Recurrences for Hypergeometric Moments and Binomial-Harmonic Sums
arXiv:2607.10135
The paper investigates hypergeometric series with coefficients involving Pochhammer symbols, introducing a shift parameter to study sums with linear denominators and deriving recurrences that yield closed‑form identities for ordinary, alternating, and product‑binomial/harmonic‑number sums.
Abstract
Let \[ F(a,b;c;x)={}_2F_1(a,b;c;x), \qquad Φ_{m,\varepsilon}(λ;a,b,c) = \int_0^1 x^{m+λ}F(a,b;c;\varepsilon x)\,dx, \quad \varepsilon=\pm1. \] We study these moments by deriving and solving a first-order recurrence in . This recurrence leads to formulas for higher powers of the denominator and for denominators of the form , with applications to product-binomial series and moments of complete elliptic integrals. Differentiation with respect to gives corresponding recurrences for harmonic-number weights, whose initial values are described using Bell polynomials, logarithms, zeta values, and Dirichlet -values. Finally, comparison with a terminating formula also gives finite hypergeometric and binomial--harmonic identities.
19 pages