2023/02/09 by John M. Campbell, JOHN M. CAMPBELL
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Mathematical functions and polynomials
paper · pdf · doi:10.1017/s0004972723000060
Abstract In 2011, Guillera [‘A new Ramanujan-like series for 1/π 2 ’, Ramanujan J. 26 (2011), 369–374] introduced a remarkable rational 7F6( \frac 2764 ) -series for 1/π 2 using the Wilf–Zeilberger (WZ) method, and Chu and Zhang later proved this evaluation using an acceleration method based on Dougall’s 5F4 -sum. Another proof of Guillera’s 7F6( \frac 2764 ) -series was given by Guillera in 2018, and this subsequent proof used a recursive argument involving Dougall’s sum together with the WZ method. Subsequently, Chen and Chu introduced a q -analogue of Guillera’s 7F6( \frac 2764 ) -series. The many past research articles concerning Guillera’s 7F6( \frac 2764 ) -series for 1/π 2 naturally lead to questions about similar results for other mathematical constants. We apply a WZ-based acceleration method to prove new rational 7F6( \frac 2764 ) - and 6F5( \frac 2764 ) -series for √ 2 .