Construction of a generalization of the Leibnitz numbers and their properties

TL;DR

This paper introduces a new generalization of Leibnitz numbers using Beta functions and Bernstein basis functions, explores their properties via p-adic integrals, and provides computational methods and identities involving related special numbers.

Contribution

It presents a novel generalization of Leibnitz numbers, derives their properties through generating functions and p-adic integrals, and offers computational formulas and new identities.

Findings

New generalization of Leibnitz numbers introduced

Derived properties and identities involving related special numbers

Provided computational implementation and tables for the generalized numbers

Abstract

The aim of this paper is to give a novel generalization of the Leibnitz numbers derived from application of the Beta function to the modification for the Bernstein basis functions. We also give some properties of the Leibnitz numbers with the aid of their generating functions derived from the Volkenborn integral on the set of $p$-adic integers. We also give some novel identities and relations involving the Leibnitz numbers, the Daehee numbers, the Changhee numbers, inverse binomial coefficients, and combinatorial sums. Finally, by coding computation formula for the generalization of the Leibnitz numbers in Mathematica 12.0 with their implementation, we compute few values of these numbers with their tables. Finally, by using the applications of Volkenborn integral to Mahler coefficients, we derive some novel formulas involving the Leibnitz numbers.