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Summing series using recurrence relations

Erik Talvila

TL;DR

The paper develops a unified framework for summing power series whose terms satisfy linear recurrences with polynomial coefficients by converting recurrences into differential or algebraic equations for the generating function $S(x)=\sum a_n x^n$, enabling closed forms. When recurrence coefficients are linear in $n$, the method yields a first-order linear differential equation for $S(x)$; higher-degree coefficients lead to higher-order equations, with solutions often expressible via integrals or standard special functions. The approach is illustrated via a cosine example, and then applied to two American Mathematical Monthly problems, the binomial theorem, and a generating function for Bessel functions, highlighting connections to erf, Jacobi polynomials, and hypergeometric functions. Collectively, the results demonstrate how discrete recurrences translate into continuous differential equations to achieve exact sums and reveal deep links between summation methods and classical special-function identities.

Abstract

Power series in which the summand satisfies a linear recurrence relation with polynomial coefficients are shown to be the solution of a linear differential or algebraic equation. Solving the associated differential or algebraic equation yields a closed form for the series. This method is used to sum several series and to solve two {\it American Mathematical Monthly} problems.

Summing series using recurrence relations

TL;DR

The paper develops a unified framework for summing power series whose terms satisfy linear recurrences with polynomial coefficients by converting recurrences into differential or algebraic equations for the generating function , enabling closed forms. When recurrence coefficients are linear in , the method yields a first-order linear differential equation for ; higher-degree coefficients lead to higher-order equations, with solutions often expressible via integrals or standard special functions. The approach is illustrated via a cosine example, and then applied to two American Mathematical Monthly problems, the binomial theorem, and a generating function for Bessel functions, highlighting connections to erf, Jacobi polynomials, and hypergeometric functions. Collectively, the results demonstrate how discrete recurrences translate into continuous differential equations to achieve exact sums and reveal deep links between summation methods and classical special-function identities.

Abstract

Power series in which the summand satisfies a linear recurrence relation with polynomial coefficients are shown to be the solution of a linear differential or algebraic equation. Solving the associated differential or algebraic equation yields a closed form for the series. This method is used to sum several series and to solve two {\it American Mathematical Monthly} problems.
Paper Structure (7 sections, 2 theorems, 45 equations)

This paper contains 7 sections, 2 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Cosine example
  3. Recurrence relations with polynomial coefficients
  4. American Mathematical Monthly Problem 10977
  5. Binomial theorem
  6. Bessel functions
  7. American Mathematical Monthly Problem 11274

Key Result

Theorem 5.1

Let $a,z\in{\mathbb C}$ with $\lvert z\rvert<1$. Then $(1-z)^{-a}=\frac{1}{\Gamma(a)}\sum_{n=0}^\infty \frac{\Gamma(n+a)z^n}{n!}$.

Theorems & Definitions (5)

  • Example 2.1
  • Theorem 5.1
  • proof
  • Theorem 6.1
  • proof