Some New Results Involving the Generalized Bose–Einstein and Fermi–Dirac Functions

Abstract

In this paper, we obtain a new series representation for the generalized Bose−Einstein and Fermi−Dirac functions by using fractional Weyl transform. To achieve this purpose, we obtain an analytic continuation for these functions by generalizing the domain of Riemann zeta functions from ( 0 < ℜ ( s ) < 1 ) to ( 0 < ℜ ( s ) < μ ) . This leads to fresh insights for a new generalization of the Riemann zeta function. The results are validated by obtaining the classical series representation of the polylogarithm and Hurwitz−Lerch zeta functions as special cases. Fractional derivatives and the relationship of the generalized Bose−Einstein and Fermi−Dirac functions with Apostol−Euler−Nörlund polynomials are established to prove new identities.

Keywords

• Fermi–Dirac function
• Bose–Einstein function
• Weyl transform
• series representation