A Parametric Type of Cauchy Polynomials with Higher Level

Takao Komatsu

doi:10.3390/axioms10030207

Abstract

There are many kinds of generalizations of Cauchy numbers and polynomials. Recently, a parametric type of the Bernoulli numbers with level 3 was introduced and studied as a kind of generalization of Bernoulli polynomials. A parametric type of Cauchy numbers with level 3 is its analogue. In this paper, as an analogue of a parametric type of Bernoulli polynomials with level 3 and its extension, we introduce a parametric type of Cauchy polynomials with a higher level. We present their characteristic and combinatorial properties. By using recursions, we show some determinant expressions.

Highlights

We propose still a different kind of generalization of Cauchy numbers and polynomials
As an analogue of a parametric type of Bernoulli polynomials with level 3 and its extension, we introduce a parametric type of Cauchy polynomials with a higher level
We show some determinant expressions

Summary

Introduction

In [29], along with ideas about parametric Bernoulli polynomials mentioned above, a parametric type of Cauchy numbers with levels 2 and 3 is introduced and studied. Such levels can be raised by an extension. As an analogue of a parametric type of Bernoulli polynomials with level 3 and its extension, we introduce a parametric type of Cauchy polynomials with a higher level We give their characteristic and combinatorial properties. By using these formulas, we give determinant expressions of bivariate Cauchy polynomials with higher levels. We can get determinant expressions of the classical Cauchy polynomials and numbers

Basic Properties

Determinants

Discussion