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

• Main Authors: Rekha Srivastava, Humera Naaz, Sabeena Kazi, Asifa Tassaddiq
• Format: Article
• Language: English
• Published: MDPI AG 2019-05-01
• Series: Axioms
• Subjects: Fermi–Dirac function; Bose–Einstein function; Weyl transform; series representation
• Online Access: https://www.mdpi.com/2075-1680/8/2/63
• ISSN: 2075-1680

In this paper, we obtain a new series representation for the generalized Bose&#8722;Einstein and Fermi&#8722;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>&lt;</mo> <mo>&real;</mo> <mrow> <mo>(</mo> <mi mathvariant="normal">s</mi> <mo>)</mo> </mrow> <mo>&lt;</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>&lt;</mo> <mo>&real;</mo> <mrow> <mo>(</mo> <mi mathvariant="normal">s</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <mi mathvariant="sans-serif">&#956;</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&#8722;Lerch zeta functions as special cases. Fractional derivatives and the relationship of the generalized Bose&#8722;Einstein and Fermi&#8722;Dirac functions with Apostol&#8722;Euler&#8722;N&#246;rlund polynomials are established to prove new identities.