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Touchard's identity and a detailed determination of the radius of convergence of the Catalan series

Jean-Christophe Pain

Abstract

While the value of the radius of convergence of the generating series of the Catalan numbers is well-known, obtaining it solely from recurrence relations is less immediate. It is sometimes considered that no known proof establishes that the radius $R$ equals 1/4 without relying on the explicit closed formula for the Catalan numbers. In particular, it has been shown that one can obtain, at the cost of substantial technical effort and without resorting to the main Segner recursion relation or to the explicit formula for the Catalan number, the lower bound $R\geq 1/6$. In this work, we prove that Touchard's recurrence alone yields the optimal exponential upper bound $\limsup_{n\to\infty} C_n^{1/n} \le 4$, which implies $R \ge 1/4$. Combined with the classical lower estimate $\limsup_{n\to\infty} C_n^{1/n} \ge 4$, obtained from central binomial coefficients, this gives $R = 1/4$.

Touchard's identity and a detailed determination of the radius of convergence of the Catalan series

Abstract

While the value of the radius of convergence of the generating series of the Catalan numbers is well-known, obtaining it solely from recurrence relations is less immediate. It is sometimes considered that no known proof establishes that the radius equals 1/4 without relying on the explicit closed formula for the Catalan numbers. In particular, it has been shown that one can obtain, at the cost of substantial technical effort and without resorting to the main Segner recursion relation or to the explicit formula for the Catalan number, the lower bound . In this work, we prove that Touchard's recurrence alone yields the optimal exponential upper bound , which implies . Combined with the classical lower estimate , obtained from central binomial coefficients, this gives .
Paper Structure (6 sections, 4 theorems, 52 equations)

This paper contains 6 sections, 4 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. The Touchard identity
  3. Upper exponential bound using Touchard's identity
  4. Lower exponential bound
  5. Conclusion
  6. On the Segner recursion relation and the radius of convergence

Key Result

Theorem 2.1

For every $n\ge0$, the Catalan numbers satisfy the recursion relation

Theorems & Definitions (8)

  • Theorem 2.1: Touchard
  • proof
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2: Lower bound via Touchard
  • proof