  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.
