A New Linear Inversion Formula for a class of Hypergeometric polynomials

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 $$A_{n,k}(x,\nu ;\alpha ,\beta ,\gamma )=(-1{)}^k(\genfrac{}{}{0ex}{}{n+\alpha }{k+\alpha })\cdot F(k-n,-(\beta +n)\nu ;-(\gamma +n);x)$$ for $1⩽k⩽n$, depending on the Hypergeometric polynomials $F(-n,\cdot ;\cdot ;x)$, $n\in {\mathbb{N}}^{\ast }$. 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,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 $1⩽k⩽n$, thus providing a new class of linear inversion formulas. Functional relations for the generating functions of related sequences $S$ and $T$, that is, $T=\mathbf{A}(x,\nu ;\alpha ,\beta ,\gamma )S⟺S=\mathbf{B}(x,\nu ;\alpha ,\beta ,\gamma )T$, are also provided.

16 pages. arXiv admin note: substantial text overlap with arXiv:1904.08283, arXiv:1909.09694