Finite bivariate biorthogonal I -- Konhauser polynomials

TL;DR

This paper introduces a finite set of biorthogonal polynomials in two variables derived from Konhauser polynomials, exploring their properties, transformations, and modifications to enable fractional calculus and semigroup properties.

Contribution

It presents a novel finite biorthogonal polynomial family in two variables with new operational, integral, and fractional calculus properties, including parameter modifications for semigroup behavior.

Findings

Derived a finite biorthogonal polynomial set using Konhauser polynomials

Established operational, integral, and Laplace transform properties

Modified the set to include parameters for fractional calculus and semigroup

Abstract

In this paper, a finite set of biorthogonal polynomials in two variables is produced using Konhauser polynomials. Some properties containing operational and integral representation, Laplace transform, fractional calculus operators of this family are studied. Also, computing Fourier transform for the new set, a new family of biorthogonal functions are derived via Parseval's identity. On the other hand, this finite set is modified by adding two new parameters in order to have semigroup property and construct fractional calculus operators. Further, integral equation and integral operator are also derived for the modified version.