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

Abstract: We study hypergeometric series with coefficients $a_n(α)=\frac{(α)<em>n(1-α)_n}{(n!)<sup>2},$ where $0< α< 1$. The main idea is to introduce a shift parameter in the linear denominator and consider $$Φ</em>{m,\varepsilon}(λ;α) = \sum_{n=0}^{{\infty}\varepsilonⁿ} \frac{a_n(α)}{n+m+1+λ}, \qquad \varepsilon\in{1,-1}.$$ Expanding this expression in powers of $λ$ produces sums with denominator powers $(n+m+1)<sup>{-K}$. 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 $Φ_{m,\varepsilon}$. 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 $(dn+m+1)<sup>K$ 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.

• The paper introduces a novel recurrence method that rigorously reduces hypergeometric sums with binomial-harmonic components into closed finite forms.
• It employs shifted generating functions and recurrence relations to derive explicit formulas for series including 1/π and central binomial sums.
• The framework links hypergeometric functions, harmonic series, and combinatorial identities, broadening analytic tools for special function theory.

Hypergeometric Sums with Binomial-Harmonic Structure: Formal Reductions via Shifted Recurrences

Introduction and Problem Formulation

This work rigorously analyzes the class of hypergeometric-type sums whose summands involve coefficients $$a_n(\alpha) = \frac{(\alpha)_n (1-\alpha)_n}{(n!)^2}$$ for $0 < \alpha < 1$. These coefficients are structurally embedded in special cases of the Gauss hypergeometric series ${}_2F_1(\alpha, 1-\alpha; 1; x)$, and recurrently appear in combinatorial binomial identities, Ramanujan-type series, and series for $1/\pi$.

The central object of the study is the evaluation and reduction of power-denominator sums

$$A_m^{(K)}(\alpha) = \sum_{n=0}^\infty a_n(\alpha) \frac{1}{(n+m+1)^K}, \quad m \geq 0, \quad K \geq 1,$$

incorporating, as key subcases, product-binomial summations, alternating sign extensions, and sums multiplied by harmonic numbers.

Recurrence Relations and Shifted Generating Functions

A unifying methodological advancement of the paper is the introduction of the shifted parameter

$$\Phi_{m,\varepsilon}(\lambda;\alpha) = \sum_{n=0}^\infty \varepsilon^n a_n(\alpha) \frac{1}{n + m + 1 + \lambda}, \qquad \varepsilon \in \{1, -1\},$$

which interpolates all denominator powers via the Taylor expansion in $\lambda$. The main technical achievement is the derivation of a first-order recurrence in $m$ for $\Phi_{m,\varepsilon}(\lambda;\alpha)$, based on the underlying hypergeometric differential equation and its shifted variants.

The recurrence, specialized for rational $\alpha$, yields

$0 < \alpha < 1$0

where $0 < \alpha < 1$1 and $0 < \alpha < 1$2 encodes the boundary value contributions, distinct in the ordinary and alternating cases. The derivation leverages the analytic properties of $0 < \alpha < 1$3 and asymptotic endpoint analysis using the Euler reflection formula and related special function identities.

Closed Reductions for Denominator Powers

By extracting coefficients in the Taylor expansion with respect to $0 < \alpha < 1$4, the paper supplies explicit, finite reductions for $0 < \alpha < 1$5 for all positive integer $0 < \alpha < 1$6. These reductions express the hypergeometric sum with arbitrary denominator power as a sum of "initial constants" and a finite, recursively computable collection of lower-power sums, each with rational-function weights determined by combinations of binomial and shifting parameters.

Key specializations include:

• Product-binomial cases: For $0 < \alpha < 1$7 (with integer $0 < \alpha < 1$8), explicit formulas are produced for sums such as $0 < \alpha < 1$9, subsuming classical series for central and multinomial coefficients.
• Alternating/Linear denominator generalization: The recurrences and finite reductions account for denominators ${}_2F_1(\alpha, 1-\alpha; 1; x)$0 for arbitrary ${}_2F_1(\alpha, 1-\alpha; 1; x)$1 and alternating signs ${}_2F_1(\alpha, 1-\alpha; 1; x)$2, via residue class decomposition and mod-${}_2F_1(\alpha, 1-\alpha; 1; x)$3 shifts.

Connections to Harmonic Sums and Parametric Differentiation

A critical extension is the treatment of binomial-harmonic sums

${}_2F_1(\alpha, 1-\alpha; 1; x)$4

using parametric differentiation in the lower parameter of the hypergeometric function (${}_2F_1(\alpha, 1-\alpha; 1; x)$5) and log-derivative manipulations of the Pochhammer symbol. This approach systematically produces closed forms for sums involving ${}_2F_1(\alpha, 1-\alpha; 1; x)$6, and, upon further differentiation, for ${}_2F_1(\alpha, 1-\alpha; 1; x)$7 and ${}_2F_1(\alpha, 1-\alpha; 1; x)$8.

For low-degree cases, the methodology recovers and generalizes classical identities involving Catalan's constant, central binomial coefficients, and Ramanujan-like series for ${}_2F_1(\alpha, 1-\alpha; 1; x)$9. Explicit identities (e.g., for $1/\pi$0) are provided, including the exact answer to an open question on central-binomial product series posed in prior literature.

Evaluation of Initial Constants and Special Cases

Via the reduction to initial constants and closed boundary values, the paper achieves:

• Representation of initial value constants in terms of gamma, digamma, and trigamma functions for rational $1/\pi$1, using Gauss's summation and reflection formulas.
• In the case of $1/\pi$2, precise evaluation of finite sums and cancellation identities involving weighted binomial coefficients and central values of associated hypergeometric functions.
• Explicit transformation formulas for half-argument and residue class cases, facilitating modular and analytic connection with other special function expansions and modular forms.

Practical and Theoretical Implications

The presented framework offers a canonical route to algorithmically generate and reduce a wide spectrum of binomial-hypergeometric-harmonic identities, with direct application to the explicit computation of series for $1/\pi$3, generating function manipulation, and evaluation of combinatorial sums in special function theory. The method provides a new perspective on the algebraic structure underlying recurrences for binomial-hypergeometric families and links the classical theory of special functions with the contemporary combinatorial and analytic toolkit.

The general approach opens a path to investigate analogous recurrences and finite reductions for more involved hypergeometric families, higher-level hypergeometric parameters, and transforms, and also suggests a systematic strategy for machine computation of binomial-harmonic identities across various parameter regimes.

Conclusion

This work formalizes a comprehensive recurrence framework governing a broad class of hypergeometric sums with product-binomial and harmonic structure, delivering closed, finite reductions for all denominator powers and wide-ranging specializations. The results unify and extend a spectrum of prior results, providing both explicit formulae for practitioners and a structural toolkit for further theoretical exploration in analytic combinatorics, $1/\pi$4-series, and special function theory.

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Reference: "On a Class of Hypergeometric Sums via Recurrences and Product Binomial-Harmonic Identities" (2607.10135)