Generating functions for generalized Stirling type numbers, Array type polynomials, Eulerian type polynomials and their applications

Abstract

The first aim of this paper is to construct new generating functions for the generalized λ-Stirling type numbers of the second kind, generalized array type polynomials and generalized Eulerian type polynomials and numbers. We derive various functional equations and differential equations using these generating functions. The second aim is to provide a novel approach to derive identities including multiplication formulas and recurrence relations for these numbers and polynomials using these functional equations and differential equations.Furthermore, we derive some new identities for the generalized λ-Stirling type numbers of the second kind, the generalized array type polynomials and the generalized Eulerian type polynomials. We also give many applications related to the class of these polynomials and numbers.

Some results on q-Hermite based hybrid polynomials

“…Several unified forms of the Apostol-type polynomials are introduced and studied by many authors, for this see [7,21,23,22,27,8,9,12,15]. We recall the definition of the Apostol type q-Frobenius-Euler polynomials H (α) n,q (x, y; u; λ) introduced and studied by Kurt in [13].…”

Identities Associated with Generalized Stirling Type Numbers and Eulerian Type Polynomials

“…In [31], we provided generating functions, related to nonnegative real parameters, for the generalized Eulerian type polynomials and numbers (generalized Apostol-type Frobenius-Euler polynomials and numbers of higher-order). We derived fundamental properties, recurrence relations and many new identities for these polynomials and numbers based on the generating functions, functional equations and differential equations.…”

“…These numbers occur in combinatorics and in the theory of partitions (cf. [31], [16], [33]). In [31], we constructed a new generating function, related to nonnegative real parameters, for the generalized  -Stirling type numbers of the second kind.…”

“…[31], [16], [33]). In [31], we constructed a new generating function, related to nonnegative real parameters, for the generalized  -Stirling type numbers of the second kind. We derived some elementary properties including recurrence relations of these numbers.…”

“…Although, in the literature, one can find extensive investigations related to the generating functions for the Bernoulli, Euler and Genocchi numbers and polynomials and also their generalizations, the generalized  -Stirling numbers of the second kind, the generalized array type polynomials and the Eulerian type polynomials, related to nonnegative real parameters, have not been studied in detail, yet (cf. [31] (cf. [16]).…”