Ramanujan's cubic transformation and generalized modular equation

2013/05/28 by Miaokun Wang, Yu‐Ming Chu, Wang, Miaokun +5
Mathematics · #11F03 #33C05 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #G.1.2 #Number Theory (math.NT) #acm:11F03 #acm:33C05 #math.CA #math.NT #msc:11F03 #msc:33C05

paper · pdf · doi:10.48550/arxiv.1305.6525

16 pages

arxiv created 2013/05/28 · openalex publication_date 2013/05/28 · arxiv updated 2013/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the quotient of hypergeometric functions μ_a^*(r)=\fracπ2sin(πa)(F(a,1-a;1;1-r³))/(F(a,1-a;1;r³)) (r∈(0,1)) in the theory of Ramanujan's generalized modular equation for a∈(0,1/2], find an infinite product formula for μ_1/3^*(r) by use of the properties of μ_a^*(r) and Ramanujan's cubic transformation. Besides, a new cubic transformation formula of hypergeometric function is given, which complements the Ramanujan's cubic transformation.

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