Some unified formulas involving generalized-Apostol-type-Gould-Hopper polynomials and multiple power sums

Abstract

The article aims to introduce a new unified class of generalized Apostol type polynomials. Further, under a slight modification on the parameters associated to the generalized Apostol type and generalized Gould-Hopper polynomials and by the use of generating method, we introduce a new class of generalized-Apostol-type-Gould-Hopper polynomials. We state some quasi-monomial properties for a new class of extensions of generalized Apostol type polynomials as well as, some summation, multiplication and explicit formulae which connect this polynomial class with the λ-Stirling numbers of second kind and generalized Hurwitz zeta function. Some general symmetry identities involving multiple power sums are also established. The new class of polynomials contains as its special cases, not only the generalized-Gould-Hopper-Apostol-Bernoulli, Euler and Genocchi polynomials but many more known smaller classes of polynomials. Finally, the these polynomials are framed within the context of generalized modified Milne-Thomson's polynomials.

Certain results for unified Apostol type-truncated exponential-Gould-Hopper polynomials and their relatives

“…For y = 0, the UATyTEGHP reduce to the truncated-exponential based Apostol-type polynomials considered in [25] and for z = 0 becomes the Gould-Hopper based Apostol-type polynomials [3]. It should be noted that the UATyTEGHP eH P (α,m,r) n,β (x, y, z; k, a, b) include the following special cases: Remark 1.1.…”

Unification of Two‐Variable Family of Apostol‐Type Polynomials with Applications

“…Araci et al [10] introduced and investigate a new unified class of generalized Apostol type polynomials by the following generating function:…”

“…e generalized Apostol-type Gould-Hopper polynomials, [10] 2 α i � −λ and φ(y, t) � c yt j , j > 2 in ( 13) pU (r) n (x, y; a, b, ], μ; −λ) � HF (r) n (x, y; a, b, c, μ, ]; λ) e generalized Apostol type polynomials of order r, [10] 3 α i � −λ, b � e and a � 1 in ( 16)…”