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Reciprocal binomial sums via Beta integrals

Jean-Christophe Pain

Abstract

We develop a systematic and fully explicit approach to the evaluation of binomial sums involving reciprocals of binomial coefficients based on Beta integral techniques. Starting from a simple integral representation, we provide a derivation of classical identities, including Frisch's formula, with all intermediate transformations rigorously justified. This framework naturally extends to parametric sums, yielding integral representations that lead to closed forms in terms of hypergeometric functions. In particular, we establish connections with terminating ${}_2F_1$ and generalized ${}_3F_2$ series, thereby linking discrete combinatorial sums with the analytic theory of special functions. We further derive explicit finite expansions suitable for symbolic and numerical computation, as well as higher-order extensions involving Pochhammer symbols. In addition, we present new families of identities, including shifted reciprocal sums and weighted sums involving powers of the summation index, which admit unified hypergeometric representations. Overall, the Beta integral method provides a versatile and unifying framework bridging combinatorial identities, integral representations, and hypergeometric analysis, and opens the way to further generalizations in combinatorics and special function theory.

Reciprocal binomial sums via Beta integrals

Abstract

We develop a systematic and fully explicit approach to the evaluation of binomial sums involving reciprocals of binomial coefficients based on Beta integral techniques. Starting from a simple integral representation, we provide a derivation of classical identities, including Frisch's formula, with all intermediate transformations rigorously justified. This framework naturally extends to parametric sums, yielding integral representations that lead to closed forms in terms of hypergeometric functions. In particular, we establish connections with terminating and generalized series, thereby linking discrete combinatorial sums with the analytic theory of special functions. We further derive explicit finite expansions suitable for symbolic and numerical computation, as well as higher-order extensions involving Pochhammer symbols. In addition, we present new families of identities, including shifted reciprocal sums and weighted sums involving powers of the summation index, which admit unified hypergeometric representations. Overall, the Beta integral method provides a versatile and unifying framework bridging combinatorial identities, integral representations, and hypergeometric analysis, and opens the way to further generalizations in combinatorics and special function theory.

Paper Structure

This paper contains 20 sections, 13 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Preliminaries: Beta integrals
  3. Inverse binomial representation
  4. Proof of Frisch's identity
  5. Parametric extension
  6. Finite expansions and weighted reciprocal sums
  7. Shifted reciprocal sums
  8. First-order weighted sums
  9. Quadratic weights
  10. Examples
  11. Example 1.
  12. Example 2.
  13. Higher-order structure
  14. Integral lifts of reciprocal binomial sums
  15. Basic lifting principle
  16. ...and 5 more sections

Key Result

Proposition 3.1

For integers $b\ge c>0$ and any non-negative integer $k$, $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (24)

  • Proposition 3.1
  • proof
  • Theorem 4.1
  • proof
  • Theorem 5.1
  • proof
  • Proposition 6.1: Hypergeometric and finite expansion
  • proof
  • Proposition 6.2: Shift invariance
  • proof
  • ...and 14 more