2020/07/05 by Nasri, Ridha, Simonian, Alain, Guillemin, Fabrice
#05A10 #05A15 #15A09 #33C05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2007.01865
Given complex parameters x, ν, α, β and γ∉ -ℕ, consider the infinite lower triangular matrix A(x,ν;α, β,γ) with elements An,k(x,ν;α,β,γ) = (-1)k\binomn+αk+α ⋅ F(k-n,-(β+n)ν;-(γ+n);x) for 1 \leqslant k \leqslant 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,ν;α, β,γ) = A(x,ν;α, β,γ)-1 is given by Bn,k(x,ν;α, β,γ) = amp; (-1)k\binomn+αk+α ⋅
amp; \biggl [ (γ+k)/(β+k) F(k-n,(β+k)ν;γ+k;x) +
amp; (β-γ)/(β+k) F(k-n,(β+k)ν;1+γ+k;x) \biggr ] for 1 \leqslant k \leqslant 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,ν;α, β,γ) S \Longleftrightarrow S = B(x,ν;α, β,γ) T, are also provided.