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.

Computational Analysis of Generalized Zeta Functions by Using Difference Equations

“…The computational analysis presented in this investigation is worthwhile to evaluate these generalized Hurwitz zeta functions that are consistent with the existing results. For such related studies please see [51][52][53][54].…”

Mathematical Analysis and Applications II

“…Web Site: http://www.math.uvic.ca/faculty/harimsri/ The present volume contains the invited, accepted and published submissions (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]) to a Special Issue of the MDPI's journal, Axioms, on the subject-area of "Mathematical Analysis and Applications II". A successful predecessor of this volume happens to be the Special Issue of the MDPI's journal, Axioms, on the subject-area of "Mathematical Analysis and Applications" (see, for details, [18]).…”