Recurrences for Hypergeometric Moments and Binomial-Harmonic Sums

2026/07/26 by Narendra Bhandari
Mathematics · #math.NT #math.CA

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Abstract

Let F(a,b;c;x)=₂F₁(a,b;c;x), Φ_m,ε(λ;a,b,c) = ∫₀¹ x^m+λF(a,b;c;ε x) dx, ε=±1. We study these moments by deriving and solving a first-order recurrence in m. This recurrence leads to formulas for higher powers of the denominator and for denominators of the form (dn+m+1)^K, with applications to product-binomial series and moments of complete elliptic integrals. Differentiation with respect to c gives corresponding recurrences for harmonic-number weights, whose initial values are described using Bell polynomials, logarithms, zeta values, and Dirichlet L-values.

Citations

• On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier-Legendre expansions
• Derivative operator and harmonic number identities
• Hypergeometric Series and Harmonic Number Identities
• Differentiably Finite Power Series

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