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Finite bivariate biorthogonal I -- Konhauser polynomials

Esra Güldoğan Lekesiz, Bayram Çekim, Mehmet Ali Özarslan

TL;DR

This work constructs a novel finite two-variable biorthogonal system grounded in Konhauser polynomials, introducing the two-variable finite polynomials $_{K}I_{k;\upsilon}^{(p,q)}(z,t)$ and $_{K}\mathcal{I}_{k;\upsilon}^{(p,q)}(z,t)$ tied to univariate finite orthogonal polynomials $I_{k}^{(p)}(t)$. It develops corresponding bivariate Mittag-Leffler functions $E_{\gamma_{3},\gamma_{4};q;\upsilon}^{(\gamma_{1};\gamma_{2})}(z,t)$ and provides integral, operational, Laplace, and Fourier transform representations that yield a Parseval based class of finite biorthogonal functions. A semigroup property is achieved by modifying the polynomials with new parameters $(\gamma,c)$, leading to modified Mittag-Leffler functions and a convolution type integral equation with an associated integral operator. The results extend multivariate biorthogonal theory, connect to fractional calculus via RL and Caputo type operators, and offer explicit transform based constructions that may enhance spectral methods and two variable analysis. The framework thus bridges finite biorthogonality, Mittag-Leffler analysis, and integral transform techniques in two variables, with potential applications in multivariate spectral problems and fractional dynamics.

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.

Finite bivariate biorthogonal I -- Konhauser polynomials

TL;DR

This work constructs a novel finite two-variable biorthogonal system grounded in Konhauser polynomials, introducing the two-variable finite polynomials and tied to univariate finite orthogonal polynomials . It develops corresponding bivariate Mittag-Leffler functions and provides integral, operational, Laplace, and Fourier transform representations that yield a Parseval based class of finite biorthogonal functions. A semigroup property is achieved by modifying the polynomials with new parameters , leading to modified Mittag-Leffler functions and a convolution type integral equation with an associated integral operator. The results extend multivariate biorthogonal theory, connect to fractional calculus via RL and Caputo type operators, and offer explicit transform based constructions that may enhance spectral methods and two variable analysis. The framework thus bridges finite biorthogonality, Mittag-Leffler analysis, and integral transform techniques in two variables, with potential applications in multivariate spectral problems and fractional dynamics.

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.
Paper Structure (12 sections, 49 theorems, 161 equations)

This paper contains 12 sections, 49 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. The finite bivariate biorthogonal I - Konhauser polynomials
  3. Integral and operational representations of the bivariate I - Konhauser Mittag-Leffler functions and the finite bivariate I - Konhauser polynomials
  4. Fractional calculus operators and Laplace transform for the bivariate I - Konhauser Mittag-Leffler functions and the finite bivariate I - Konhauser polynomials
  5. Fourier transform of I - Konhauser polynomials
  6. A class of finite biorthogonal functions derived from Fourier transform for the finite I - Konhauser polynomials
  7. A limit relation for I - Konhauser polynomials
  8. The modified finite bivariate I - Konhauser polynomials
  9. Integral and operational representation of the modified bivariate I - Konhauser Mittag-Leffler functions and the modified finite bivariate I - Konhauser polynomials
  10. Some properties for the modified I - Konhauser polynomials and the modified I - Konhauser Mittag-Leffler functions
  11. Convolution type integral equation and integral operator
  12. Conclusion

Key Result

Theorem 2

Let $d(t)$ and $f(t)$ be fundamental polynomials corresponding to polynomials $D_{s}(t)$ and$\ F_{k}(t)$, respectively, and $\rho(t)$ be the weight function over the interval $\left( \alpha_{1},\alpha_{2}\right)$. For $s,k\in\mathbb{N} _{0}$, the necessary and sufficient condition is so that and are provided Konhauser2.

Theorems & Definitions (64)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • Definition 4
  • Remark 5
  • Theorem 6
  • Corollary 7
  • Theorem 8
  • Theorem 9
  • Definition 10
  • ...and 54 more