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Open AccessArticle

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

1
Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
2
College of Computer and Information Sciences, Majmaah University, Al Majmaah 11952, Saudiarabia
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Author to whom correspondence should be addressed.
Axioms 2019, 8(2), 63; https://doi.org/10.3390/axioms8020063
Received: 28 March 2019 / Revised: 16 May 2019 / Accepted: 17 May 2019 / Published: 21 May 2019
(This article belongs to the Special Issue Mathematical Analysis and Applications II)
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 < ( s ) < 1 ) to ( 0 < ( 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. View Full-Text
Keywords: Fermi–Dirac function; Bose–Einstein function; Weyl transform; series representation Fermi–Dirac function; Bose–Einstein function; Weyl transform; series representation
MDPI and ACS Style

Srivastava, R.; Naaz, H.; Kazi, S.; Tassaddiq, A. Some New Results Involving the Generalized Bose–Einstein and Fermi–Dirac Functions. Axioms 2019, 8, 63.

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