Fourier transform representation of the extended Fermi–Dirac and Bose–Einstein functions with applications to the family of the zeta and related functions

Abstract

On the one hand, the Fermi–Dirac and Bose–Einstein functions have been extended in such a way that they are closely related to the Riemann and other zeta functions. On the other hand, the Fourier transform representation of the gamma and generalized gamma functions proved useful in deriving various integral formulae for these functions. In this paper, we use the Fourier transform representation of the extended functions to evaluate integrals of products of these functions. In particular, we evaluate some integrals containing the Riemann and Hurwitz zeta functions, which had not been evaluated before.

Keywords:

• Fourier transform
• Parseval's identity
• Riemann zeta function
• Hurwitz zeta function
• Hurwitz–Lerch zeta function
• polylogarithm function
• Fermi–Dirac function
• Bose–Einstein function

2010 Mathematics Subject Classifications :

• Primary: 46FXX
• 60B15
• 11M06
• 11M35
• Secondary: 33B15
• 33CXX

Acknowledgements

A.T. acknowledges her indebtedness to the Higher Education Commission of the Government of Pakistan for the Indigenous PhD Fellowship.