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

For any complex parameters x and ν, we provide a new class of linear inversion formulas T = A(x,ν) · S S = B(x,ν) · T between sequences S = (S_n)_n ∈N^* and T = (T_n)_n ∈N^*, where the infinite lower-triangular matrix A(x,ν) and its inverse B(x,ν) involve Hypergeometric polynomials F(·), namely { < a r r a y > . 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.

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