2021/12/18 by Berndt, Bruce C., Dixit, Atul, Gupta, Rajat +1
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2112.09819
The neglected Russian mathematician, N.~S.~Koshliakov, derived beautiful generalizations of the classical Abel--Plana summation formula through a setting arising from a boundary value problem in heat conduction. When we let the parameter p in this setting tend to infinity, his formulas reduce to the classical Abel--Plana summation formula. Rigorous formulations and proofs of these summation formulas are given. In his notebooks, Ramanujan derived different analogues of the Abel--Plana summation formula. One particular example provides a vast new generalization of the classical transformation formula for Eisenstein series, which we generalize in Koshliakov's setting.