摘要 : Page 332 of Ramanujan's Lost Notebook contains a compelling identity for zeta (1/2), which has been studied by many mathematicians over the years. On the same page, Ramanujan also recorded the series, 1(r)/exp(1(s)x) - 1 + 2(r)/ex... 展开 Page 332 of Ramanujan's Lost Notebook contains a compelling identity for zeta (1/2), which has been studied by many mathematicians over the years. On the same page, Ramanujan also recorded the series, 1(r)/exp(1(s)x) - 1 + 2(r)/exp(2(s)x) - 1 + 3(r)/exp(3(s)x) - 1 + ... , where s is a positive integer and r-s is any even integer. Unfortunately, Ramanujan doesn't give any formula for it. This series was rediscovered by Kanemitsu, Tanigawa, and Yoshimoto, although they studied it only when r-s is a negative even integer. Recently, Dixit and the second author generalized the work of Kanemitsu et al. and obtained a transformation formula for the aforementioned series with r - s is any even integer. While extending the work of Kanemitsu et al., Dixit and the second author obtained a beautiful generalization of Ramanujan's formula for odd zeta values. In the current paper, we investigate transformation formulas for an infinite series, and interestingly, we derive Ramanujan's formula for zeta (1/2), Wigert's formula for zeta(1/k) as well as Ramanujan's formula for zeta (2m +1). Furthermore, we obtain a new identity for zeta (-1/2) in the spirit of Ramanujan. (c) 2021 Elsevier Inc. All rights reserved. 收起