2021/12/17 by Shashi Chourasiya, Chourasiya, Shashi, Md Kashif Jamal +3
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Primary 11M06 #Secondary 26C10
paper · pdf · doi:10.48550/arxiv.2112.09322
openalex publication_date 2021/12/17 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
One of the celebrated formulas of Ramanujan is about odd zeta values, which has been studied by many mathematicians over the years. A notable extension was given by Grosswald in 1972. Following Ramanujan's idea, we rediscovered a Ramanujan-type identity for ζ(2k+1) that was first established by Malurkar and later by Berndt using different techniques. In the current paper, we extend the aforementioned identity of Malurkar and Berndt to derive a new Ramanujan-type identity for L(2k+1, χ1), where χ1 is the principal character modulo prime p. In the process, we encounter a new family of Ramanujan-type polynomials and we notice that a particular case of these polynomials has been studied by Lalín and Rogers in 2013. Furthermore, we establish a character analogue of Grosswald's identity and a few more interesting results inspired from the work of Gun, Murty and Rath.