Inversion formula with hypergeometric polynomials and its application to an integral equation

Abstract

For any complex parameters $x$ and $\nu$, we provide a new class of linear inversion formulas $T=A(x,\nu )\cdot S\iff S=B(x,\nu )\cdot T$ between sequences $S=(S_n{)}_{n\in {\mathbb{N}}^{\ast }}$ and $T=(T_n{)}_{n\in {\mathbb{N}}^{\ast }}$, where the infinite lower-triangular matrix $A(x,\nu )$ and its inverse $B(x,\nu )$ involve Hypergeometric polynomials $F(\cdot )$, namely $$\{\begin{array}{l}{A}_{n,k}(x,\nu )=(-1{)}^k(\genfrac{}{}{0ex}{}{n}{k})F(k-n,-n\nu ;-n;x),\\ B_{n,k}(x,\nu )=(-1{)}^k(\genfrac{}{}{0ex}{}{n}{k})F(k-n,k\nu ;k;x)\end{array}$$ for $1⩽k⩽n$. 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.

22 pages, no figure