Combinatorial identities associated with new families of the numbers and polynomials and their approximation values

Abstract

Recently, the numbers $Y_n(\lambda )$ and the polynomials $Y_n(x,\lambda )$ have been introduced by the second author [22]. The purpose of this paper is to construct higher-order of these numbers and polynomials with their generating functions. By using these generating functions with their functional equations and derivative equations, we derive various identities and relations including two recurrence relations, Vandermonde type convolution formula, combinatorial sums, the Bernstein basis functions, and also some well known families of special numbers and their interpolation functions such as the Apostol--Bernoulli numbers, the Apostol--Euler numbers, the Stirling numbers of the first kind, and the zeta type function. Finally, by using Stirling's approximation for factorials, we investigate some approximation values of the special case of the numbers $Y_n\left(\lambda \right)$.

17 pages

References in corpus (1)

• $q$-Bernoulli Numbers and Polynomials Associated with Multiple $q$-Zeta Functions and Basic $L$-series

Cited by in corpus (3)

• Construction of general forms of ordinary generating functions for more families of numbers and multiple variables polynomials
• New integral formulas and identities involving special numbers and functions derived from certain class of special combinatorial sums
• Construction of a generalization of the Leibnitz numbers and their properties