Abstract
In this paper, the degenerate poly-Cauchy polynomials with a q parameter of the first and the second kind are introduced and their properties are studied. For these polynomials, some explicit formulas, recurrence relations, and connections with a few previously known families of polynomials are established.
Highlights
1 Introduction Throughout the paper assume that n, k ∈ Z and = q ∈ R
E–qλt – qλ Umbral calculus has been used in numerous problems of mathematics and applied mathematics; for example, see [, – ] and references therein
Let us start by presenting several explicit formulas for the degenerate poly-Cauchy polynomials with a q parameter, namely Cn(k,q)(λ, x) and Cn(k,q)(λ, x)
Summary
Cn(k,q)( ) are, respectively, called the poly-Cauchy numbers with a q parameter of the first kind and of the second kind. The degenerate poly-Cauchy polynomials with a q parameter of the first kind Cn(k,q)(λ, x) and of the second kind Cn(k,q)(λ, x) are, respectively, given by. When x = , Cn(k,q)(λ, ) and Cn(k,q)(λ, ) are, respectively, called the degenerate poly-Cauchy numbers with a q parameter of the first kind and of the second kind. The aim of this paper is to use umbral calculus techniques (see [ , ]) in order to derive some properties, recurrence relations, and identities for the degenerate poly-Cauchy polynomials with a q parameter of the first kind and of the second kind. Q e–qt – e–qλt – qλ Umbral calculus has been used in numerous problems of mathematics and applied mathematics; for example, see [ , – ] and references therein
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