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, attached to Dirichlet character. We derive various functional equations and differential equations using these generating functions. The second aim is provide a novel approach to deriving identities including multiplication formulas and recurrence relations for these numbers and polynomials using these functional equations and differential equations. Furthermore, by applying p-adic Volkenborn integral and Laplace transform, 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.

32 pages

References in corpus (6)

• $q$-Bernoulli Numbers and Polynomials Associated with Multiple $q$-Zeta Functions and Basic $L$-series
• Behavior as $t\to \infty$ of Solutions of a problem in Mathematical Physics
• Unification of the three families of generalized Apostol type polynomials on the Umbral algebra
• Interpolation function of the genocchi type polynomials
• New q-Euler numbers and polynomials associated with p-adic q-integrals
• Some identities on the q-Bernstein polynomials, q-Stirling number and q-Bernoulli numbers

Cited by in corpus (8)

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• Explicit formulas for p-adic integrals: approach to p-adic distributions and some families of special numbers and polynomials
• Combinatorial applications of the special numbers and polynomials
• A finite sum involving generalized falling factorial polynomials and degenerate Eulerian polynomials
• Identities on the k-ary Lyndon words related to a family of zeta functions
• Central factorial numbers associated with sequences of polynomials
• A Second Type Of Higher Order Generalised Geometric Polynomials and Higher Order Generalised Euler Polynomials