Explicit formulae for all higher order exponential lacunary generating functions of Hermite polynomials

Abstract

For a sequence P = (p n (x)) ∞ n=0 of polynomials p n (x), we study the K-tuple and L-shifted exponential lacunary generating functions G K,L (λ; x) := ∞ n=0 λ n n! p n•K+L (x), for K = 1, 2 . . . and L = 0, 1, 2 . . . . We establish an algorithm for efficiently computing G K,L (λ; x) for generic polynomial sequences P . This procedure is exemplified by application to the study of Hermite polynomials, whereby we obtain closed-form expressions for G K,L (λ; x) for arbitrary K and L, in the form of infinite series involving generalized hypergeometric functions. The basis of our method is provided by certain resummation techniques, supplemented by operational formulae. Our approach also reproduces all the results previously known in the literature.

Operational Methods in the Study of Sobolev-Jacobi Polynomials

“…Section 5.1). We have achieved not only a deeper understanding of the Sobolev-Jacobi polynomials for parameters α = β = −1 as umbral images of the two-variable Hermite polynomials H n (x, y) (see Theorem 1), but were in consequence in particular able to derive explicit formulae for all K-tuple L-shifted lacunary exponential generating functions (see Theorem 2) for these SJ polynomials (based on a previous work [41] on lacunary EGFs for the bi-variate Hermite polynomials). A crucial step in this derivation consisted of a technical Lemma called "Pochhammer Proliferation Lemma" (See Lemma 4), which highlights the strengths of our integral transform techniques and fully explains the explicit structure of the lacunary EGFs as formal power series over certain generalized hypergeometric functions.…”

“…However, the presence of the operator Î enriches this technique in a substantial way. For comparison, we refer the interested readers to our recent work [41] for an alternative, but equivalent definition of the lacunary dilatation operator L (K) .…”

“…While it may not be immediately evident, the true merit of the definition of the lacunary dilatation and shift operators L (K) and ∂ λ is revealed upon applying them to computations of the EGF forms described in (74b) for the case of an EGF G(λ; x). Some of the authors of the present paper [41] recently applied these ideas to the computation of explicit formulae for all lacunary generating functions H K,L (λ; x, z) (with K ∈ Z ≥1 and L ∈ Z ≥0 ) of the two-variable Hermite polynomials H n (x, y). We quote the respective results in Appendix C for the interested readers' convenience.…”

Dual Numbers and Operational Umbral Methods

“…It will prove particularly useful in the following to note that according to the definition of the Hermite polynomials H n (x, y) as given in (22b), an alternative interpretation of ( 32) is provided in terms of the double lacunary exponential generating function H 2,0 (λ; x, y) of the polynomials H n (x, y), where we employ notations as in [34] e…”

“…Consequently, by invoking the operational identity (34), the paraxial evolution of a flattened beam may be expressed in the form…”