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A Family of Generating Functions for Reciprocal Binomial Coefficients and Its Applications

Dmitry Kruchinin, Vladimir Kruchinin

TL;DR

This work constructs an explicit two-variable generating function $A(x,y)=\sum_{n\ge0}\sum_{m\ge0}{\binom{n}{m}}^{-1}x^n y^m$ for reciprocal binomial coefficients, providing a rational–logarithmic closed form and establishing its convergence domain. Building on Riordan arrays and bivariate composition, it derives a comprehensive family of related functions $I(x,y),J(x,y),K(x,y)$ and parity-modified variants that encode weighted sums, diagonals, and row/column-parity subsets of the reciprocal triangle, enabling systematic generation of sums and identities. The paper then applies this framework to obtain generating functions for a wide array of sums, proves new connections to harmonic numbers and Fibonacci numbers, and derives infinite-sum representations involving dilogarithms, with rigorous discussion of convergence and analytic continuation. The results offer a unified toolkit for analyzing sums of reciprocal binomial coefficients, illuminate links to classical sequences, and pave the way for extensions to generalized binomial coefficients and related combinatorial problems in probability and physics.

Abstract

A generating function for reciprocal binomial coefficients is written down, integral representations of this function are obtained, generating functions for sums of reciprocal binomial coefficients are derived, new identities are obtained, including identities connecting reciprocal binomial coefficients with harmonic numbers and Fibonacci numbers. The application of the found functions for evaluating infinite numerical sequences involving reciprocal binomial coefficients is demonstrated.

A Family of Generating Functions for Reciprocal Binomial Coefficients and Its Applications

TL;DR

This work constructs an explicit two-variable generating function for reciprocal binomial coefficients, providing a rational–logarithmic closed form and establishing its convergence domain. Building on Riordan arrays and bivariate composition, it derives a comprehensive family of related functions and parity-modified variants that encode weighted sums, diagonals, and row/column-parity subsets of the reciprocal triangle, enabling systematic generation of sums and identities. The paper then applies this framework to obtain generating functions for a wide array of sums, proves new connections to harmonic numbers and Fibonacci numbers, and derives infinite-sum representations involving dilogarithms, with rigorous discussion of convergence and analytic continuation. The results offer a unified toolkit for analyzing sums of reciprocal binomial coefficients, illuminate links to classical sequences, and pave the way for extensions to generalized binomial coefficients and related combinatorial problems in probability and physics.

Abstract

A generating function for reciprocal binomial coefficients is written down, integral representations of this function are obtained, generating functions for sums of reciprocal binomial coefficients are derived, new identities are obtained, including identities connecting reciprocal binomial coefficients with harmonic numbers and Fibonacci numbers. The application of the found functions for evaluating infinite numerical sequences involving reciprocal binomial coefficients is demonstrated.
Paper Structure (50 sections, 11 theorems, 143 equations, 1 table)

This paper contains 50 sections, 11 theorems, 143 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Standard Generating Functions (Reference)
  5. Riordan Arrays and the Main Theorem
  6. Compositions of Bivariate Generating Functions
  7. Diagonal Sum
  8. Weighted Row Sum
  9. A Family of Generating Functions for Reciprocal Binomial Coefficients
  10. Generating Function for Reciprocal Binomial Coefficients $\binom{n}{m}^{-1}$
  11. Generating Functions for Reciprocal Binomial Coefficients $\binom{2n}{m}^{-1}$, $\binom{n}{2m}^{-1}$, $\binom{2n-1}{m}^{-1}$, $\binom{n}{2m-1}^{-1}$
  12. Extracting Even and Odd Rows
  13. Extracting Even and Odd Columns
  14. Explicit Expressions for the Obtained Functions
  15. Combinatorial Interpretation
  16. ...and 35 more sections

Key Result

Theorem 2.2

Let $Q(x)=\sum_{k\ge 0} q_k x^k$ be an arbitrary formal power series. Then the coefficients of the generating function are expressed via the Riordan array $R(n,k)$:

Theorems & Definitions (25)

  • Definition 2.1
  • Theorem 2.2: Composition Theorem for Riordan Arrays Shapiro
  • Theorem 3.1
  • proof
  • Theorem 6.1
  • proof
  • Theorem 6.2
  • proof
  • Theorem 6.3
  • proof
  • ...and 15 more