A method of composition orthogonality and new sequences of orthogonal polynomials and functions for non-classical weights

TL;DR

This paper introduces a novel composition orthogonality method to generate new orthogonal polynomial sequences, interpret classical polynomials within this framework, and analyze their properties with respect to complex weight functions involving Bessel functions.

Contribution

It develops a new composition orthogonality approach, enabling the creation and analysis of new orthogonal polynomial sequences for non-classical weights.

Findings

New sequences of orthogonal polynomials with specific weight functions are constructed.

Classical orthogonal polynomials are reinterpreted through composition orthogonality.

Explicit formulas, recurrence relations, and differential properties are derived for these polynomials.

Abstract

A new method of composition orthogonality is introduced. It is applied to generate new sequences of orthogonal polynomials and functions. In particular, classical orthogonal polynomials are interpreted in the sense of composition orthogonality. Finally, new sequences of orthogonal polynomials with respect to the weight function $x^\alpha \rho^2_\nu(x), \rho_{\nu}(x)= 2 x^{\nu/2} K_\nu(2\sqrt x),\ x >0, \nu \ge 0, \alpha > -1$, where $K_\nu(z)$ is the modified Bessel function or Macdonald function, are investigated. Differential properties, recurrence relations, explicit representations, generating functions and Rodrigues-type formulae are obtained. The corresponding multiple orthogonal polynomials are exhibited.