and α+1 shows that γn = α+2n+1 for all n is necessary. The sufficiency was proved already.

4 Change of Basis and Meixner Polynomials

Since there is another obvious orthogonal basis for L2 xα+1dx , the stan- dard Laguerre functions Lα+1 (cx) e−cx/2 : m = 0, 1, 2,... , with some scal- m  ing parameter c, the computation of the transformation coefficients will  give rise to an infinite orthogonal matrix. The entries will be expressed as Meixner polynomials: for parameters γ > 0 and 0

−1 Mn (x; γ, c)= 2F1 −n, −x; γ; 1 − c , with orthogonality relations  ∞ (γ) δ n! x cxM (x; γ, c) M (x; γ, c)= mn , (9) x! n m (γ) cn (1 − c)γ x=0 n X satisfying the second order difference equation

c (x + γ) f (x + 1) − (x + (x + γ) c) f (x)+ xf (x − 1) = n (c − 1) f (x) , (10) and the three-term recurrence c (γ + n) xM (x; γ, c)= M (x; γ, c) + (11) n c − 1 n+1 n + (n + γ) c n M (x; γ, c)+ M − (x; γ, c) 1 − c n c − 1 n 1

7 (see the handbook of Koekoek and Swarttouw [6], p.45). Now consider the two orthogonal bases {q0,n : n ≥ 0} and {q1,m : m ≥ 0} 2 α+1 α+1 −βx/2 for L (0, ∞) ,x dx where q0,n (x) = Ln (βx) e and q1,m (x) = Lα+1 ((2 − β) x) e−(2−β)x/2 (for 0 <β< 2, again not identical to previous m  usage of this symbol). By use of the expansion (7) for β 6= 1 it follows easily that ∞ −1 α+1 Γ (α + 2) q0,n (x) q1,m (x) x dx 0 Z ∞ −1 α+1 α+1 −x α+1 = Γ (α + 2) Ln (βx) Lm ((2 − β) x) e x dx Z0 (α + 2) (α + 2) β (β − 2) = (1 − β)n+m (−1)m n m F −n, −m; α + 2; ; n! m! 2 1 2  (1 − β)  2 and the 2F1-sum equals Mn m; α + 2, (1 − β) . Of course this calculation is only for illustration and is not novel; the orthogonality of Meixner poly- nomials is a simple consequence of that of the Laguerre polynomials and −1 ∞ 2 α+1 −α−2 (α+2)n the values Γ (α + 2) 0 (q0,n (x)) x dx = β n! and the similar result for q . 1,m R 4.1 The transformation matrix We consider the analogous computation for the two bases α+1 x −x/(2(α+1)) Lm α+1 e : m ≥ 0 and {φn : n ≥ 0} (from definition 1). The scalingn  for the former is chosen too force equality at m = n = 0.

1 1 1 n Proposition 3 For m,n ≥ 0 let An = 2 α+2n+1 + α+1 , un = α+n+1 and ∞   − x x − I (n,m)=Γ(α + 2) 1 Lα Lα+1 e Anx xα+1 dx, n α + 2n + 1 m α + 1 Z0     α+2 1 then I (0, 0) = (α + 1) ; I (0,n)=0= I (n, 0) for n> 0; An = (α+1)(1+un) and

(α + 3) − (α + 2) − − I (n,m) = ((α + 1) (1 + u ))α+2 m 1 n 1 (−1)m 1 1 − u2 n (m − 1)! (n − 1)! n m+n−3 −2  × un 2F1 1 − n, 1 − m; α + 3; 1 − un .

Proof. In I (n,m) change the integration variable to s = Anx and let −1 α+1 β = (An (α + 2n + 1)) = α+n+1 , then ∞ −1 −α−2 α α+1 −s α+1 I (n,m)=Γ(α + 2) An Ln (βs) Lm ((2 − β) s) e s ds. Z0 8 By formulae (7) and (6)

n (α +2+ j) − − − − Lα (βs)= n 1 j βj (1 − β)n 1 j Lα+1 (s) n (n − j)! j Xj=0 × ((1 − β) (α + n + 1) − (n − j)) n j (α +2+ j) − − − − = n 1 j βj (1 − β)n 1 j Lα+1 (s) , (n − j)! j Xj=1 α for n> 0 while L0 (βs) = 1. By formula (7) m (α +2+ j) − − Lα+1 ((2 − β) s)= m j (2 − β)j (β − 1)m j Lα+1 (s) . m (m − j)! j Xj=0 − Multiply the two sums and integrate with respect to Γ (α + 2) 1 e−ssα+1ds and by use of the orthogonality relations obtain

−α−2 m I (n,m)= An (α + 2)m (α + 2)n−1 (−1) min(m,n) j β (β − 2) j × 2 , (n − j)! (m − j)!j! (α + 2)j (1 − β) Xj=1   −α−2 α+2 for n> 0; and I (0, 0) = A0 = (α + 1) ,I (0,m)=0for m> 0. When n −1 n> 0 let un = α+n+1 then An = ((1+ un) (α + 1)) , β = 1 − un, 2 − β = i 1+un. In the sum replace j by i+1 (and (n − j)! by (n − 1)!/ (1 − n)i (−1) and similarly for (m − 1)! ) to obtain the claimed result. α+1 Certainly the integral could be computed for any scaling of the Lm basis, but our particular choice gives the neatest formulae. Motivated by  the value of I (n,m) we write the following:

n Definition 4 For α ≥ 0 and n = 0, 1, 2, 3 . . . (and un = α+n+1 ) define the function

α x 2 hn (x)= un+1 (x + 1) Mn x; α + 3, un+1 for x ≥−1.  Corollary 5 For n> 0,

(α + 3) − (α + 2) − I (n,m) = ((α + 1) (1 + u ))α+2 m 1 n 1 n m! (n − 1)! m−1 2 n−2 α × (−1) 1 − un un hn−1 (m − 1) .  9 Let

∞ 2 − x − N = Γ (α + 2) 1 Lα+1 e x/(α+1) xα+1 dx m m α + 1 Z0   (α + 2) = (α + 1)α+2 m , m!

2 α+3 (α+2)n−1 and recall from Theorem 2 that ||φn|| = (α + 2n + 1) n! (divid- ing by the normalizing Γ (α + 2)). By the orthogonality of the two bases ∞ I (n,m) the matrix is orthogonal. The entries in row 0 and 1/2 (||φn|| Nm )n,m=0 α+2 1/2 column 0 are zero except I (0, 0) = (α + 1) = ||φ0|| N0 . The row or- α thogonality demonstrates the orthogonality of the functions {hn}; indeed ∞ −1 2 m=0 Nm I (n,m) I (l,m) = δnl||φn|| . After some manipulation this im- plies for n, l ≥ 1 that P ∞ (α + 2)m α α h − (m − 1) h − (m − 1) m! n 1 l 1 m=1 X (n − 1)! (α + 2)2 (α + n + 1)2α+2n+4 = δnl . α+4 α+3 2n−3 (α + 2)n−1 (α + 1) (α + 2n + 1) n For n = l this can be verified from the Meixner polynomial relations: the sum equals

∞ (α + 3) 2x 2 2 (α + 2) x (x + 1) u M − x; α + 3, u ; x! n n 1 n x=0 X  2 then (x + 1) Mn−1 x; α + 3, un can be expanded in terms of 2 Mj x; α + 3, un : j = n − 2,n − 1,n by use of formula (11). The coeffi- n(α+n+1)  cient of Mn−1 is  α+1 and together with (9) this produces the desired α ∞ result. The orthogonality of {hn}n=0 can also be established by exhibiting these functions as eigenfunctions of a self-adjoint operator.

4.2 A self-adjoint operator

2 −x First let Mn−1 x; α + 3; un = un g (x) and substitute in the second-order difference equation (10) to obtain:  −1 (x + α + 3) g (x + 1) + (α + 1) g (x)+ xg (x − 1) = un + un (x + 1) g (x) ;  10 f(x) replace g (x) by x+1 then f satisfies

x + α + 3 α + 1 − f (x +1)+ f (x)+ f (x − 1) = u + u 1 f (x) . x + 2 x + 1 n n  It is easy to see that the operator

x + α + 3 α + 1 Df (x)= f (x +1)+ − 2 f (x)+ f (x − 1) x + 2 x + 1   defined for functions on x = −1, 0, 1, 2,... such that f (−1) = 0 is sym- ∞ (α + 2)x+1 metric for the inner product hf1,f2i = f1 (x) f2 (x). Further x=0 (x + 1)! 2 2 2 α (un − 1) α P (un − 1) (α + 1) Dhn−1 = hn−1 (for n ≥ 1), and = , and un un n (α + n + 1) α so here is another proof of the orthogonality of the set {hn : n ≥ 0}. Note that D can be written as

Df (x) = (f (x + 1) − 2f (x)+ f (x − 1)) α + 1 1 1 + (f (x + 1) − f (x)) + (α + 1) + f (x) , x + 2 x + 1 x + 2   which is a discrete analogue of the radial Coulomb equation (1). It is clear that the difference equation

Df (x)= −λf (x) (12) with initial conditions f (0) = 1 and f (−1) = 0 has a unique solution for any λ. To find a formal expression for the solution set

∞ x f (x)= u (x + 1) aj (−x)j Xj=0 with u, aj to be determined, and substitute in the equation (the series ter- minates for each x = 0, 1, 2,... ). After the usual manipulations we obtain (for j ≥ 0):

2 −u (j + 1) (j + α + 3) aj+1

+u (u (α + 2j +3)+ α + 1 − (2 − λ) (j + 1)) aj 2 − (u − 1) + λu aj−1 = 0.  

11 To make this a two-term recurrence, solve u2 − (2 − λ) u +1=0 for u; when λ < 0 select the solution with 0

2 −u (j + 1) (j + α + 3) aj+1 +(1+ u) {j (u − 1) + u (α + 2) − 1} aj = 0. (13)

First we handle the special case λ = 0, u = 1 with the solution aj+1 = 2(α+1) α (j+1)(j+α+3) aj and denote the corresponding solution f of (12) by h∞ where

∞ j α (−x)j (2 (α + 1)) h∞ (x) = (x + 1) , (α + 3)j j! Xj=0 a hypergeometric series of type 1F1 (and resembling a Laguerre polynomial). 1 − u (α + 2) Second, for u 6= 1 let γ = then equation (13) becomes u − 1

2 2 −u (j + 1) (j + α + 3) aj+1 + u − 1 (j − γ) aj = 0, with the corresponding solution of (12) 

x −2 f (x)= u (x + 1) 2F1 −x, −γ; α + 3; 1 − u , generally only meaningful for x = −1, 0, 1, 2,... . In this parametrization, 1+ γ (α + 1)2 u = and λ = − . (The terminating solutions α +2+ γ (γ + 1) (α + γ + 2) α of course were found already, γ = n for hn.) The asymptotic behavior of Meixner polynomials and their zeros has been investigated by Jin and Wong [4],[5], a typical situation being the properties of Mn (nx; γ, c) as n →∞ for fixed x,γ,c. But here c depends on n, and a simple pointwise result is valid:

α α Proposition 6 lim h (x)= h∞ (x) for each x ∈ R. n→∞ n

α −2 (α+1)(α+2n+3) Proof. In h the argument of the F is 1 − u = − 2 . n 2 1 n+1 (n+1) Then un+1 → 1 and

(−x) (−n) (α + 1) (α + 2n + 3) j (−x) j j − → j (2 (α + 1))j , (α + 3) j! 2 (α + 3) j! j  (n + 1)  j

j for each j as n → ∞. Since (−n)j /n ≤ 1 the dominated convergence theorem applies to prove the claim.

12 α For fixed k = 1, 2, 3,... the first (in increasing order) k zeros of hn tend α to the first k zeros of h∞. In this paper we gave an elementary proof of the strange orthogonality of Laguerre functions associated with the wave functions of the Coulomb potential and found an analogous orthogonality for Meixner polynomials.

References

[1] Beckers, J. and Debergh, N.: On a parastatistical hydrogen atom and its supersymmetric properties, Phys. Lett. A 178 (1993), 43-46.

[2] Blaive, B. and Cadilhac, M.: Upper limit of the discrete hydrogen-like wave functions: expansion in the inverse principal quantum number n−1, J. Math. Phys. 38 (1997), 6061-6071.

[3] Dehesa, J., Y´a˜nez, R., Aptekarev, A. and Buyarov, V.: Strong asymptotics of Laguerre polynomials and information entropies of two- dimensional harmonic oscillator and one-dimensional Coulomb poten- tials, J. Math. Phys. 39 (1998), 3050-3060.

[4] Jin, X-S. and Wong, R.: Uniform asymptotic expansions for Meixner polynomials, Constr. Approx. 14 (1998), 113-150.

[5] Jin, X-S. and Wong, R.: Asymptotic formulas for the zeros of the Meixner polynomials, J. Approx. Theory 96 (1999), 281-300.

[6] Koekoek, R. and Swarttouw, R.: The of Hypergeometric and its q-Analogue, Report 98-17, Delft Univer- sity of Technology, 1998.

[7] Robinett, R.: , Oxford University Press, New York, Oxford, 1997.

[8] Szeg¨o, G.: Orthogonal Polynomials 3rd ed., Colloquium Publications vol. 23, Amer. Math. Soc., Providence, 1967.

[9] Xu, B-W. and Kong, F-M.: Factorization of the radial Schr¨odinger equa- tion of the hydrogen atom, Phys. Lett. A 259 (1999), 212-214.

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