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arXiv · 2607.10135

Recurrences for Hypergeometric Moments and Binomial-Harmonic Sums

Narendra Bhandari
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Abstract

Let \[ F(a,b;c;x)={}_2F_1(a,b;c;x), \qquad \Phi_{m,\varepsilon}(\lambda;a,b,c) = \int_0^1 x^{m+\lambda}F(a,b;c;\varepsilon x)\,dx, \quad \varepsilon=\pm1. \] 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. Finally, comparison with a terminating ${}_3F_2(1)$ formula also gives finite hypergeometric and binomial--harmonic identities.

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Narendra Bhandari. 2026-07-11. Recurrences for Hypergeometric Moments and Binomial-Harmonic Sums. https://arxiv.org/abs/2607.10135

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