A New Linear Inversion Formula for a class of Hypergeometric polynomials

Given complex parameters $x$, $ν$, $α$, $β$ and $γ\notin -\mathbb{N}$, consider the infinite lower triangular matrix $\mathbf{A}(x,ν;α, β,γ)$ with elements $A_{n,k}(x,ν;α,β,γ) = \displaystyle (-1)^k\binom{n+α}{k+α} \cdot F(k-n,-(β+n)ν;-(γ+n);x)$ 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 $\mathbf{B}(x,ν;α, β,γ) = \mathbf{A}(x,ν;α, β,γ)^{-1}$ is given by \begin{align} B_{n,k}(x,ν;α, β,γ) = & \; \displaystyle (-1)^k\binom{n+α}{k+α} \; \cdot \nonumber \\ & \; \biggl [ \; \frac{γ+k}{β+k} \, F(k-n,(β+k)ν;γ+k;x) \; + \nonumber \\ & \; \; \; \frac{β-γ}{β+k} \, F(k-n,(β+k)ν;1+γ+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, $T = \mathbf{A}(x,ν;α, β,γ) \, S \Longleftrightarrow S = \mathbf{B}(x,ν;α, β,γ) \, T$, are also provided.

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