An inversion formula with hypergeometric polynomials and application to singular integral operators
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Given parameters x ∉R^- ∪{1} and ν, Re(ν) < 0, and the space H_0 of entire functions in C vanishing at 0, we consider the family of operators L = c_0 ·δ∘M with constant c_0 = ν(1-ν)x/(1-x), δ = z d/dz and integral operator M defined by Mf(z) = ∫_0^1 e^- z/xt^-ν(1-(1-x)t) f ( z/x t^-ν(1-t) ) dt/t, z ∈C, for all f ∈H_0. Inverting L or M proves equivalent to solve a singular Volterra equation of the first kind. The inversion of operator L on H_0 leads us to derive 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 relations finally enable us to derive the integral representation L^-1f(z) = 1-x/2iπ x e^z∫_(0+)^1 e^-xtz/t(1-t) f ( xz (-t)^ν(1-t)^1-ν ) dt, z ∈C, for the inverse L^-1 of operator L on H_0, where the integration contour encircles the point 0.
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