Degenerate Bell Polynomials Research Articles

Simplifying coefficients in differential equations related to generating functions of reverse Bessel and partially degenerate Bell polynomials

In the paper, by virtue of the Fa\'a di Bruno formula and identities for the Bell polynomials of the second kind, the author simplifies coefficients in a family of ordinary differential equations related to generating functions of reverse Bessel and partially degenerate Bell polynomials.

A note on degenerate derangement polynomials and numbers

In this paper, we study the degenerate derangement polynomials and numbers, investigate some properties of those polynomials and numbers and explore their connections with the degenerate gamma distributions. In more detail, we derive their explicit expressions, recurrence relations and some identities involving the degenerate derangement polynomials and numbers and other special polynomials and numbers, which include the fully degenerate Bell polynomials, the degenerate Fubini polynomials and the degenerate Stirling numbers of both kinds. We also show that those polynomials and numbers are connected with the moments of some variants of the degenerate gamma distributions.

Degenerate Bell polynomials associated with umbral calculus

Carlitz initiated a study of degenerate Bernoulli and Euler numbers and polynomials which is the pioneering work on degenerate versions of special numbers and polynomials. In recent years, studying degenerate versions regained lively interest of some mathematicians. The purpose of this paper is to study degenerate Bell polynomials by using umbral calculus and generating functions. We derive several properties of the degenerate Bell polynomials including recurrence relations, Dobinski-type formula, and derivatives. In addition, we represent various known families of polynomials such as Euler polynomials, modified degenerate poly-Bernoulli polynomials, degenerate Bernoulli polynomials of the second kind, and falling factorials in terms of degenerate Bell polynomials and vice versa.

On generalized degenerate Gould-Hopper based fully degenerate Bell polynomials

On generalized degenerate Gould-Hopper based fully degenerate Bell polynomials

Degenerate polyexponential functions and degenerate Bell polynomials

In recent years, studying degenerate versions of some special polynomials, which was initiated by Carlitz in an investigation of the degenerate Bernoulli and Euler polynomials, regained lively interest of many mathematicians. In this paper, as a degenerate version of polyexponential functions introduced by Hardy, we study degenerate polyexponential functions and derive various properties of them. Also, we introduce new type degenerate Bell polynomials, which are different from the previously studied partially degenerate Bell polynomials and arise naturally in the recent study of degenerate zero-truncated Poisson random variables, and deduce some of their properties. Furthermore, we derive some identities connecting the polyexponential functions and the new type degenerate Bell polynomials.

Some Identities of Degenerate Bell Polynomials

The new type degenerate of Bell polynomials and numbers were recently introduced, which are a degenerate version of Bell polynomials and numbers and are different from the previously introduced partially degenerate Bell polynomials and numbers. Several expressions and identities on those polynomials and numbers were obtained. In this paper, as a further investigation of the new type degenerate Bell polynomials, we derive several identities involving those degenerate Bell polynomials, Stirling numbers of the second kind and Carlitz’s degenerate Bernoulli or degenerate Euler polynomials. In addition, we obtain an identity connecting the degenerate Bell polynomials, Cauchy polynomials, Bernoulli numbers, Stirling numbers of the second kind and degenerate Stirling numbers of the second kind.

On fully degenerate Bell numbers and polynomials

Recently, the partially degenerate Bell numbers and polynomials were introduced as a degenerate version of Bell numbers and polynomials. In this paper, as a further degeneration of them, we study fully degenerate Bell numbers and polynomials. Among other things, we derive various expressions for the fully degenerate Bell numbers and polynomials.

A Note on Some Identities of New Type Degenerate Bell Polynomials

Recently, the partially degenerate Bell polynomials and numbers, which are a degenerate version of Bell polynomials and numbers, were introduced. In this paper, we consider the new type degenerate Bell polynomials and numbers, and obtain several expressions and identities on those polynomials and numbers. In more detail, we obtain an expression involving the Stirling numbers of the second kind and the generalized falling factorial sequences, Dobinski type formulas, an expression connected with the Stirling numbers of the first and second kinds, and an expression involving the Stirling polynomials of the second kind.

Degenerate central Bell numbers and polynomials

Here we study the degenerate central Bell numbers and polynomials as a degenerate version of the recently introduced central Bell numbers and polynomials, which are motivated by Zhang’s work ‘Some identities involving the Euler and the central factorial numbers’ (Zhang in Fibonacci Quart 36(2):154–157, 1998) and we derive some properties, identities, and recurrence relations for these numbers and polynomials. In particular, we find various expressions for the degenerate central Bell numbers and polynomials.

Some identities for umbral calculus associated with partially degenerate Bell numbers and polynomials

Some identities for umbral calculus associated with partially degenerate Bell numbers and polynomials

On degenerate Bell numbers and polynomials

Recently, several authors have studied the degenerate Bernoulli and Euler polynomials and given some intersting identities of those polynomials. In this paper, we consider the degenerate Bell numbers and polynomials and derive some new identities of those numbers and polynomials associated with special numbers and polynomials. In addition, we investigate some properties of the degenerate Bell polynomials which are derived by using the notion of composita. From our investigation, we give some new relations between the degenerate Bell polynomials and the special polynomials.