On poly-Bell numbers and polynomials

Abstract

This paper aims to construct a new family of numbers and polynomials which are related to the Bell numbers and polynomials by means of the confluent hypergeometric function. We give various properties of these numbers and polynomials (generating functions, explicit formulas, integral representations, recurrence relations, probabilistic representation,...). We also derive some combinatorial sums including the generalized Bernoulli polynomials, lower incomplete gamma function, generalized Bell polynomials. Finally, by applying Cauchy formula for repeated integration, we introduce poly-Bell numbers and polynomials.

A note on truncated degenerate Bell polynomials

“…Many researchers have studied Bell numbers in connection with various areas (see [1,2,3,8,15,18]). Recently, the degenerate Stirling numbers of the first kind are defined as the coefficients in the expansion [1,11]).…”