On a Class of Hypergeometric Sums via Recurrences and Product Binomial-Harmonic Identities

Abstract

We study hypergeometric series with coefficients $a_n(α)=\frac{(α)_n(1-α)_n}{(n!)^2},$ where $0< α< 1$. The main idea is to introduce a shift parameter in the linear denominator and consider $$ Φ_{m,\varepsilon}(λ;α) = \sum_{n=0}^{\infty}\varepsilon^n \frac{a_n(α)}{n+m+1+λ}, \qquad \varepsilon\in\{1,-1\}. $$ Expanding this expression in powers of $λ$ produces sums with denominator powers $(n+m+1)^{-K}$. We first discuss analytic interpolation in the denominator exponent and explain why positive integer exponents lead to terminating recurrences. Using Euler&#39;s hypergeometric differential equation, we derive a first-order recurrence in $m$ for $Φ_{m,\varepsilon}$. Solving this recurrence gives finite reductions for ordinary and alternating sums, and coefficient extraction yields formulas for all positive denominator powers. The same framework also treats linear denominators $(dn+m+1)^K$ of arbitrary positive parameter by separating $m$ into residue classes modulo $d$. We then specialize the results to product-binomial cases, especially $α=1/R$ with $R=2,3,4$. Finally, we apply the same framework to harmonic-number sums by differentiating with respect to a lower hypergeometric parameter.