Some representations of the extended Fermi-Dirac and Bose-Einstein functions with applications

Abstract

The familiar Fermi-Dirac (FD) and Bose-Einstein (BE) functions are of importance not only for their role in Quantum Statistics, but also for their several interesting mathematical properties in themselves. Here, in my present investigation, I have ex- tended these functions by introducing an extra parameter in a way that gives new insights into these functions and their relationship to the family of zeta functions. This thesis gives applications of their transform and distributional representations. The Weyl and Mellin transform representations are used to derive mathematical prop- erties of these extended functions. The series representations and difference equations presented led to various new results for the FD and BE functions. It is demonstrated that the domain of the real parameter x involved in the definition of the FD and BE functions can be extended to a complex z. These extensions are dual to each other in a sense that is explained in this thesis. Some identities are proved here for each of these general functions and their relationship with the general Hurwitz-Lerch zeta function Φ(z, s, a) is exploited to derive some new identities. A closely related function to the eFD and eBE functions is also introduced here, which is named as the generalized Riemann zeta (gRZ) function. It approximates the trivial and non- trivial zeros of the zeta function and shows that the original FD and BE functions are related with the Riemann zeta function in the critical strip. Its relation with the Hurwitz zeta functions is used to derive a new series representation for the eBE and the Hurwitz-Lerch zeta functions. ivThe integrals of the zeta function and its generalizations can be of interest in the proof of the Riemann hypothesis (one of the famous problem in mathematics) as well as in Number Theory. The Fourier transform representation is used to derive various integral formulae involving the eFD, eBE and gRZ functions. These are obtained by using the properties of the Fourier and Mellin transforms. Distributional repre- sentation extends some of these formulae to complex variable and yields many new results. In particular, these representations lead to integrals involving the Riemann zeta function and its generalizations. It is also suggested that the Fourier transform and distributional representations of other special functions can be used to evaluate new integrals involving these functions. As an example, I have considered the gen- eralized gamma function. Some of the integrals of products of the gamma function with zeta-related functions can not be expressed in a closed form without defining the eFD, eBE and gRZ functions. It proves the natural occurrence of these general- izations in mathematics. This study led to various new results for the classical FD and BE functions. Integrals of the gamma function and its generalizations are used in engineering mathematics while integrals of the zeta-related functions are essential in Number Theory. Both classes of integrals have been combined first time in this thesis. This in turn gives integrals of product of the modified Bessel functions and zeta-related functions. Further, whereas complex distributions had been defined ear- lier, and in fact used for different applications, there has been no previous utilization of them for Special Functions in general and for the zeta family in particular. This is provided for the first time in this thesis. An important feature of the approach used is the remarkable simplicity of the proofs by using integral transforms.