Abstract
In this paper, we consider the poly-Cauchy polynomials and numbers of the second kind which were studied by Komatsu. We note that the poly-Cauchy polynomials of the second kind are the special generalized Bernoulli polynomials of the second kind. The purpose of this paper is to give various identities of the poly-Cauchy polynomials of the second kind which are derived from umbral calculus.
Highlights
1 Introduction As is well known, the Bernoulli polynomials of the second kind are defined by the generating function to be t ( + t)x = log ( + t) tn bn(x) n!
We investigate the properties of the poly-Cauchy numbers and polynomials of the second kind with umbral calculus viewpoint
The purpose of this paper is to give various identities of the poly-Cauchy polynomials of the second kind which are derived from umbral calculus
Summary
1 Introduction As is well known, the Bernoulli polynomials of the second kind are defined by the generating function to be t ( + t)x = log ( + t) tn bn(x) n! Let Lifk(x) be the polylogarithm factorial function, which is defined by The poly-Cauchy polynomials of the second kind c(nk)(x) (k ∈ Z, n ∈ Z≥ ) are defined by the generating function to be When x = , c(nk) = c(nk)( ) are called the poly-Cauchy numbers of the second kind, defined by
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