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New biorthogonal sequences generated by index integrals of the weight functions

Semyon Yakubovich

TL;DR

This work constructs and analyzes new biorthogonal sequences associated with nonclassical weight functions arising from index integrals of the modified Bessel and gamma functions, using composition orthogonality under the Kontorovich-Lebedev transform. It develops operator-based representations with the core operator $A= x^2- x d/dx x d/dx$ and derives moments $\\mu_n(x)$, leading to explicit recurrence structures and integral representations that connect to Lebedev-type kernels and Wilson integrals via Meijer $G$-functions. The paper then introduces three polynomial families $P_n$, $Q_n$, and $R_n$ orthogonal to different nonclassical weights, including a quasi-orthogonality result for $[\\mathrm{Im}K_{1+i\\tau}(x)]^2$ and interrelations between the families. Finally, it generalizes Wilson polynomials to a broader class $W_n(\\tau^2,[a]_{p+1})$ with a gamma-ratio weight, developing composition orthogonality, inverse Kontorovich-Lebedev transforms, hypergeometric coefficient formulas, and a differential-recurrence framework. Overall, the paper broadens the landscape of nonclassical orthogonal polynomials tied to transform methods, with potential implications for spectral theory and integral-transform techniques.

Abstract

We exhibit new biorthogonal sequences generated by index integrals of the squares of the modified Bessel functions and gamma functions. The composition orthogonality, involving differential operators is employed. Generalized Wilson polynomials are introduced. Some properties are investigated.

New biorthogonal sequences generated by index integrals of the weight functions

TL;DR

This work constructs and analyzes new biorthogonal sequences associated with nonclassical weight functions arising from index integrals of the modified Bessel and gamma functions, using composition orthogonality under the Kontorovich-Lebedev transform. It develops operator-based representations with the core operator and derives moments , leading to explicit recurrence structures and integral representations that connect to Lebedev-type kernels and Wilson integrals via Meijer -functions. The paper then introduces three polynomial families , , and orthogonal to different nonclassical weights, including a quasi-orthogonality result for and interrelations between the families. Finally, it generalizes Wilson polynomials to a broader class with a gamma-ratio weight, developing composition orthogonality, inverse Kontorovich-Lebedev transforms, hypergeometric coefficient formulas, and a differential-recurrence framework. Overall, the paper broadens the landscape of nonclassical orthogonal polynomials tied to transform methods, with potential implications for spectral theory and integral-transform techniques.

Abstract

We exhibit new biorthogonal sequences generated by index integrals of the squares of the modified Bessel functions and gamma functions. The composition orthogonality, involving differential operators is employed. Generalized Wilson polynomials are introduced. Some properties are investigated.
Paper Structure (3 sections, 341 equations)

This paper contains 3 sections, 341 equations.

Table of Contents

  1. Introduction and preliminary results
  2. Orthogonal polynomials for the weights $K^2_{i\tau}(x), \newline \left[ {\rm Re} \left[ K_{1+i\tau}(x) \right]\right]^2, \left[ {\rm Im} \left[ K_{1+i\tau}(x) \right]\right]^2$
  3. Generalized Wilson polynomials

Theorems & Definitions (2)

  • proof
  • proof