Some new formulas for π

2001/10/31 by Gert Almkvist, Christian Krattenthaler, Joakim Petersson
Mathematics · #math.NT #math.CA #msc:40A25 #msc:11B65 #msc:11Y60 #msc:65B10

paper · pdf

published as Experiment. Math. 12 (2003), 441-456. · 28 pages, LaTeX; some important references added

arxiv created 2002/10/10 · arxiv updated 2009/11/30

Abstract

We show how to find series expansions for π of the form π=∑_n=0^∞ S(n)/\binommnpnaⁿ, where S(n) is some polynomial in n (depending on m,p,a). We prove that there exist such expansions for m=8k, p=4k, a=(-4)^k, for any k, and give explicit examples for such expansions for small values of m, p and a.

Related

• Some new series for 1/π motivated by congruences
• New type series for powers of π
• Series expansions for 1/π^m and π^m
• A new family of series expansions for 1/π and a binomial identity
• New identities obtained from Gegenbauer series expansion