Polynomial Sequences Associated with the Moments of Hypergeometric Weights

We present some families of polynomials related to the moments of weight functions of hypergeometric type. We also consider different types of generating functions, and give several examples.


Introduction
Let {µ n } be a sequence of complex numbers and L : C[x] → C be a linear functional defined by L x n = µ n , n = 0, 1, . . . .
Then, L is called the moment functional determined by the formal moment sequence {µ n }. The number µ n is called the moment of order n. The task of finding an explicit representation for the functional L is called a moment problem [1,24,32]. A sequence {Π n (x)} ⊂ C[x], with deg(Π n ) = n is called an orthogonal polynomial sequence with respect to L provided that [4] L(Π n Π m ) = K n δ n,m , n, m = 0, 1, . . . , where K n = 0 and δ n,m is Kronecker's delta. The moments play a fundamental role in the theory of orthogonal polynomials since, among other results, we have the determinantal representation Π n (x) = C n µ 0 µ 1 · · · µ n µ 1 µ 2 · · · µ n+1 . . . . . . . . . . . . µ n−1 µ n · · · µ 2n−1 1 x · · · x n , for some normalization constant C n = 0, with µ 0 µ 1 · · · µ n µ 1 µ 2 · · · µ n+1 . . . Given their importance, it is very striking that they are not explicitly listed in the standard books on orthogonal polynomials, or even in encyclopedic texts such as [23]. In fact, the only place where we found a comprehensive enumeration of the moments of classical orthogonal polynomials was the recent article [28], based on the results obtained in the Ph.D. Thesis of the first author [27] 1 .
Note that we have Hence, the weight function ρ(x; α, β, c) satisfies the Pearson equation (see [30] or [31, (6.3)]) is the forward difference operator. If we define s by the sequence of polynomials orthogonal with respect to L is called semiclassical of class s [18,25]. Weight functions of the form (2) are also related to discrete probability distributions (Poisson, Pascal, binomial, hypergeometric, etc.) [21] 2 . The Generalized Hypergeometric probability distributions were studied by Adrienne W. Kemp in her Ph.D. Thesis [22] 3 . An excellent reference outlining the connections between the theory of probability and orthogonal polynomials is [31].
In [5], it was pointed out that since one has where the differential operator ϑ is defined by [29, (16.8.2)] Successive applications of (5) give and it follows that the first moment µ 0 determines the whole sequence {µ n }. If we use the operational formula [26] where n k denote the Stirling numbers of the second kind defined by [29, (26.8)] From (2), we have where p F q is the generalized hypergeometric function [29, (16.2.1)]. Depending on the values of p and q, we have to consider three different cases: 1. If p < q + 1, µ 0 (c) is an entire function of c.
3. If p > q + 1, the series (9) diverges for c = 0, unless one or more of the top parameters α i is a negative integer. If we take α 1 = −N , with N ∈ N, then µ 0 (c) becomes a polynomial of degree N .
Using the formula [29, (16 Although (10) seems to give an explicit formula for the moments, this type of sums (to our knowledge) can't be evaluated in closed form. An alternative is to consider generalized moments, defined by [2] ν n = L(ϕ n ), Since [29, (26.8.10)] we have and we recover (10).
In a series of papers [7,8,10,13,14,16], we studied polynomial solutions of differentialdifference equations of the form where P 0 (x) = 1, and A n (x), B n (x) are polynomials of degree at most 2 and 1 respectively. In this article, we consider some extensions of (12) to the multidimensional case, with P n (x) replaced by a vector P n (x), and B n (x) replaced by a matrix B n (x).
In [19] and [20], it was shown that some families of orthogonal polynomials can be represented as moments of probability measures. In this work, we do the opposite and express the moments µ n (c) in terms of some polynomials P n (c).
The paper is organized as follows: in Section 2, we derive a differential-difference equation for the polynomials P n (c) associated to the moments µ n (c). We also find formulas for the exponential generating functions and Stieltjes transforms of µ n (c) and P n (c). In Section 3, we apply our results to most of the families of discrete semiclassical orthogonal polynomials of class 1 studied in [15], except for limiting and c = 1 cases. Finally, in Section 4, we outline some conclusions and possible future directions.

Main results
The function µ 0 (c) satisfies the differential equation [29, (16.8 where ϑ was defined in (6). We can rewrite the ODE (13) as where the coefficients σ k (c) are linear functions of c.
Introducing the quantities we can rewrite (14) as If we define the (ξ + 1) × (ξ + 1) matrix M(c) by We can now state our main result.

Generating functions
Let's consider the exponential generating function for the moments, defined by the formal power series Given that we conclude that Using (18), we see that the exponential generating function for the polynomials P n (c) or, using (19),

Stieltjes transform
A different type of generating function for the moments that is very important in the theory of orthogonal polynomials is the Stieltjes transform (or Z transform), that can be defined by the formal Laurent series We have and using (5), we get From (4), we have S µ (0, z) = 1 z and solving (22) we obtain Using the recurrence relation for the Gamma function, we can write and therefore where we have used the integral representation [29, (16.5 We derived (23) in [12] using a different approach. If we define the Stieltjes transform of P n (c) by then it follows from (18) that

Examples
In [15] we studied all families of semiclassical polynomials of class s ≤ 1 orthogonal with respect to (1). When s = 0, we have three canonical cases (the discrete classical polynomials): where φ(x) and η(x) were defined in (3). These polynomials are associated with the Poisson, Pascal, and hypergeometric probability distributions, and results about their moments have appeared in many places before (see [21], for instance).
In [28], the authors used (11)  The moments of these polynomials have not (to our knowledge) been studied before.
In this case, we have with P n (c) defined by P 0 (c) = 1 and P n+1 = c dP n dc + cP n .
The polynomials satisfying (25) are known as Bell (or Touchard, or exponential) polynomials. It is well known that they have the explicit representation [11] P n (c) = n k=0 n k c k , and therefore (see also [28, equation (44)]) µ n (c) = e c n k=0 n k c k , in agreement with (10). Using (19) and (20), we see that the generating functions of µ n (c) and P n (c) are given by (see also [28,

Proposition 1 gives
with P n (c) defined by (16) and To obtain the Stieltjes transform of P n (c), let's write From (34), we have Q n+1 = cQ n + cR n , R n+1 = cR n − βR n + Q n , with Q 0 = 1, R 0 = 0. Using (21), we get The solution of the system (35) is given by In this case and since 1 we see that (24) is satisfied.
From (13) we see that µ 0 (c) satisfies the ODE which implies and therefore Proposition 1 gives µ n (c) = P n (c) · µ(c), with P n (c) defined by (16) and To obtain the Stieltjes transform of P n (c), let's write From (34), we have with Q 0 = 1, R 0 = 0. Using (21), we get If we represent the functions U , V as v n (z)c n , then (36) gives zu n − δ n,0 = nu n + αv n−1 , zv n = nv n + v n−1 − βv n + u n .

From (38), it follows that
Thus, In this case and since we see that (24) is satisfied.

Proposition 1 gives
with P n (c) defined by (16) and
Remark 1. Except for the Bell polynomials (26), all the other families P n (c) associated to the moments µ n (c) seem to be new.

Remark 2.
Stieltjes transforms of the generalized Krawtchouk and generalized Hahn polynomials of type I have been omitted because of the complexity of the formulas.

Conclusion
We have developed a technique for computing the moments of weight functions of hypergeometric type. We have shown that the moments are linear combinations of the first ξ + 1 moments with polynomial coefficients in the parameter c. We have also constructed generating functions for both the moments and the polynomials associated with them. All the results in Sections 3.3-3.5 are new, and give efficient ways of computing the moments of the discrete semiclassical polynomials of class 1. The same method can be used to find the moments of polynomials of class s > 1.
In a previous work [16], we found the asymptotic zero distribution of polynomial families satisfying first-order differential-recurrence relations of the form (12). It would be interesting to know if our results could be extended to include the polynomials P n (c) studied in this paper.