2019/09/19 by Ridha Nasri, Nasri, R., Alain Simonian +3
Mathematics · Physics and Astronomy · #33C05 #45E99 #47G10 #Algebraic and Geometric Analysis #Classical Analysis and ODEs (math.CA) #Electromagnetic Scattering and Analysis #FOS: Computer and information sciences #FOS: Mathematics #Mathematical functions and polynomials #Performance (cs.PF)
paper · pdf · doi:10.48550/arxiv.1909.09694
openalex publication_date 2019/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given parameters x ∉ ℝ- ∪ \1\ and ν, Re(ν) < 0, and the space \mathscrH0 of entire functions in ℂ vanishing at 0, we consider the family of operators \mathfrakL = c0 ⋅ δ∘ \mathfrakM with constant c0 = ν(1-ν)x/(1-x), δ= z d/dz and integral operator \mathfrakM defined by \mathfrakMf(z) = ∫01 e^- (z)/(x)t-ν(1-(1-x)t) f ( (z)/(x) t-ν(1-t) ) (dt)/(t), z ∈ ℂ, for all f ∈ \mathscrH0. Inverting \mathfrakL or \mathfrakM proves equivalent to solve a singular Volterra equation of the first kind. The inversion of operator \mathfrakL on \mathscrH0 leads us to derive 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 relations finally enable us to derive the integral representation \mathfrakL-1f(z) = (1-x)/(2iπx) ez ∫(0+)1 \frace-xtzt(1-t) f ( xz (-t)ν(1-t)1-ν ) dt, z ∈ ℂ, for the inverse \mathfrakL-1 of operator \mathfrakL on \mathscrH0, where the integration contour encircles the point 0.