Some polynomials defined by generating relations

Abstract

ABSTRACT. In an attempt to present a unified treatment of the various polynomial systems introduced from time to time, new generating functions are given for the sets of polynomials {S" ¿ (A.; x)} and {7"¿ ^ '(\; x)}, defined respectively by (6) and (29)

Addendum to: “Some polynomials defined by generating relations” (Trans. Amer. Math. Soc. 205 (1975), 360–370)

“…Let the parameters a, ß and Xx.Xr, and the coefficients C(kx,..., kr), kj; > 0, V/' G {1,..., r}, be arbitrary complex numbers independent of n. Also let the sequence of polynomials AÍa,/3)[A1 (... ,Xr; qx,... ,qr;zx.z,] be defined by equation (40) Remark. Evidently, in the general case considered in [1], the above equations (41*) and (42*) would replace our earlier relationships (41) and (42), respectively.…”

“…Certain constraints are explicitly specified for the validity of a recent result involving a multivariate generating function, due to the present authors [1, p. 369, Theorem 6]. It is also indicated how this result can be further generalized.…”

“…It should be pointed out that Theorem 6 (p. 369) of [1] holds as stated if and only if (t) À,. = À and qj = q, V/ G {1,...,r),…”

Some Generalizations of Pochhammer’s Symbol and their Associated Families of Hypergeometric Functions and Hypergeometric Polynomials

“…The existing literature on generating functions is abundant in results that are based essentially upon the Lagrange expansion theorem as well as the three formulas (15), (16) and (18) (see, for details, [42,Chapter 7]; see also [10], [2], [32], [33], [34], [36], [43], [46] and [54], and as well as many references to other closely-related investigations cited in each of these works). With a view to applying it to derive generating functions for a certain class of generalized incomplete hypergeometric polynomials, we recall here a general result on generating functions asserted by Lemma …”

Connection coefficients between Boas–Buck polynomial sets

“…. A corresponding generating function is given by [31] ∞ n=0 α α + (β + 1)n (3.12) where ξ is defined by (2.20) and…”