The r-central factorial numbers with even indices

F A Shiha

doi:10.1186/s13662-020-02763-1

Abstract

In this paper, we introduce the r-central factorial numbers with even indices of the first and second kind as extended versions of the central factorial numbers with even indices of both kinds. We obtain several fundamental properties and identities related to these numbers. The connections between the new numbers and the Stirling numbers are presented. In addition, we give the probability distribution of the unsigned r-central factorial numbers with even indices. Finally, we consider the r-central factorial matrices and study some of their properties.

Highlights

1 Introduction The Stirling numbers of the first kind s(n, k) and of the second kind S(n, k), which are the coefficients of the expansions of factorials into powers and of powers into factorials, respectively, were introduced by J
We show in the theorem the connection between the r-central factorial numbers with even indices of both kinds and the Stirling numbers
6 Conclusions In this paper, we introduced the r-central factorial numbers with even indices of both kinds as extended versions of the central factorial numbers with even indices of both kinds

Summary

Introduction

The Stirling numbers of the first kind s(n, k) and of the second kind S(n, k), which are the coefficients of the expansions of factorials into powers and of powers into factorials, respectively, were introduced by J. For any nonnegative integer r, Kim et al [20] defined the extended central factorial numbers of the second kind T(r)(n, k): t e2 Recall that the central factorial numbers with even indices of the first and second kind, respectively, are denoted by u(n, k) = t(2n, 2k) and U(n, k) = T(2n, 2k) (see [8]).

Results

Conclusion