Kanagala Ajay Acharya, Kanagala Jayaram Tej, Alok Kumar Samanta
F-603, Panchavati Plaza, Sector5, Ghansoli,
Navi Mumbai, 400701, INDIA
Theoretical Chemistry Section
Bhabha Atomic Research Centre
Mumbai, 400085, INDIA

Abstract. We have shown here for the first time that the completeness relation provides a simple unified theoretical framework for deriving different kinds of new recurrence formulae for Riemann Zeta Functions, Dirichlet series and Other Allied Series by selecting only different forms of complete set of orthonormal function (CSOF) in contrast to the expansion method (EM) where one needs to select not only different kinds of CSOF but also suitable arbitrary function. A new proof is also given by selecting only orthogonal Bessel functions for the well known identity corresponding to the sum of squares of the reciprocals of zeros for the m-th order Bessel function. In addition, we have shown here that, in comparison to the EM and other methods, our present method has far-reaching implications, viz. (i) All proofs are based on completeness relation and proper selection of orthogonal functions without selecting any arbitrary functions. (ii) Simpler proofs are possible for new identities corresponding to infinite number of infinite series, sum of each having a fixed value pi ( ). (iii) New proofs emerge not only for the identities corresponding to the Associated Clausen functions but also the sum of new classes of infinite series which resemble the associated Clausen functions.

Received: October 21, 2013

AMS Subject Classification: 11M41, 30B50, 30B60

Key Words and Phrases: completeness relation, Dirac delta function, Rieman Zeta functions, Dirichlet Series, Clausen Functions, Orthonormal functions

Download paper from here.