2015/06/30 by Muhammet Cihat Dağlı, M. Cihat Dağlı, Mümün Can · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebra over a field #Analytic Number Theory Research #Bernoulli number #Bernoulli's principle #Class (philosophy) #Dedekind cut #Dedekind eta function #Dedekind sum #Eisenstein series #Generating function #Mathematical analysis #Mathematics #Modular form #Physics #Pure mathematics #Reciprocity (cultural anthropology) #Reciprocity law #Series (stratigraphy) #Transformation (genetics) #math.NT #msc:11B68 #msc:11F20 #msc:11M36
paper · pdf · doi:10.1007/s11139-016-9808-y
arxiv created 2016/01/06 · openalex created_date 2016/06/24 · openalex publication_date 2017/02/08 · arxiv updated 2017/02/10 · openalex updated_date 2026/08/05
In this paper, a transformation formula under modular substitutions is derived for a large class of generalized Eisenstein series. Appearing in the transformation formulae are generalizations of Dedekind sums involving the periodic Bernoulli function. Reciprocity theorems are proved for these Dedekind sums. Furthermore, as an application of the transformation formulae, relations between various infinite series and evaluations of several infinite series are deduced. Finally, we consider these sums for some special cases.