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

We study hypergeometric series with coefficients $a_n(\alpha)=\frac{(\alpha)_n(1-\alpha)_n}{(n!)^2},$ where $0< \alpha < 1$. The main idea is to introduce a shift parameter in the linear denominator and consider $$\Phi_{m,\varepsilon}(\lambda;\alpha) = \sum_{n=0}^{\infty}\varepsilon^n \frac{a_n(\alpha)}{n+m+1+\lambda}, \qquad \varepsilon\in\{1,-1\}.$$ Expanding this expression in powers of $\lambda$ 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's hypergeometric differential equation, we derive a first-order recurrence in $m$ for $\Phi_{m,\varepsilon}$. 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 $\alpha=1/R$ with $R=2,3,4$. Finally, we apply the same framework to harmonic-number sums by differentiating with respect to a lower hypergeometric parameter.