A New Linear Inversion Formula for a class of Hypergeometric polynomials
Abstract
Given complex parameters x, , α, β and γ -N, consider the infinite lower triangular matrix A(x,;α, β,γ) with elements An,k(x,;α,β,γ) = (-1)kn+αk+α · F(k-n,-(β+n);-(γ+n);x) for 1 ≤slant k ≤slant n, depending on the Hypergeometric polynomials F(-n,·;·;x), n ∈ N*. After stating a general criterion for the inversion of infinite matrices in terms of associated generating functions, we prove that the inverse matrix B(x,;α, β,γ) = A(x,;α, β,γ)-1 is given by align Bn,k(x,;α, β,γ) = & \; (-1)kn+αk+α \; · \\ & \; [ \; γ+kβ+k \, F(k-n,(β+k);γ+k;x) \; + \\ & \; \; \; β-γβ+k \, F(k-n,(β+k);1+γ+k;x) \; ] align for 1 ≤slant k ≤slant n, thus providing a new class of linear inversion formulas. Functional relations for the generating functions of related sequences S and T, that is, T = A(x,;α, β,γ) \, S S = B(x,;α, β,γ) \, T, are also provided.