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On Touchard's Identity: Generalizations and Related Results

Kunle Adegoke

TL;DR

The paper addresses generalizations of Touchard's identity for Catalan numbers $C_k$. It develops two complementary approaches: a Beta-function–driven integral method and a Stirling-number–based method, to produce parameterized identities that extend the classical case and connect to broader combinatorial structures. It then introduces a binomial-transform framework that yields a network of related equalities and corollaries, including self-inverse transforms and harmonic-number variants. The results generalize existing identities, provide new closed forms for Catalan-weighted sums, and deepen connections among Catalan numbers, binomial transforms, and Stirling numbers of the second kind.

Abstract

Starting with a known polynomial identity, we derive two generalizations of Touchard's identity concerning Catalan numbers; one obtained using the Beta function and the other via a connection with Stirling numbers of the second kind. We subsequently establish several new combinatorial identities.

On Touchard's Identity: Generalizations and Related Results

TL;DR

The paper addresses generalizations of Touchard's identity for Catalan numbers . It develops two complementary approaches: a Beta-function–driven integral method and a Stirling-number–based method, to produce parameterized identities that extend the classical case and connect to broader combinatorial structures. It then introduces a binomial-transform framework that yields a network of related equalities and corollaries, including self-inverse transforms and harmonic-number variants. The results generalize existing identities, provide new closed forms for Catalan-weighted sums, and deepen connections among Catalan numbers, binomial transforms, and Stirling numbers of the second kind.

Abstract

Starting with a known polynomial identity, we derive two generalizations of Touchard's identity concerning Catalan numbers; one obtained using the Beta function and the other via a connection with Stirling numbers of the second kind. We subsequently establish several new combinatorial identities.
Paper Structure (4 sections, 16 theorems, 68 equations)

This paper contains 4 sections, 16 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. A generalization of Touchard's identity and related results
  3. A generalization involving Stirling numbers of the second kind
  4. Related results via binomial transform

Key Result

Lemma 1

If $r$ and $s$ are complex numbers, then and

Theorems & Definitions (32)

  • Lemma 1
  • Theorem 1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • ...and 22 more