q-Analogues of two Ramanujan-type formulas for 1/π

2018/02/06 by Victor J. W. Guo, Guo, Victor J. W., Ji-Cai Liu +1 · 1 citation
Mathematics · #05A10 #11B65 #33D15 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1802.01944

openalex publication_date 2018/02/06 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

Abstract

We give q-analogues of the following two Ramanujan-type formulas for 1/π: ∑_k=0^∞ (6k+1)(((1)/(2))_k³)/(k!³ 4^k) =\frac4π \quadand ∑_k=0^∞ (-1)^k(6k+1)(((1)/(2))_k³)/(k!³ 8^k ) =\frac2√(2)π. Our proof is based on two q-WZ pairs found by the first author in his earlier work.

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