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On two-term hypergeometric recursions with free lower parameters

John M. Campbell, Paul Levrie

Abstract

Let $F(n,k)$ be a hypergeometric function that may be expressed so that $n$ appears within initial arguments of inverted Pochhammer symbols, as in factors of the form $\frac{1}{(n)_{k}}$. Only in exceptional cases is $F(n, k)$ such that Zeilberger's algorithm produces a two-term recursion for $\sum_{k = 0}^{\infty} F(n, k)$ obtained via the telescoping of the right-hand side of a difference equation of the form $p_{1}(n) F(n + r, k) + p_{2}(n) F(n, k) = G(n, k+1) - G(n, k)$ for fixed $r \in \mathbb{N}$ and polynomials $p_{1}$ and $p_{2}$. Building on the work of Wilf, we apply a series acceleration technique based on two-term hypergeometric recursions derived via Zeilberger's algorithm. Fast converging series previously given by Ramanujan, Guillera, Chu and Zhang, Chu, Lupaş, and Amdeberhan are special cases of hypergeometric transforms introduced in our article.

On two-term hypergeometric recursions with free lower parameters

Abstract

Let be a hypergeometric function that may be expressed so that appears within initial arguments of inverted Pochhammer symbols, as in factors of the form . Only in exceptional cases is such that Zeilberger's algorithm produces a two-term recursion for obtained via the telescoping of the right-hand side of a difference equation of the form for fixed and polynomials and . Building on the work of Wilf, we apply a series acceleration technique based on two-term hypergeometric recursions derived via Zeilberger's algorithm. Fast converging series previously given by Ramanujan, Guillera, Chu and Zhang, Chu, Lupaş, and Amdeberhan are special cases of hypergeometric transforms introduced in our article.
Paper Structure (14 sections, 5 theorems, 75 equations, 1 table)

This paper contains 14 sections, 5 theorems, 75 equations, 1 table.

Table of Contents

  1. Introduction
  2. Motivating results and background
  3. A Zeilberger-based series acceleration method
  4. Main results
  5. An acceleration based on $\frac{(a)_k (b)_k}{ (n)_k^2}$
  6. An acceleration based on $\frac{(a)_k (b)_k}{(n)_k (2 n)_k}$
  7. Series of convergence rate $\frac{27}{256}$
  8. Series of convergence rate $\frac{108}{3125}$
  9. An acceleration based on $\frac{(a)_k (b)_k}{(n)_k (a+n)_k}$
  10. An acceleration based on $\frac{(a)_k (b)_k}{(n)_k (a+2 n)_k}$
  11. An acceleration based on $\frac{(a)_k (b)_k}{(2 n)_k (a+n)_k}$
  12. Series of convergence rate $-\frac{1}{4}$
  13. Series of convergence rate $-\frac{1}{27}$
  14. Conclusion

Key Result

Theorem 4.1

Letting $f(n)$ denote the hypergeometric series $\sum_{k=0}^{\infty} F(n, k)$, where $F(n, k)$ is as in Fakbknknk, the recursion holds if the above series converge.

Theorems & Definitions (46)

  • Definition 3.1
  • Theorem 4.1
  • Example 4.2
  • Example 4.3
  • Example 4.4
  • Example 4.5
  • Example 4.6
  • Example 4.7
  • Example 4.8
  • Example 4.9
  • ...and 36 more