Umbral calculus associated with Frobenius-type Eulerian polynomials

Abstract

In this paper, we study some properties of several polynomials arising from umbral calculus. In particular, we investigate the properties of orthogonality type of the Frobeniustype Eulerian polynomials which are derived from umbral calculus. By using our properties, we can derive many interesting identities of special polynomials associated with Frobeniustype Eulerian polynomials. An application to normal ordering is presented.

On Finite Times Degenerate Higher-Order Cauchy Numbers and Polynomials

“…In [5,13], D. S. Kim, T. Kim and T. Mansour have studied some properties of Bernoulli polynomials of the second kind associated with special polynomials arising from umbral calculus.…”

“…Cauchy polynomials are also called Bernoulli polynomials of the second kind and these polynomials are very important to study mathematical physics (see [5,13]). In [5,13], D. S. Kim, T. Kim and T. Mansour have studied some properties of Bernoulli polynomials of the second kind associated with special polynomials arising from umbral calculus.…”

“…Kim introduced the degenerate Cauchy numbers and polynomials which are derived from the degenerate function e t (see [10,13]). In this paper, we try to degenerate higher-order Cauchy numbers and polynomials k-times and investigate some properties of these k-times degenerate higher-order Cauchy numbers and polynomials.…”

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

“…As is well-known, the Stirling numbers of the second kind are defined by the generating function (see [12,14,16])…”

Differential equations for Changhee polynomials and their applications

“…Recently, several authors have studied Changhee polynomials and numbers (see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]). In this paper, we consider differential equations derived from the generating function of Changhee polynomials and give some new and explicit formulae for the Changhee polynomials by using our results on differential equations.…”