A New Linear Inversion Formula for a class of Hypergeometric polynomials

TL;DR

This paper introduces a new linear inversion formula for a class of hypergeometric polynomials, providing explicit inverse matrices and functional relations for related generating functions, expanding the toolkit for hypergeometric function analysis.

Contribution

The paper derives an explicit inverse for a matrix constructed from hypergeometric polynomials, establishing a new class of linear inversion formulas and related functional relations.

Findings

Explicit inverse matrix formula for hypergeometric polynomial-based matrices

New class of linear inversion formulas established

Functional relations for generating functions derived

Abstract

Given complex parameters $x$, $\nu$, $\alpha$, $\beta$ and $\gamma \notin -\mathbb{N}$, consider the infinite lower triangular matrix $\mathbf{A}(x,\nu;\alpha, \beta,\gamma)$ with elements $$ A_{n,k}(x,\nu;\alpha,\beta,\gamma) = \displaystyle (-1)^k\binom{n+\alpha}{k+\alpha} \cdot F(k-n,-(\beta+n)\nu;-(\gamma+n);x) $$ for $1 \leqslant k \leqslant n$, depending on the Hypergeometric polynomials $F(-n,\cdot;\cdot;x)$, $n \in \mathbb{N}^*$. After stating a general criterion for the inversion of infinite matrices in terms of associated generating functions, we prove that the inverse matrix $\mathbf{B}(x,\nu;\alpha, \beta,\gamma) = \mathbf{A}(x,\nu;\alpha, \beta,\gamma)^{-1}$ is given by \begin{align} B_{n,k}(x,\nu;\alpha, \beta,\gamma) = & \; \displaystyle (-1)^k\binom{n+\alpha}{k+\alpha} \; \cdot \nonumber \\ & \; \biggl [ \; \frac{\gamma+k}{\beta+k} \, F(k-n,(\beta+k)\nu;\gamma+k;x) \; + \nonumber \\ & \; \; \; \frac{\beta-\gamma}{\beta+k} \, F(k-n,(\beta+k)\nu;1+\gamma+k;x) \; \biggr ] \nonumber \end{align} for $1 \leqslant k \leqslant n$, thus providing a new class of linear inversion formulas. Functional relations for the generating functions of related sequences $S$ and $T$, that is, $T = \mathbf{A}(x,\nu;\alpha, \beta,\gamma) \, S \Longleftrightarrow S = \mathbf{B}(x,\nu;\alpha, \beta,\gamma) \, T$, are also provided.