  For any complex parameters $x$ and $\nu$, we provide a new class of linear
inversion formulas $T = A(x,\nu) \cdot S \Leftrightarrow S = B(x,\nu) \cdot T$
between sequences $S = (S_n)_{n \in \mathbb{N}^*}$ and $T = (T_n)_{n \in
\mathbb{N}^*}$, where the infinite lower-triangular matrix $A(x,\nu)$ and its
inverse $B(x,\nu)$ involve Hypergeometric polynomials $F(\cdot)$, namely $$
  \left\{
  \begin{array}{ll}
  A_{n,k}(x,\nu) = \displaystyle (-1)^k\binom{n}{k}F(k-n,-n\nu;-n;x),
  \\
  B_{n,k}(x,\nu) = \displaystyle (-1)^k\binom{n}{k}F(k-n,k\nu;k;x)
  \end{array} \right. $$ for $1 \leqslant k \leqslant n$. Functional relations
between the ordinary (resp. exponential) generating functions of the related
sequences $S$ and $T$ are also given.
  These new inversion formulas have been initially motivated by the resolution
of an integral equation recently appeared in the field of Queuing Theory; we
apply them to the full resolution of this integral equation. Finally, matrices
involving generalized Laguerre polynomials polynomials are discussed as
specific cases of our general inversion scheme.
