About polynomials related to a quadratic equation

TL;DR

This paper explores polynomials associated with a specific quadratic equation in a C-Algebra, revealing their connection to Bernoulli polynomials and providing formulas, generating functions, and applications in integral transforms.

Contribution

It introduces a new class of polynomials linked to a quadratic equation in a C-Algebra, with explicit formulas and applications, expanding the understanding of polynomial sequences related to Bernoulli polynomials.

Findings

Derived generating functions and explicit formulas for the polynomials.

Established connections between these polynomials and Bernoulli, cosecant, and tangent numbers.

Applied the polynomials to compute certain integral transforms.

Abstract

We consider here a particular quadratic equation linking two elements of a C-Algebra. By analysing powers of the unknowns, it appears a double sequence of polynomials related to classical Bernoulli polynomials. We get the generating functions, integral forms and explicit formulas for the coefficients involving cosecant and tangent numbers. We also study the use of these polynomials for the calculation of some integral transforms.