Some New Results Involving the Generalized Bose–Einstein and Fermi–Dirac Functions
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 function...
Main Authors: | , , , |
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Format: | Article |
Language: | English |
Published: |
MDPI AG
2019-05-01
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Series: | Axioms |
Subjects: | |
Online Access: | https://www.mdpi.com/2075-1680/8/2/63 |
Summary: | 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 <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo><</mo> <mo>ℜ</mo> <mrow> <mo>(</mo> <mi mathvariant="normal">s</mi> <mo>)</mo> </mrow> <mo><</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula> to <inline-formula> <math display="inline"> <semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo><</mo> <mo>ℜ</mo> <mrow> <mo>(</mo> <mi mathvariant="normal">s</mi> <mo>)</mo> </mrow> <mo><</mo> <mi mathvariant="sans-serif">μ</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mrow> </semantics> </math> </inline-formula> 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. |
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ISSN: | 2075-1680 |