2019/04/16 by Nasri, Ridha, Simonian, Alain, Guillemin, Fabrice
#15B99 #33C05 #45B99 #Classical Analysis and ODEs (math.CA) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Performance (cs.PF)
paper · doi:10.48550/arxiv.1904.08283
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 = (Sn)n ∈ ℕ^* and T = (Tn)n ∈ ℕ^*, where the infinite lower-triangular matrix A(x,ν) and its inverse B(x,ν) involve Hypergeometric polynomials F(⋅), namely \ An,k(x,ν) = (-1)k\binomnkF(k-n,-nν;-n;x),
Bn,k(x,ν) = (-1)k\binomnkF(k-n,kν;k;x) . 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.