On the interplay between hypergeometric series, Fourier-Legendre expansions and Euler sums

Original · EN

In this work we continue the investigation about the interplay between hypergeometric functions and Fourier-Legendre (FL) series expansions. In the section "Hypergeometric series related to π,π² and the lemniscate constant", through the FL-expansion of [x(1-x)]μ (with μ+1∈1/4N) we prove that all the hypergeometric series ∑ₙ≥ ₀(-1)ⁿ(4n+1)/p(n)[1/4ⁿ2nn]³, ∑ₙ≥ ₀(4n+1)/p(n)[1/4ⁿ2nn]⁴, ∑ₙ≥ ₀(4n+1)/p(n)²[1/4ⁿ2nn]⁴, ∑ₙ≥ ₀1/p(n)[1/4ⁿ2nn]³, ∑ₙ≥ ₀1/p(n)[1/4ⁿ2nn]² return rational multiples of 1π,1/π² or the lemniscate constant, as soon as p(x) is a polynomial fulfilling suitable symmetry constraints. Additionally, by computing the FL-expansions of x√x and related functions, we show that in many cases the hypergeometric ₚ₊₁ Fₚ(, z) function evaluated at z=± 1 can be converted into a combination of Euler sums. In particular we perform an explicit evaluation of ∑ₙ≥ ₀1/(2n+1)²[1/4ⁿ2nn]², ∑ₙ≥ ₀1/(2n+1)³[1/4ⁿ2nn]². In the section "Twisted hypergeometric series" we show that the conversion of some ₚ₊₁ Fₚ(,± 1) values into combinations of Euler sums, driven by FL-expansions, applies equally well to some twisted hypergeometric series, i.e. series of the form ∑ₙ≥ ₀ aₙ bₙ where aₙ is a Stirling number of the first kind and ∑ₙ≥ ₀bₙ zⁿ = ₚ₊₁ Fₚ(;z).