Computation Methods for combinatorial sums and Euler type numbers related to new families of numbers

TL;DR

This paper introduces new families of special numbers related to combinatorial sums and Euler numbers, providing identities, computational algorithms, and combinatorial interpretations to enhance understanding and calculation of these mathematical entities.

Contribution

It defines new families of numbers linked to classical combinatorial and special numbers, along with identities, algorithms, and interpretations that extend current mathematical knowledge.

Findings

Derived new identities involving the new numbers

Provided computational algorithms and numerical tables

Established connections with classical combinatorial numbers

Abstract

The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order. We can show that these numbers are related to the well-known numbers and polynomials such as the Stirling numbers of the second kind and the central factorial numbers, the array polynomials, the rook numbers and polynomials, the Bernstein basis functions and others. In order to derive our new identities and relations for these numbers, we use a technique including the generating functions and functional equations. Finally, we not only give a computational algorithm for these numbers, but also some numerical values of these numbers and the Euler numbers of negative order with tables. We also give some combinatorial interpretations of our new numbers.