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

Abstract: 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.


Introduction
The importance of the Fermi-Dirac and Bose-Einstein functions emerges from their fundamental presence in quantum physics and related sciences. Unlike the classical mechanics of particles, where the Maxwell distribution is used to study the velocity of classical gas molecules, the quantum gas is analyzed by using the Fermi-Dirac and Bose-Einstein functions. The distinct particles obey Fermi-Dirac statistics, while the indistinct particles follow Bose-Einstein statistics. All particles have a spin in relation to the usual theory. Fermions have half-integer spin and bosons have integer spin. The Fermi-Dirac and Bose-Einstein distribution functions are used to analyze them in the language of mathematics and physics. Indistinguishable particles that are not categorized through either of the aforementioned types are called anyons. The extensions of the Bose-Einstein and Fermi-Dirac functions interpolate between the two. Therefore, Chaudhry et al. [1] proposed that the extensions of the Bose-Einstein and Fermi-Dirac functions may help to describe anyons. In this paper, we generalize the results of Chaudhry and Qadir [2] by proving a general representation theorem to establish a new series representation of the generalized Bose-Einstein and Fermi-Dirac functions. However, we also discuss the fractional derivative, and the relationship of the generalized Bose-Einstein and Fermi-Dirac functions with Apostol-Euler-Nörlund polynomials. Before we provide our research results, it is necessary to enlist all the basic definitions and preliminaries that are required to present and understand this work.

Generalized Bose-Einstein and Fermi-Dirac Functions
During the course of our investigation, we consider the subsequent usual notations: In addition, Z is the set of integers, R denotes the set of real numbers, R + denotes the set of positive numbers, and C is the set of complex numbers, s = σ + iτ. Gamma function Γ(s) as a generalization of factorials is also used here as a basic special function. For a detailed study of gamma and related functions, we refer the interested reader to [3,4].
For further study of the Fermi-Dirac and Bose-Einstein functions, we refer the interested reader to [7][8][9]. The reduction and duality theorems for these functions are given by ( [5], (p. 12-13)) respectively, where R 1 (M , m, −ν) and R(M , m, −ν) are the polynomials having explicit representations in terms of Stirling numbers. For examples and details see Carlitz [10,11]. More recently, Tassaddiq [12,13] considered the λ-generalized extended Fermi-Dirac functions and λ-generalized extended Bose-Einstein functions as a transformed form of Srivastava's λ-generalized Hurwitz-Lerch zeta functions ( [14], (p. 1487), Equation (1.14)). In this research, we generalize the results of Chaudhry and Qadir [2]. To achieve this goal, it is important to briefly highlight their relationship with the zeta functions. It should be noted that for x = 0, the Bose-Einstein and Fermi-Dirac functions are related to the Riemann zeta functions respectively.
The polylogarithm function is an important function in the study of theory of polymers that was introduced and investigated by Truesdell [15] Li s (z) := ∞ n=1 z n n s (s ∈ C, |z|< 1; (s) >1, |z| = 1). (15) It generalizes the Riemann zeta function, as we have and it can also be represented as an integral In our present analysis, we are especially interested in Lindelöf's representation of these functions given by ( [15], (p. 149), Equation (13)), The Hurwitz-Lerch zeta function ( [16], (p. 27)), as a generalization of the polylogarithm, is given by It has a meromorphic extension to the whole complex s-plane, while it has a simple singularity at s = 1 of residue 1. It is also represented by ( [16], (p. 27), Equation (1.6)) Apart from other applications, the Hurwitz-Lerch zeta function is the most general function in the original zeta family. For example, different values of the involved parameters in (19)(20) yield the following relationships with the polylogarithm, Hurwitz, and Riemann zeta functions, respectively: For our purposes, it is important to note that the Hurwitz-Lerch zeta function has a series representation ( [16], (pp. 28-29)) that generalizes Lindelöf's representation (18). Further to all of the above discussion, Chaudhry et al. [17] defined a new generalization of the Riemann zeta function in the critical strip by The Riemann hypothesis is a well-known unsolved problem in analytic number theory [18]. It states that "all the non-trivial zeros of the zeta function exist on the line s = 1/2". These zeros seem to be complex conjugates and are hence symmetrical on this line. The Riemann zeta function in the critical strip is defined and studied in [18] which can be obtained as a special case of Equation (25) by putting x = a = 0.

A Class ∞ (A, P, δ) of Functions and the Representation Theorem
More recently Chaudhry and Qadir [19] discussed some important classes of functions. The statements of this section are taken from [19][20][21].
Note that W s (s ∈ C) satisfies the multiplicative group property. For further detailed study of Weyl and related integral transforms, we refer the interested reader to [22][23][24].

Example 1.
Define Note that ϕ ∈ ∞ ( π 2 , ln(1/2π), δ) and Hence, we have an expansion which is the standard result. Using we can rewrite Φ(s; x) in terms of Bernoulli numbers.

Application of the General Representation Theorem to the Generalized Bose-Einstein and Fermi-Dirac and Related Functions
In this section, we first evaluate the fractional Weyl transform for the function involved in the integrand of generalized Bose-Einstein functions and then analytically continued this function in the interval (0 < (s) < µ), namely the generalized critical strip.

Remark 1.
To apply the general representation theorem, we first discussed analytic continuation of the Bose-Einstein function in the critical strip. The integral representation (5) of the generalized Bose-Einstein function Ψ ν (s, µ; 0) can be continued to the domain, 0 < (s) < µ, where a particular case of this domain 0 < (s) < 1 is known as the critical strip for the zeta function. For (s) > µ, we may write in the usual sense as we write for the zeta function ( [18], (p. 37)) which is true by analytic continuation for (s) > 0. For these values 0 < (s) < µ, we get such that we can write Putting ν = 0 in Equation (53), we get the representation Putting ν = 0; µ = 1 in Equation (53), the classical representation (26) for the Riemann zeta function is recovered. (24) for the Hurwitz-Lerch function is proved in ( [16], (p. 28)) by using the following steps.

1.
Using the contour integral to state the involved function 2.
Using the Cauchy residue theorem from complex analysis 3.
Using the following identity known as Hurwitz formula [16] ζ(s, ν) In this section, we have obtained a new series representation for the generalized Bose-Einstein and Fermi-Dirac functions. We have shown that the above stated results (18) and (24) for the polylogarithm and Hurwitz-Lerch functions are special cases by using the fractional Weyl transform. (56) Proof. The generalized Fermi-Dirac function (1) can be written as which leads to the required result by using Equations (10) and (39).
It is important to notice from integral representations (1) and (5) that the functions Θ ν (s, µ; x) and Ψ ν (s, µ; x) are in effect a Riemann-Liouville fractional derivative of the Fermi-Dirac and Bose-Einstein functions respectively given by Remark 5. The Apostol-Euler-Nörlund polynomials E (µ) n (x; λ) [27,28] are defined by the generating function and Bernoulli-Nörlund polynomials [27,28] are defined by It is important to further mention that the relation of the generalized Fermi-Dirac and Bose-Einstein functions with Apostol--Euler-Nörlund [27,28] polynomials can be established in view of integral representations (1) and (5), respectively, as follows.

Concluding Remarks
One important aspect in relation to the analysis of special functions is to study their representations. These special functions can be studied in different regions by using their series, asymptotic, and integral representations. This fact is also important when writing simpler mathematical proofs of known results. Here, we have provided a new series representation of the generalized Bose-Einstein and Fermi-Dirac functions by using a general representation theorem. To accomplish this work, we discussed an analytic continuation for these functions by generalizing the Riemann zeta function from (0 < (s) < 1) to (0 < (s) < µ). This gives new insights for a possible generalization of the Rieman zeta function and will be discussed in more detail in our future research. Our results were validated by obtaining known series representations for the polylogarithm and the Hurwitz-Lerch zeta functions as special cases. A comparison of the known proof of their series representation was given with this new proof. It is hoped that the general representation theorem can also be applied to analyze other special functions.
Author Contributions: R.S. did project administration and supervision of this research. All the authors (R.S., H.N., S.K., and A.T.) participated equally in the methodology and conceptualization of this research. H.N. and A.T. wrote, reviewed, and edited the manuscript. S.K. checked the validation of the results. All the authors finalized the manuscript after its internal evaluation and contributed substantially to the work reported.
Funding: This research received no external funding.