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

TL;DR

This paper introduces higher-order versions of the recently defined numbers and polynomials, deriving identities, relations, and approximation properties using generating functions and connections to well-known special numbers.

Contribution

It constructs higher-order versions of the numbers and polynomials, deriving new identities and relations, and explores their approximation values and connections to classical special numbers.

Findings

Derived recurrence relations and convolution formulas.

Established connections with classical special numbers.

Investigated approximation values using Stirling's approximation.

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) $.