A note on truncated degenerate Bell polynomials

TL;DR

This paper introduces and explores properties of truncated degenerate Bell polynomials and numbers, including explicit formulas, identities, and applications involving moments of beta distributions.

Contribution

It is the first to define truncated degenerate Bell polynomials and derive their key properties and relations, expanding the theory of special polynomials.

Findings

Derived explicit expressions and identities for truncated degenerate Bell polynomials.

Established integral representations and generating functions involving differential operators.

Connected truncated degenerate Bell numbers to moments of beta random variables.

Abstract

The aim of this paper is to introduce truncated degenerate Bell polynomials and numbers and to investigate some of their properties. In more detail, we obtain explicit expressions, identities involving other special polynomials, integral representations, Dobinski-like formula and expressions of the generating function in terms of differential operators and linear incomplete gamma function. In addition, we introduce truncated degenerate modified Bell polynomials and numbers and get similar results for those polynomials. As an application of our results, we show that the truncated degenerate Bell numbers can be expressed as a finite sum involving moments of a beta random variable with certain parameters.