Next Article in Journal
Fractional Order Forced Convection Carbon Nanotube Nanofluid Flow Passing Over a Thin Needle
Next Article in Special Issue
Fractional Telegraph Equation and Its Solution by Natural Transform Decomposition Method
Previous Article in Journal
Handling Semantic Complexity of Big Data using Machine Learning and RDF Ontology Model
Previous Article in Special Issue
Some General Classes of q-Starlike Functions Associated with the Janowski Functions
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Some Difference Equations for Srivastava’s λ-Generalized Hurwitz–Lerch Zeta Functions with Applications

College of Computer and Information Sciences Majmaah University, Al Majmaah 11952, Saudi Arabia
Symmetry 2019, 11(3), 311; https://doi.org/10.3390/sym11030311
Submission received: 25 December 2018 / Revised: 6 February 2019 / Accepted: 19 February 2019 / Published: 1 March 2019
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

Abstract

:
In this article, we establish some new difference equations for the family of λ-generalized Hurwitz–Lerch zeta functions. These difference equations proved worthwhile to study these newly defined functions in terms of simpler functions. Several authors investigated such functions and their analytic properties, but no work has been reported for an estimation of their values. We perform some numerical computations to evaluate these functions for different values of the involved parameters. It is shown that the direct evaluation of involved integrals is not possible for the large values of parameter s ; nevertheless, using our new difference equations, we can evaluate these functions for the large values of s . It is worth mentioning that for the small values of this parameter, our results are 100% accurate with the directly computed results using their integral representation. Difference equations so obtained are also useful for the computation of some new integrals of products of λ-generalized Hurwitz–Lerch zeta functions and verified to be consistent with the existing results. A derivative property of Mellin transforms proved fundamental to present this investigation.

1. Introduction

In this paper, we practice the customary symbolizations:
N     { 1 ,   2 ,   ·   ·   ·   } ;   N 0   N     { 0 } ;   Z   { 1 , 2 ,   ·   ·   ·   } ;   Z 0   Z   { 0 } ,
where Z   is the set of integers. The involved symbols R ,   R + , and C represent the set of real, positive real, and complex numbers, consistently.
The Hurwitz–Lerch zeta function has always been a topic of motivation for several researchers due to its impact in analytic number theory and other applied sciences. Recently, Srivastava presented a considerably new universal family of Hurwitz–Lerch zeta functions defined by [1] (p. 1487, Equation (1.14)):
Φ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , s , a ; b , λ )     = 1 Γ ( s ) 0 t s 1 exp ( a t b t λ ) p Ψ * q [ ( λ 1 ,   ρ 1 ) , , ( λ p ,   ρ p ) ( μ 1 , σ 1 ) , , ( μ p , σ p ) ; z e t ] d t ;                    ( m i n [ R ( a ) , R ( s ) ] > 0 ; R ( b ) 0 ; λ 0 )
so that, evidently, one can get the subsequent connection with the extended Hurwitz–Lerch zeta functions Φ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , s , a ) defined by the authors of [2] (p. 503, Equation (6.2)) (see also References [3,4]):
Φ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , s , a ; 0 , λ ) = Φ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , s , a ) = e b Φ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , s , a ; b , 0 ) .
In the above Equation (2), p Ψ * q where (p, q 0) is the standard Fox–Wright function defined by the authors of [4] (p. 2219, Equation (1)) (see also References [3] (p. 516, Equation (1)) and [2] (p. 493, Equation (2.1)):
p Ψ * q [ ( λ 1 ,   ρ 1 ) , , ( λ p ,   ρ p ) ( μ 1 , σ 1 ) , , ( μ q , σ q ) ; z   ] = χ = 0 ( [ λ p ] ) ρ p χ ( [ μ q ] ) σ q χ z χ χ ! .
Pochhammer symbols ( [ λ p ] ) ρ p n   : =   [ λ 1 ] ρ p n   ·   ·   ·   [ λ p ] ρ p n symbolize the shifted factorial defined in terms of the basic Gamma function as follows:
( λ   ) ρ = Γ ( λ + ρ   ) Γ ( λ ) =   { 1                                       ( ρ = 0 , λ   { 0 } ) λ ( λ + 1 ) ( λ + χ 1 )   ( ρ = χ   ; λ   ) ,
Δ j = 1 q σ j j = 1 p ρ j   and   :   =   ( j = 1 p ρ j ρ j ) . ( j = 1 q σ j σ j ) .
The series given by Equation (4) converges in the entire complex z-plane for Δ   >   1 ; and if Δ   =   0 , the series (Equation (4)) converges only for | z |   <   . For more detailed discussion of such functions, we refer the interested reader to also see References [5,6,7,8,9].
The analysis of Srivastava’s λ-generalized Hurwitz–Lerch zeta functions and its different forms have attracted noteworthy concern, and many papers have subsequently appeared on this subject. Jankov et al. [10] and Srivastava et al. [3] discussed some inequalities for different cases of λ-generalized Hurwitz–Lerch zeta functions. Srivastava et al. [11] introduced a nonlinear operator related with the λ-generalized Hurwitz–Lerch zeta functions to analyze the inclusion properties of definite subclass of special type of meromorphic functions. Srivastava and Gaboury [12] deliberated on new expansion formulas for such functions (see, for details, References [13,14]; see also the further thoroughly associated studies cited in each of these publications). Luo and Raina [4] discussed some new inequalities involving Srivastava’s λ-generalized Hurwitz–Lerch zeta functions and obtained the following series representation [4] (p. 2221, Equation (6)):
Φ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , s , a ; b , λ ) = 1 λ Γ ( s ) χ = 0 ( [ λ p ] ) ρ p χ ( [ μ q ] ) σ q χ Z 1 λ s λ ( a + χ ) λ b z χ χ ! ( χ + a ) s , ( λ j R ( j = 1 , . . ,   p ) a n d   µ j R \ Z 0   ( j = 1 , ,   q ) ; ρ j > 0 ( j , ,   p ) ; σ j > 0 ( j = 1 , ,   q ) ; 1 + Δ 0 ) .
Srivastava beautifully described important results about the zeta and related functions in an expository article [15]. Choi et al. [16] further discussed these functions by introducing one more variable. Srivastava et al. [17] presented an innovative integral transform connected with the λ-extended Hurwitz–Lerch zeta function. More recently, Tassaddiq [18] obtained a new representation for this family of the λ -generalized Hurwitz–Lerch zeta functions in terms of complex delta functions such that the definition of these functions is formalized over the space of entire test functions denoted by Z . The author also listed and discussed all the possible special cases of Srivastava’s λ-generalized Hurwitz–Lerch zeta functions [18] (p. 4) in the form of a table. For the purposes of our present investigation, this table is given on the next page. For any use of the special cases of the generalized Hurwitz–Lerch zeta functions, the reader is referred to this table. For more detailed study of zeta and related functions, we refer the interested reader to References [19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40] and further bibliography cited therein.
In this research, our focus is to establish some new difference equations for the family of λ-generalized Hurwitz–Lerch zeta functions and its special cases by following the approach of Tassaddiq and Qadir [33]. From the above discussion and Table 1, we can notice that several authors presented and studied worthwhile generalizations of the Hurwitz–Lerch zeta functions. They obtained various analytic formulas, integral, and series representations. However, as we deeply study Riemann zeta functions, we know their values, their graphs, and several other important aspects. We could not develop this approach for these generalizations. Bayad and Chiki [43] obtained reduction and duality formulas of the generalized Hurwitz–Lerch zeta functions. Their results contain the earlier obtained results of Choi [47]. These reduction formulas were concerned with the reduction of one parameter that represent the generalized Hurwitz–Lerch zeta Φ μ * ( z , s , a ) and Hurwitz zeta functions ζ μ * ( s , a ) in terms of Hurwitz–Lerch zeta Φ ( z , s , a ) and Hurwitz zeta ζ ( s , a ) functions, respectively. The difference equations presented here have the advantage of reducing the generalized Hurwitz–Lerch zeta Φ μ * ( z , s , a ) and the generalized Hurwitz zeta functions ζ μ * ( s , a ) in terms of basic polylogarithm Li s ( z ) and zeta functions ζ ( s ) , respectively. That means we have reduced one more parameter and our results are simple enough to evaluate these functions for different values of the involved parameters. By following the approach developed in this paper, we can initiate a deeper analysis of these functions that will enhance their applications. The Riemann hypothesis is a well-known unsolved problem in analytic number theory [22]. It states that “all the non-trivial zeros of the zeta function exist on the real line s = 1 2 ”. These zeros seem to be complex conjugates and hence symmetric on this line. The integrals of the zeta function and its generalizations are vital in the study of Riemann hypothesis and for the investigation of zeta functions themselves. The study of distributions in statistical inference and reliability theory [1,48,49] also involves such integrals. Difference equations obtained in this investigation are worthwhile to evaluate integrals of products of the family of λ-generalized Hurwitz–Lerch zeta functions that are consistent with the existing results.
The plan of the paper as follows: We present some new difference equations involving the λ-generalized Hurwitz–Lerch zeta functions in Section 2 and obtain similar results for other related functions. We discuss some applications of these difference equations in Section 3 by evaluating some special cases of the function. Based upon the results of Section 2, we evaluate new integrals of products of these functions in Section 4. We conclude our results in the last Section 5 by highlighting some future directions of this work.
Throughout this investigation, conditions on the parameters will be considered standard as given in Equations (1)–(6) and Table 1 unless otherwise stated.

2. Results

New Difference Equation of the λ-Generalized Hurwitz–Lerch Zeta Functions
Theorem 1.
Prove that λ-Generalized Hurwitz–Lerch zeta functions satisfy the following relation:
Γ ( s ) Φ ( λ 1 + ρ 1 , , λ p ρ p , μ 1 + σ 1 , , μ q + σ q ) λ 1 , , λ p , μ 1 , , , μ q ( z , s , a + 1 ; b , λ ) = ( μ 1 ) σ 1 . . ( μ q ) σ q z . ( λ 1 ) ρ 1 . . ( λ p ) ρ p [ b λ Γ ( s λ 1 ) Φ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p σ 1 , , σ q ) ( z , s λ 1 , a , b ) + Γ ( s ) [ Φ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p σ 1 , , σ q ) ( z , s 1 , a ; b , λ ) a Φ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p σ 1 , , σ q ) ( z , s , a ; b , λ ) ] ] .
Proof: 
Consider the function:
f ( t ) = exp ( at b t λ )   p Ψ * q [ ( λ 1 ,   ρ 1 ) , , ( λ p ,   ρ p ) ( μ 1 , σ 1 ) , , ( μ q , σ q ) ; ze t ]
and differentiate Equation (8) to get:
d dt [ exp ( at b t λ ) p Ψ * q [ ( λ 1 , ρ 1 ) , , ( λ p , ρ p ) ( μ 1 , σ 1 ) , , ( μ q , σ q ) ; ze t ] ] = a . exp ( at b t λ ) p Ψ * q [ ( λ 1 , ρ 1 ) , , ( λ p , ρ p ) ( μ 1 , σ 1 ) , , ( μ q , σ q ) ; ze t ] + b λ exp ( at b t λ ) t λ + 1 p Ψ * q [ ( λ 1 , ρ 1 ) , , ( λ p , ρ p ) ( μ 1 , σ 1 ) , , ( μ q , σ q ) ; ze t ] z ( λ 1 ) ρ 1 . . ( λ p ) ρ p ( μ 1 ) σ 1 . . ( μ q ) q exp ( ( a + 1 ) t b t λ ) p Ψ * q [ ( λ 1 + ρ 1 , ρ 1 ) , , ( λ p + ρ p , ρ p ) ( μ 1 + σ 1 , σ 1 ) , , ( μ q + σ q , σ q ) ; ze t ]
so that:
f ( t ) = exp ( at b t λ ) p Ψ * q [ ( λ 1 , ρ 1 ) , , ( λ p , ρ p ) ( μ 1 , σ 1 ) , , ( μ q , σ q ) ze t ] [ a + b λ t λ + 1 ] z ( λ 1 ) ρ 1 . . ( λ p ) ρ p ( μ 1 ) σ 1 . . ( μ q ) q exp ( ( a + 1 ) t b t λ ) p Ψ * q [ ( λ 1 + ρ 1 , ρ 1 ) , , ( λ p + ρ p , ρ p ) ( μ 1 + σ 1 , σ 1 ) , , ( μ q + σ q , σ q ) ; ze t ] ,
where we have used the usual differentiation and the derivative property, which is obtained on the same lines as given by Reference [1] (p. 1492, Equation (3.1)):
d dt [ p Ψ * q [ ( λ 1 , ρ 1 ) , , ( λ p , ρ p ) ( μ 1 , σ 1 ) , , ( μ q , σ q ) ; ze t ] ] = z e t ( λ 1 ) ρ 1 . . ( λ p ) ρ p ( μ 1 ) σ 1 . . ( μ q ) q p Ψ * q [ ( λ 1 + ρ 1 , ρ 1 ) , , ( λ p + ρ p , ρ p ) ( μ 1 + σ 1 , σ 1 ) , , ( μ q + σ q , σ q ) ; ze t ] .
Taking Mellin transform on both sides of Equation (8) and using the defining integral representation as given in Equation (2), we can write:
Γ ( s ) Φ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , s , a ; b , λ ) = M   [ exp ( at b t λ ) p Ψ * q [ ( λ 1 ,   ρ 1 ) , , ( λ p ,   ρ p ) ( μ 1 , σ 1 ) , , ( μ p , σ p ) ze t ] ; s ] .
Using the derivative property of Mellin transform given by, see [50] (Chapter 10):
M [ u ( y ) ;   τ ] =   ( τ 1 ) M [ u ( y ) ;   τ 1 ]
we obtain the following equation:
Γ ( s ) [ a Φ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , s , a ; b , λ ) Φ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , s 1 , a ; b , λ ) ] = b λ Γ ( s λ 1 ) Φ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , s λ 1 , a , b ) ( λ 1 ) ρ 1 . . ( λ p ) ρ p ( μ 1 ) σ 1 . . ( μ q ) q z Γ ( s ) Φ ( λ 1 + ρ 1 , , λ p ρ p , μ 1 + σ 1 , , μ q + σ q ) λ 1 , , λ p , μ 1 , , , μ q ( z , s , a + 1 ; b , λ )
which leads to:
( λ 1 ) ρ 1 . . ( λ p ) ρ p ( μ 1 ) σ 1 . . ( μ q ) q z Γ ( s ) Φ ( λ 1 + ρ 1 , , λ p ρ p , μ 1 + σ 1 , , μ q + σ q ) λ 1 , , λ p , μ 1 , , , μ q ( z , s , a + 1 ; b , λ ) = b λ Γ ( s λ 1 ) Φ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , s λ 1 , a , b ) + Γ ( s ) [ Φ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , s 1 , a ; b , λ ) a Φ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , s , a ; b , λ ) ] .
After some simple modifications, one can arrive at the required result of Equation (7). □
Remark 1.
We can obtain similar results for other related functions as listed in Table 1 by considering different parameter values in the resulting corollaries.
Corollary 1.
λ-Generalized Extended Fermi–Dirac functions have the following representation:
Γ ( s ) Θ λ 1 + ρ 1 , , λ p + ρ p , μ 1 + σ 1 , , , μ q σ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( x , s , a + 1 ; b , λ ) = e x · ( μ 1 ) σ 1 . . ( μ q ) σ q ( λ 1 ) ρ 1 . . ( λ p ) ρ p [ Γ ( s ) [ a Θ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( x , s , a ; b , λ ) Θ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( x , s 1 , a ; b , λ ) ] b λ Γ ( s λ 1 ) Θ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( x , s λ 1 , a , b , λ ) ]
and λ-Generalized Extended Bose–Einstein functions have the following representation:
Γ ( s ) Ψ λ 1 + ρ 1 , , λ p + ρ p , μ 1 + σ 1 , , , μ q + σ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( x , s , a + 1 ; b , λ )           = e x . ( μ 1 ) σ 1 . . ( μ q ) σ q ( λ 1 ) ρ 1 . . ( λ p ) ρ p [ b λ Γ ( s λ 1 ) Ψ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( x , s λ 1 , a , b , λ )           + Γ ( s ) [ Ψ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( x , s 1 , a ; b , λ ) a Ψ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( x , s , a ; b , λ ) ] ] .
Proof. 
The results follow directly from Equation (7) upon replacing z ± e x   and using the parallel case given in row 2 and column 2 of Table 1. □
Corollary 2.
λ-Generalized Fermi–Dirac functions have the following representation:
Γ ( s ) Θ λ 1 + ρ 1 , , λ p + ρ p , μ 1 + σ 1 , , , μ q + σ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( x , s , 2 ; b , λ ) = e x . ( μ 1 ) σ 1 . . ( μ q ) σ q ( λ 1 ) ρ 1 . . ( λ p ) ρ p [ b λ Γ ( s λ 1 ) F λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( x , s λ 1 , b , λ ) + Γ ( s ) [ F λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( x , s 1 ; b , λ ) F λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( x , s ; b , λ ) ] ]
and λ-Generalized Bose–Einstein functions have the following representation:
Γ ( s ) Ψ λ 1 + ρ 1 , , λ p + ρ p , μ 1 + σ 1 , , , μ q + σ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( x , s , 2 ; b , λ )           = e x . ( μ 1 ) σ 1 . . ( μ q ) σ q ( λ 1 ) ρ 1 . . ( λ p ) ρ p [ b λ Γ ( s λ 1 ) B λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( x , s λ 1 , b , λ )           + Γ ( s ) [ B λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( x , s 1 ; b , λ ) B λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( x , s ; b , λ ) ] ] .
Proof. 
The results follow directly from Equation (7) upon replacing z ± e x   ; a 1   and taking the item from Table 1 corresponding to these parameter values. □
Corollary 3.
λ-Generalized Polylogarithm functions have the following representation:
Γ ( s ) Li λ 1 + ρ 1 , , λ p + ρ p , μ 1 + σ 1 , , , μ q + σ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , s , 2 ; b , λ ) = ( μ 1 ) σ 1 . . ( μ q ) σ q z . ( λ 1 ) ρ 1 . . ( λ p ) ρ p [ b λ Γ ( s λ 1 ) Li λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , s λ 1 , b , λ ) + Γ ( s ) [ Li λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , s 1 ; b , λ ) Li λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , s ; b , λ ) ] ] .
Proof. 
The result follows directly from Equation (7) upon replacing a 1 and considering the specific case of these parameter values from Table 1. □
Corollary 4.
λ-Generalized Hurwitz zeta functions have the following representation:
Γ ( s ) ζ λ 1 + ρ 1 , , λ p + ρ p , μ 1 + σ 1 , , , μ q + σ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( s , a + 1 ; b , λ )           = ( μ 1 ) σ 1 . . ( μ q ) σ q ( λ 1 ) ρ 1 . . ( λ p ) ρ p [ b λ Γ ( s λ 1 ) ζ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( s λ 1 , a ; b , λ )           + Γ ( s ) [ ζ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( s 1 , a ; b , λ ) a ζ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( s , a ; b , λ ) ] ] .
Proof. 
The result follows directly from Equation (7) upon replacing z 1 and in view of the defined item from Table 1 dependable on these parameter values. □
Corollary 5.
λ-Generalized Riemann zeta functions have the following representation:
ζ λ 1 + ρ 1 , , λ p + ρ p , μ 1 + σ 1 , , , μ q + σ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( s , 2 ; b , λ ) = ( μ 1 ) σ 1 . . ( μ q ) σ q ( λ 1 ) ρ 1 . . ( λ p ) ρ p [ b λ Γ ( s λ 1 ) ζ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( s λ 1 ; b , λ ) + Γ ( s ) [ ζ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( s 1 ; b , λ ) ζ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( s ; b , λ ) ] ] .
Proof. 
The result follows directly from Equation (7) upon replacing z 1 ; a 1 and with reference to the definite element from Table 1 stable with these parameter values. □
Remark 2.
We can get similar representations for other special cases of these functions by considering different parameter variations in view of Table 1 column-wise.
Note that by taking b = 0 in the above results, we can get the following formulae for unified extended Hurwitz–Lerch zeta functions Φ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , s , a ; 0 , λ ) :
Φ λ 1 + ρ 1 , , λ p + ρ p , μ 1 + σ 1 , , , μ q + σ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , s , a + 1 ) = ( μ 1 ) σ 1 . . ( μ q ) σ q z . ( λ 1 ) ρ 1 . . ( λ p ) ρ p [ Φ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , s 1 , a ) a Φ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , s , a ) ] .
Next, by selecting p 1 = q = 0 ; λ 1 = μ 0 ;   1 = ρ 1 , in the above results, we can get the following result for unified Hurwitz–Lerch zeta functions Φ μ * ( z , s , a ) :
0 t s 1 e ( a + 1 ) t ( 1 ze t ) μ + 1 dt = Γ ( s ) Φ μ + 1 * ( z , s , a + 1 ) = Γ ( s ) μ z [ Φ μ * ( z , s 1 , a ) a Φ μ * ( z , s , a ) ] .
Next, we note that by taking μ = 1 , we get for Hurwitz–Lerch zeta functions:
0 t s 1 e ( a + 1 ) t ( 1 ze t ) 2 dt = Γ ( s ) Φ 2 * ( z , s , a + 1 ) = Γ ( s ) [ Φ ( z , s 1 , a ) a Φ ( z , s , a ) ] z .
If we consider the same parameter values as above but with b 0 , then we can find the following new results for the extended Riemann and Hurwitz zeta functions:
μ z Γ ( s ) Θ μ + 1 λ ( z , s , a + 1 , b )          = b λ Γ ( s λ 1 ) Θ μ λ ( z , s λ 1 , a , b )          + Γ ( s ) [ Θ μ λ ( z , s 1 , a , b ) a Θ μ λ ( z , s , a , b ) ]
0 t s 1 e ( a + 1 ) t   b t ( 1 z e t ) 2 d t = Γ ( s ) Φ 2 * ( z , s , a + 1 ; b , 1 )         = b Γ ( s 2 ) z Φ b ( z , s 2 , a ) + Γ ( s ) z [ Φ b ( z , s 1 , a ) a Φ b ( z , s , a ) ]
0 t s 1 e 2 t   b t ( 1 e t ) 2 d t = Γ ( s ) ζ 2 * ( s , 2 ; b , 1 ) = b Γ ( s 2 ) ζ b ( s 2 ) + Γ ( s ) [ ζ b ( s 1 ) ζ b ( s ) ] .

3. Some applications of the difference equation

In this section, we consider some interesting special cases of difference equations. On one side, these are useful to know the values of generalized Hurwitz zeta functions Φ μ * ( z , s , a ) in terms of zeta functions, and on the other, they lead to the computation of some elementary integrals that are nontrivial to obtain for small values of μ = 2 , 3 , 4 , 5 and the large values of s.
Taking μ = 2 and a a + 1 in Equation (24), we get:
0 t s 1 e ( a + 2 ) t ( 1 z e t ) 3 d t = Γ ( s ) Φ 3 * ( z , s , a + 2 ) = Γ ( s ) 1.2. z [ Φ 2 * ( z , s 1 , a + 1 ) ( a + 1 ) Φ 2 * ( z , s , a + 1 ) ] .
Next, making use of Equation (25) on the right-hand side of the above Equation (29) leads to the following form of Φ 3 * ( z , s , a + 2 ) in terms of the Hurwitz–Lerch zeta function:
0 t s 1 e ( a + 2 ) t ( 1 z e t ) 3 d t = Γ ( s ) 1.2. z 2 [ Φ ( z , s 2 , a ) a Φ ( z , s 1 , a ) ( a + 1 ) Φ ( z , s 1 , a ) + a ( a + 1 ) Φ ( z , s , a ) ] = Γ ( s ) 1.2. z 2 [ Φ ( z , s 2 , a ) ( 2 a + 1 ) Φ ( z , s 1 , a ) + a ( a + 1 ) Φ ( z , s , a ) ] .
Now we consider some interesting special cases of the above Equation (30).
For z = 1 , it leads to the following representation in terms of the Hurwitz zeta function:
0 t s 1 e ( a + 2 ) t ( 1 e t ) 3 d t = Γ ( s ) 1.2 [ ζ ( s 2 , a ) ( 2 a + 1 ) ζ ( s 1 , a ) + a ( a + 1 ) ζ ( s , a ) ] ;
0 t s 1 e ( a + 2 ) t ( 1 e t ) 3 d t = Γ ( s ) 1.2 [ ζ ( s 2 , a ) ( 2 a + 1 ) ζ ( s 1 , a ) + a ( a + 1 ) ζ ( s , a ) ] .
For a = 1 , we get the following representation in terms of the polylogarithm function:
0 t s 1 e 3 t ( 1 z e t ) 3 d t = Γ ( s ) 1.2. z 3 [ L i s 2 ( z ) 3 L i s 1 ( z ) + 2 L i s ( z ) ] .
For a = 1 ,   z = 1 , it leads to the following relation in terms of the zeta function:
0 t s 1 e 3 t ( 1 e t ) 3 d t = Γ ( s ) 1.2 [ 2 ζ ( s ) + ζ ( s 2 ) 3 ζ ( s 1 ) ] ; ( s 1 , 2 , 3 ) .
For s = 4 in Equation (34), we get the following integral:
0 t 3 e 3 t ( 1 e t ) 3 d t = Γ ( 4 ) 1.2 [ 2 ζ ( 4 ) + ζ ( 2 ) 3 ζ ( 3 ) ] .
Similarly, by considering different values of s , we can produce the following Table 2 and Table 3 of values. These computations show that Mathematica is unable to compute the involved integral on a commonly available computer for the large values of s , but it can be done using these new difference equations. For small values of s , our results are 100% accurate with the direct computed results
Putting μ = 3 , a a + 2 in Equation (24), we get:
0 t s 1 e ( a + 3 ) t ( 1 z e t ) 3 + 1 d t = Γ ( s ) Φ 3 + 1 * ( z , s , a + 3 ) = Γ ( s ) 3. z [ Φ 3 * ( z , s 1 , a + 2 ) ( a + 2 ) Φ 3 * ( z , s , a + 2 ) ] .
Next, combining the above two results (Equations (30) and (36)), we get the following representation in terms of the Hurwitz–Lerch zeta function
0 t s 1 e ( a + 3 ) t ( 1 ze t ) 4 dt = Γ ( s ) 1.2.3. z 3 [ Φ ( z , s 3 , a ) 3 ( a + 1 ) Φ ( z , s 2 , a ) + ( 3 a 2 + 6 a + 2 ) Φ ( z , s 1 , a ) a ( a + 1 ) ( a + 2 ) Φ ( z , s , a ) ] .
Some interesting special cases: For z = 1 , it leads to the following representation in terms of the Hurwitz zeta function:
0 t s 1 e ( a + 3 ) t ( 1 e t ) 4 d t = Γ ( s ) ζ 4 * ( s , a + 3 ) = Γ ( s ) 1.2.3 [ ζ ( s 3 , a ) 3 ( a + 1 ) ζ ( s 2 , a ) + ( 3 a 2 + 6 a + 2 ) ζ ( s 1 , a ) a ( a + 1 ) ( a + 2 ) ζ ( s , a ) ] .  
For a = 1 , we get the following representation in terms of the polylogarithm function:
0 t s 1 e 4 t ( 1 z e t ) 4 d t = Γ ( s ) Φ 4 * ( z , s , 4 ) = Γ ( s ) 1.2.3. z 4 [ L i s 3 ( z ) 6 L i s 2 ( z ) + 11 L i s 1 ( z ) 6 L i s ( z ) ] .
For a = 1 ,   z = 1 , it leads to the following relation in terms of the zeta function:
0 t s 1 e 4 t ( 1 e t ) 4 d t = Γ ( s ) ζ 4 * ( s , 4 ) = Γ ( s ) 1.2.3 [ ζ ( s 3 ) 6 ζ ( s 2 ) + 11 ζ ( s 1 ) 6 ζ ( s ) ] ;   ( s 1 , 2 , 3 , 4 ) .
For s = 5, we have:
0 t 4 e 4 t ( 1 e t ) 4 d t = Γ ( 5 ) ζ 4 * ( 4 , 4 ) = Γ ( 5 ) 1.2.3 [ ζ ( 2 ) 6 ζ ( 3 ) + 11 ζ ( 4 ) 6 ζ ( 5 ) ] .
Similarly, putting μ = 4 , a a + 3 in Equation (24), we get:
0 t s 1 e ( a + 4 ) t ( 1 z e t ) 4 + 1 d t = Γ ( s ) Φ 4 + 1 * ( z , s , a + 4 ) = Γ ( s ) 4. z [ Φ 4 * ( z , s 1 , a + 3 ) ( a + 3 ) Φ 4 * ( z , s , a + 3 ) ] .
Next, combining the above two results (Equations (38) and (42)), we get:
0 t s 1 e ( a + 4 ) t ( 1 z e t ) 5 d t = Γ ( s ) 1.2.3.4. z 4 [ Φ ( z , s 3 , a ) + ( 6 a 2 + 18 a + 11 ) Φ ( z , s 2 , a ) ( 4 a 3 + 18 a 2 + 22 a + 6 ) Φ ( z , s 2 , a ) + a ( a + 1 ) ( a + 2 ) ( a + 3 ) Φ ( z , s , a ) ] .
Some interesting special cases:
For z = 1:
0 t s 1 e ( a + 4 ) t ( 1 e t ) 5 d t = Γ ( s ) ζ 5 * ( s , a + 4 ) = Γ ( s ) 1.2.3.4 [ ζ ( s 4 , a ) 2 ( 2 a + 3 ) ζ ( s 3 , a ) + ( 6 a 2 + 18 a + 11 ) ζ ( s 2 , a ) ( 4 a 3 + 18 a 2 + 22 a + 6 ) ζ ( s 2 , a ) + a ( a + 1 ) ( a + 2 ) ( a + 3 ) ζ ( s , a ) ] .      
For a = 1:
0 t s 1 e 5 t ( 1 z e t ) 5 d t = Γ ( s ) Φ 5 * ( z , s , 5 ) = Γ ( s ) 1.2.3.4. z 5 [ L i s 4 ( z ) 10 L i s 3 ( z ) + 37 L i s 2 ( z ) 50 L i s 2 ( z ) + 24 L i s ( z ) ] .          
For a = 1 ,   z = 1 :
0 t s 1 e 5 t ( 1 e t ) 5 d t = Γ ( s ) ζ 5 * ( s , 5 ) = Γ ( s ) 1.2.3.4 [ ζ ( s 4 ) 10 ζ ( s 3 ) + 37 ζ ( s 2 ) 50 ζ ( s 2 ) + 24 ζ ( s ) ] ; ( s 1 , 2 , 3 , 4 , 5 ) .          
Now put s = 6 :
0 t 5 e 5 t ( 1 e t ) 5 d t = Γ ( 6 ) ζ 5 * ( 6 , 5 ) = Γ ( 6 ) 1.2.3.4 [ ζ ( 2 ) 10 ζ ( 3 ) + 37 ζ ( 4 ) 50 ζ ( 5 ) + 24 ζ ( 6 ) ] .                                      
Continuing in this way, by putting μ = 5 , a a + 4 in Equation (24), we get:
0 t s 1 e ( a + 5 ) t ( 1 z e t ) 5 + 1 d t = Γ ( s ) Φ 5 + 1 * ( z , s , a + 5 ) = Γ ( s ) 5. z [ Φ 5 * ( z , s 1 , a + 4 ) ( a + 4 ) Φ 5 * ( z , s , a + 4 ) ] .
Next, combining the above two results of Equations (43) and (48), we can get Φ 6 * ( z , s , a + 4 ) in terms of Hurwitz–Lerch zeta functions. Similarly, for nonzero values of z, for example, z = 0.3 ; a = 1 ; μ = 3 in Equation (33) we have:
0 ( e 3   t t 4 ) ( 1   3   e t 10 ) 3 d t =   0.125061
Γ ( 5 ) ( L i 3 ( 0.3 )   3   L i 4 ( 0.3 ) +   2   L i 5 ( 0.3 ) ) 2   ( 0.3 ) 3 =   0.125061
and so on and so forth.

4. Integrals of products of the family of λ-Generalized Hurwitz–Lerch zeta functions

By means of the basic Parseval’s identity of Mellin transform [50] (Chapter 10) and difference equations obtained in Section 2, we can get the following integral formulae in view of Equation (2) and column 3 of Table 1. For example, for the generalized Hurwitz–Lerch zeta functions Θ μ λ ( z , s , a , b ) :
1 2 π i c i c + i Γ ( s ) Γ ( w s ) Θ μ λ ( z , s , a , b ) Θ δ λ ( z , w s , a , b ) =   0 t w 1 e 2 a t   2 b t λ ( 1 z e t ) μ + δ d t
that leads to the following by simply replacing μ μ + δ 1 in Equation (26):
Γ ( ω ) Θ μ + δ λ ( z , ω , 2 a , 2 b ) = 1 z ( μ 1 + δ ) [ 2 b λ Γ ( ω λ 1 ) Θ μ 1 + δ λ ( z , ω λ 1 , 2 a 1 , 2 b ) + Γ ( ω ) Θ μ 1 + δ λ ( z , ω 1 , 2 a 1 , 2 b ) ( 2 a 1 ) Γ ( ω ) Θ μ 1 + δ λ ( z , w , 2 a 1 , 2 b ) ] .
Therefore, we get the following new integral formulae for other related cases given in column 3 of Table 1 and Equations (51) and (52):
1 2 π i c i c + i Γ ( s ) Γ ( w s ) Θ μ λ ( z , s , a , b ) Θ δ λ ( z , w s , a , b ) = 0 t w 1 e 2 at   2 b t λ ( 1 + e x e t ) μ + δ dt = e x μ 1 + δ [ Γ ( ω ) [ ( 2 a 1 ) Θ μ 1 + δ λ ( x , w , 2 a 1 , 2 b ) Θ μ 1 + δ λ ( x , ω 1 , 2 a 1 , 2 b ) ] 2 b λ Γ ( ω λ 1 ) Θ μ 1 + δ λ ( x , ω λ 1 , 2 a 1 ,   2 b ) ]
1 2 π i c i c + i Γ ( s ) Γ ( w s ) F μ λ ( x , s , b ) F δ λ ( x , w s , b ) =   0 t w 1 e 2 t   2 b t λ ( 1 + e x e t ) μ + δ dt = e x μ 1 + δ [ Γ ( ω ) [ F μ 1 + δ λ ( x , w , 2 b ) F μ 1 + δ λ ( x , ω 1 , 2 b ) ] 2 b λ Γ ( ω λ 1 ) F μ 1 + δ λ ( x , ω λ 1 ,   2 b ) ]
1 2 π i c i c + i Γ ( s ) Γ ( w s ) Ψ μ λ ( z , s , a , b ) Ψ δ λ ( z , w s , a , b ) = 0 t w 1 e 2 at   2 b t λ ( 1 e x e t ) μ + δ dt = e x μ 1 + δ [ 2 b λ Γ ( ω λ 1 ) Ψ μ 1 + δ λ ( x , ω λ 1 , 2 a 1 ,   2 b ) + Γ ( ω ) [ Ψ μ 1 + δ λ ( x , ω 1 , 2 a 1 , 2 b ) ] ( 2 a 1 ) Ψ μ 1 + δ λ ( x , w , 2 a 1 , 2 b ) ]
1 2 π i c i c + i Γ ( s ) Γ ( w s ) B μ λ ( x , s , b ) B δ λ ( x , w s , b ) = 0 t w 1 e 2 t   2 b t λ ( 1 e x e t ) μ + δ dt = e x μ 1 + δ [ 2 b λ Γ ( ω λ 1 ) B μ 1 + δ λ ( x , ω λ 1 ,   2 b ) + Γ ( ω ) [ B μ 1 + δ λ ( x , ω 1 , 2 b ) B μ 1 + δ λ ( x , w , 2 b ) ] ]
1 2 π i c i c + i Γ ( s ) Γ ( w s ) Li μ λ ( z , s , b ) Li δ λ ( z , w s , b ) = 0 t w 1 e 2 t   2 b t λ ( 1 ze t ) μ + δ dt = 1 ( μ 1 + δ ) z [ 2 b λ Γ ( ω λ 1 ) Li μ 1 + δ λ ( x , ω λ 1 ,   2 b ) + Γ ( ω ) [ Li μ 1 + δ λ ( x , ω 1 , 2 b ) Li μ 1 + δ λ ( x , w , 2 b ) ] ]
1 2 π i c i c + i Γ ( s ) Γ ( w s ) ζ μ λ ( s , a ; b ) ζ δ λ ( w s , a ; b ) = 0 t w 1 e 2 at   2 b t λ ( 1 e t ) μ + δ dt = 1 ( μ 1 + δ ) [ 2 b λ Γ ( ω λ 1 ) ζ μ 1 + δ λ ( ω λ 1 , 2 a 1 ,   2 b ) + Γ ( ω ) [ ζ μ 1 + δ λ ( w 1 , 2 a 1 , 2 b ) ( 2 a 1 ) ζ μ 1 + δ λ ( w , 2 a 1 , 2 b ) ] ]
1 2 π i c i c + i Γ ( s ) Γ ( w s ) ζ μ λ ( s ; b ) ζ δ λ ( w s ; b ) = 0 t w 1 e 2 t   2 b t λ ( 1 e t ) μ + δ dt = 1 ( μ 1 + δ ) [ 2 b λ Γ ( ω λ 1 ) ζ μ 1 + δ λ ( ω λ 1 , 2 b ) + Γ ( ω ) [ ζ μ 1 + δ λ ( w 1 , 2 b ) ζ μ 1 + δ λ ( w , 2 b ) ] ] .
Next, for b = 0 , we can get the following new formulae in view of column 8 of Table 1 and Equation (53):
1 2 π i c i c + i Γ ( s ) Γ ( w s ) Φ μ * ( z , s , a )   Φ δ * ( z , w s , a ) =   0 t w 1 e 2 a t ( 1 z e t ) μ + δ d t = Γ ( ω ) ( μ 1 + δ ) z [ Φ μ 1 + δ *   ( z , ω 1 , 2 a 1 ) ( 2 a 1 ) Φ μ 1 + δ * ( z , w , 2 a 1 ) ]
1 2 π i c i c + i Γ ( s ) Γ ( w s ) Θ μ * ( x , s , a ) Θ δ * ( x , w s , a ) = 0 t w 1 e 2 a t ( 1 + e x e t ) μ + δ d t = e x Γ ( ω ) μ 1 + δ [ ( 2 a 1 ) Θ μ 1 + δ * ( x , w , 2 a 1 ) Θ μ 1 + δ * ( x , ω 1 , 2 a 1 ) ]
1 2 π i c i c + i Γ ( s ) Γ ( w s ) F μ * ( x , s ) F δ * ( x , w s ) = 0 t w 1 e 2 t ( 1 + e x e t ) μ + δ d t = e x Γ ( ω ) μ 1 + δ [ F μ 1 + δ * ( x , w ) F μ 1 + δ * ( x , ω 1 ) ]
1 2 π i c i c + i Γ ( s ) Γ ( w s ) Ψ μ * ( x , s , a ) Ψ δ * ( x , w s , a ) = 0 t w 1 e 2 a t ( 1 e x e t ) μ + δ d t = e x Γ ( ω ) μ 1 + δ [ Ψ μ 1 + δ * ( x , ω 1 , 2 a 1 ) ( 2 a 1 ) Ψ μ 1 + δ * ( x , w , 2 a 1 ) ]
1 2 π i c i c + i Γ ( s ) Γ ( w s ) B μ * ( x , s ) B δ * ( x , w s ) = 0 t w 1 e 2 t ( 1 e x e t ) μ + δ d t = e x Γ ( ω ) μ 1 + δ [ B μ 1 + δ * ( x , ω 1 ) B μ 1 + δ * ( x , w ) ]
1 2 π i c i c + i Γ ( s ) Γ ( w s ) Li μ * ( z , s )   Li δ * ( z , w s ) =   0 t w 1 e 2 t ( 1 z e t ) μ + δ d t = Γ ( ω ) ( μ 1 + δ ) z [ L i μ 1 + δ *   ( z , ω 1 ) L i μ 1 + δ * ( z , w ) ]
1 2 π i c i c + i Γ ( s ) Γ ( w s ) ζ μ * ( s , a )   ζ δ * ( w s , a ) d s =   0 t w 1 e 2 a t ( 1 e t ) μ + δ d t = Γ ( ω ) ( μ 1 + δ ) [ ζ μ 1 + δ *   ( ω 1 , a ) ( 2 a 1 ) ζ μ 1 + δ * ( w , a ) ]
1 2 π i c i c + i Γ ( s ) Γ ( w s ) ζ μ * ( s )   ζ δ * ( w s ) d s =   0 t w 1 e 2 t ( 1 e t ) μ + δ d t = Γ ( ω ) ( μ 1 + δ ) [ ζ μ 1 + δ *   ( ω 1 ) ζ μ 1 + δ * ( w ) ] .
Next, if we consider   δ = μ =   λ = 1 ; b 0 in Equation (53), we can obtain the following new integral formulae in view of column 6 of Table 1:
1 2 π i c i c + i Γ ( s ) Γ ( w s ) Φ b ( z , s , a )   Φ b ( z , w s , a ) d s =   0 t w 1 e 2 a t   2 b t ( 1 z e t ) 2 d t = 2 b Γ ( ω 2 ) Φ 2 b ( z , ω 2 , 2 a 1 ) + Γ ( ω ) [ Φ 2 b ( z , ω 1 , 2 a 1 ) ( 2 a 1 ) Φ 2 b ( z , w , 2 a 1 ) ] z
1 2 π i c i c + i Γ ( s ) Γ ( w s ) Li b ( z , s )   L i b ( z , w s ) d s =   0 t w 1 e 2 t   2 b t ( 1 z e t ) 2 d t = 2 b Γ ( ω 2 ) Li 2 b ( z , ω 2 ) + Γ ( ω ) [ Li 2 b ( z , ω 1 ) Li 2 b ( z , w ) ] z
1 2 π i c i c + i Γ ( s ) Γ ( w s ) ζ b ( s , a )   ζ b ( w s , a ) d s =   0 t w 1 e 2 a t   2 b t ( 1 e t ) 2 d t = 2 b Γ ( ω 2 ) ζ 2 b ( ω 2 , 2 a 1 ) + Γ ( ω ) [ ζ 2 b ( ω 1 , 2 a 1 ) ( 2 a 1 ) ζ 2 b ( w , 2 a 1 ) ]
1 2 π i c i c + i Γ ( s ) Γ ( w s ) ζ b ( s )   ζ b ( w s ) d s =   0 t w 1 e 2 t   2 b t ( 1 e t ) 2 d t = 2 b Γ ( ω 2 ) ζ 2 b ( ω 2 ) + Γ ( ω ) [ ζ 2 b ( ω 1 ) ζ 2 b ( w ) ] .
Similarly, by considering the different parameter values consistent with the results obtained in Section 2, one can obtain more integral formulae for the family of zeta and associated functions. One model is the following by means of Theorem 1:
1 2 π i c i c + i Γ ( s ) Γ ( w s ) Φ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , s , a , b ; λ ) Φ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , w s , a , b ; λ ) ds = ( μ 1 ) σ 1 . . ( μ q ) σ q z ( λ 1 ) ρ 1 . . ( λ p ) ρ p [ 2 b λ Φ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) Γ ( w λ 1 ) ( z , w λ 1 , 2 a 1 , 2 b ) + Γ ( ω ) [ Φ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , w 1 , 2 a 1 ; 2 b , λ ) ( 2 a 1 ) Φ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , w , 2 a 1 ; 2 b , λ ) ] ]
and for b = 0 , it leads to:
1 2 π i c i c + i Γ ( s ) Γ ( w s ) Φ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , s , a ) Φ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , w s , a ) ds = Γ ( ω ) ( μ 1 ) σ 1 . . ( μ q ) σ q z ( λ 1 ) ρ 1 . . ( λ p ) ρ p [ Φ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , w 1 , 2 a 1 ; 2 b , λ ) ( 2 a 1 ) Φ λ 1 , , λ p , μ 1 , , , μ q ( ρ 1 , , ρ p , σ 1 , , σ q ) ( z , w , 2 a 1 ; 2 b , λ ) ] .

5. Discussion and Future Directions

In this study, we obtained some recurrence relations for the newly defined family of the λ -generalized Hurwitz–Lerch zeta functions using the familiar Mellin transforms. These relations proved valuable to acquire new integral formulae involving the family of zeta functions. The outcomes were also confirmed with the previous obtained results as special cases. It is remarkable that the recurrence relations obtained in this research work are worthwhile to achieve simple relations such as Equations (34) and (40) that express special cases of λ -generalized Hurwitz–Lerch zeta functions in terms of Riemann zeta functions, so that we can evaluate the values of these functions. By following the method, we can obtain significant new results by considering the further specific values of the involved parameters. This is useful for the further analysis of these functions by plotting the graphs and deriving different series and asymptotic representations, etc. This work is in progress and would be a part of some future research.
λ -generalized Hurwitz–Lerch zeta functions analytically generalize the functions of the zeta family and offer consideration for some further presumable new members of this family that are not discussed in the literature. This aspect is most suitable for attaining new consequences from one key result. Our foremost results produce simultaneously important new results for a class of well-studied functions by applying the new difference equations. The Bose–Einstein and Fermi–Dirac functions are of fundamental importance in quantum statistics that contracts by means of two specific categories of spin symmetry, that is, fermions and bosons. Fitting together these functions here with the λ -generalized Hurwitz–Lerch zeta functions yields substantial new identities for them that provides clues regarding the forthcoming applications of these difference equations in quantum physics and associated fields. This practice to acquire the outcomes by making use of new difference equations explores the required simplicity that always inspires hope. We have discussed here the direct consequences of our results. It is remarked that the method established in this research is in fact noteworthy for the analysis and study of these higher transcendental functions.

Funding

This research was funded by Deanship of Scientific Research at Majmaah University, Project Number No. 1440-56.

Acknowledgments

The author would like to thank Deanship of Scientific Research at Majmaah University for supporting this work under Project Number No. 1440-56. The author is also thankful to the anonymous reviewers for their useful comments. They really improved the quality of this manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Srivastava, H.M. A new family of the λ-generalized Hurwitz-Lerch Zeta functions with applications. Appl. Math. Inf. Sci. 2014, 8, 1485–1500. [Google Scholar] [CrossRef]
  2. Srivastava, H.M.; Saxena, R.K.; Pogany, T.K.; Saxena, R. Integral and computational representations of the extended Hurwitz-Lerch Zeta function. Integral Transforms Spec. Funct. 2011, 22, 487–506. [Google Scholar] [CrossRef]
  3. Srivastava, H.M.; Jankov, D.; Pogany, T.K.; Saxena, R.K. Two-sided inequalities for the extended Hurwitz-Lerch Zeta function. Comput. Math. Appl. 2011, 62, 516–522. [Google Scholar] [CrossRef]
  4. Luo, M.-J.; Raina, R.K. New Results for Srivastava’s λ-Generalized Hurwitz-Lerch Zeta Function. Filomat 2017, 31, 2219–2229. [Google Scholar] [CrossRef]
  5. Kilbas, A.A.; Saigo, M. H-Transforms: Theory and Applications; Chapman and Hall (CRC Press Company): Boca Raton, FL, USA; London, UK; New York, NY, USA; Washington, DC, USA, 2004. [Google Scholar]
  6. Tassaddiq, A. A new representation of the k-gamma functions. Mathematics 2019, 10, 133. [Google Scholar] [CrossRef]
  7. Mathai, A.M.; Saxena, R.K. Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences; Lecture Notes in Mathematics; Springer-Verlag: Berlin/Heidelberg, Germany; New York, NY, USA, 1973. [Google Scholar]
  8. Mathai, A.M.; Saxena, R.K.; Haubold, H.J. The H-Function: Theory and Applications; Springer: New York, NY, USA; Dordrecht, The Netherlands; Heidelberg, Germany; London, UK, 2010. [Google Scholar]
  9. Olver, F.W.J.; Lozier, D.W.; Boisvert, R.F.; Clark, C.W. (Eds.) NIST Handbook of Mathematical Functions; [With 1 CD-ROM (Windows, Macintosh and UNIX)]; U.S. Department of Commerce, National Institute of Standards and Technology: Washington, DC, USA, 2010.
  10. Jankov, D.; Pogany, T.K.; Saxena, R.K. An extended general Hurwitz-Lerch Zeta function as a Mathieu (a, λ)-series. Appl. Math. Lett. 2011, 24, 1473–1476. [Google Scholar]
  11. Srivastava, H.M.; Ghanim, F.; El-Ashwah, R.M. Inclusion properties of certain subclass of univalent meromorphic functions defined by a linear operator associated with the -generalized Hurwitz-Lerch.zeta function. Asian-Eur. J. Math. 2017, 3, 34–50. [Google Scholar]
  12. Srivastava, H.M.; Gaboury, S. New expansion formulas for a family of the λ-generalized Hurwitz-Lerch zeta functions. Int. J. Math. Math. Sci. 2014, 2014, 131067. [Google Scholar] [CrossRef] [PubMed]
  13. Srivastava, H.M.; Gaboury, S.; Ghanim, F. Certain subclasses of meromorphically univalent functions defined by a linear operator associated with the λ-generalized Hurwitz-Lerch zeta function. Integral Transforms Spec. Funct. 2015, 26, 258–272. [Google Scholar] [CrossRef]
  14. Srivastava, H.M.; Gaboury, S.; Ghanim, F. Some further properties of a linear operator associated with the λ-generalized Hurwitz-Lerch zeta function related to the class of meromorphically univalent functions. Appl. Math. Comput. 2015, 259, 1019–1029. [Google Scholar]
  15. Srivastava, H.M. Some properties and results involving the zeta and associated functions. Funct. Anal. Approx. Comput. 2015, 7, 89–133. [Google Scholar]
  16. Choi, J.; Parmar, R.K. An Extension of the Generalized Hurwitz-Lerch Zeta Function of Two Variables. Filomat 2017, 31, 91–96. [Google Scholar] [CrossRef]
  17. Srivastava, H.M.; Jolly, N.; Kumar, B.; Manish Jain, R. A new integral transform associated with the λ-extended Hurwitz–Lerch zeta function. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Series A Matemáticas 2018. [Google Scholar] [CrossRef]
  18. Tassaddiq, A. A New Representation for Srivastava’s λ-Generalized Hurwitz-Lerch Zeta Functions. Symmetry 2018, 10, 733. [Google Scholar] [CrossRef]
  19. Abramowitz, M.; Stegun, I.A. (Eds.) HandbooK of Mathematical Functions with Formulas, Graphs, and Mathematical Tables; Applied Mathematics Series; National Bureau of Standards: Washington, DC, USA, 1964; Volume 55, Reprinted by Dover Publications, New York, NY, USA, 1965. [Google Scholar]
  20. Apostol, T.M. Introduction to Analytic Number Theory; Springer-Verlag: Berlin/Heidelberg, Germany; New York, NY, USA, 1976. [Google Scholar]
  21. Chaudhry, M.A.; Qadir, A.; Tassaddiq, A. A new generalization of the Riemann zeta function. Adv. Differ. Equ. 2011, 2011. [Google Scholar] [CrossRef]
  22. Titchmarsh, E.C. The Theory of the Riemann Zeta Function; Oxford University Press: Oxford, UK, 1951. [Google Scholar]
  23. Raina, R.K.; Srivastava, H.M. Certain results associated with the generalized Riemann zeta functions. Revista Tecnica de la Facultad de Ingenieria Universidad del Zulia 1995, 18, 301–304. [Google Scholar]
  24. Srivastava, H.M. Some formulas for the Bernoulli and Euler polynomials at rational arguments. Math. Proc. Camb. Philos. Soc. 2000, 129, 77–84. [Google Scholar] [CrossRef]
  25. Srivastava, H.M. Generating relations and other results associated with some families of the extended Hurwitz-Lerch Zeta functions. SpringerPlus 2013, 2, 67. [Google Scholar]
  26. Srivastava, H.M.; Choi, J. Series Associated with the Zeta and Related Functions; Kluwer Academic Publishers: Dordrecht, The Netherlands; Boston, MA, USA; London, UK, 2001. [Google Scholar]
  27. Srivastava, H.M.; Choi, J. Zeta and q-Zeta Functions and Associated Series and Integrals; Elsevier Science Publishers: Amsterdam, The Netherlands; London, UK; New York, NY, USA, 2012. [Google Scholar]
  28. Srivastava, H.M.; Gupta, K.C.; Goyal, S.P. The H-Functions of One and Two Variables with Applications; South Asian Publishers: New Delhi, India; Madras, India, 1982. [Google Scholar]
  29. Srivastava, H.M.; Karlsson, P.W. Multiple Gaussian Hypergeometric Series; Halsted Press (Ellis Horwood Limited): Chichester, UK; John Wiley and Sons: New York, NY, USA; Chichester, UK; Brisbane, Australia; Toronto, ON, Canada, 1985. [Google Scholar]
  30. Srivastava, H.M.; Manocha, H.L. A Treatise on Generating Functions; Halsted Press (Ellis Horwood Limited): Chichester, UK; John Wiley and Sons: New York, NY, USA; Chichester, UK; Brisbane, Australia; Toronto, ON, Canada, 1984. [Google Scholar]
  31. Srivastava, H.M. Some generalizations and basic (or q-) extensions of the Bernoulli, Euler and Genocchipolynomials. Appl. Math. Inform. Sci. 2011, 5, 390–444. [Google Scholar]
  32. Tassaddiq, A. Some Representations of the Extended Fermi-Dirac and Bos-Einstein Functions with Applications. Ph.D. Dissertation, National University of Sciences and Technology Islamabad, Islamabad, Pakistan, 2011. [Google Scholar]
  33. Tassaddiq, A.; Qadir, A. Fourier transform representation of the extended Fermi-Dirac and Bose-Einstein functions with applications to the family of the zeta and related functions. Integral Transforms Spec. Funct. 2011, 22, 453–466. [Google Scholar] [CrossRef]
  34. Tassaddiq, A. A New Representation of the Extended Fermi-Dirac and Bose-Einstein Functions. Int. J. Math. Appl. 2017, 5, 435–446. [Google Scholar]
  35. Gaboury, S.; Bayad, A. Series representations at special values of generalized Hurwitz-Lerch zeta function. Abstr. Appl. Anal. 2013, 2013, 975615. [Google Scholar] [CrossRef]
  36. Garg, M.; Jain, K.; Kalla, S.L. A further study of general Hurwitz-Lerch zeta function. Algebras Groups Geom. 2008, 25, 311–319. [Google Scholar]
  37. Garg, M.; Jain, K.; Srivastava, H.M. Some relationships between the generalized Apostol-Bernoulli polynomials and Hurwitz-Lerch Zeta functions. Integral Transforms Spec. Funct. 2006, 17, 803–815. [Google Scholar] [CrossRef]
  38. Gupta, P.L.; Gupta, R.C.; Ong, S.-H.; Srivastava, H.M. A class of Hurwitz-Lerch Zeta distributions and their applications in reliability. Appl. Math. Comput. 2008, 196, 521–531. [Google Scholar] [CrossRef]
  39. Lin, S.-D.; Srivastava, H.M. Some families of the Hurwitz-Lerch Zeta functions and associated fractional derivative and other integral representations. Appl. Math. Comput. 2004, 154, 725–733. [Google Scholar] [CrossRef]
  40. Lin, S.-D.; Srivastava, H.M.; Wang, P.-Y. Some expansion formulas for a class of generalized Hurwitz-Lerch Zeta functions. Integral Transforms Spec. Funct. 2006, 17, 817–827. [Google Scholar] [CrossRef]
  41. Raina, R.K.; Chhajed, P.K. Certain results involving a class of functions associated with the Hurwitz zeta function. Acta Math. Univ. Comen. 2004, 73, 89–100. [Google Scholar]
  42. Srivastava, H.M.; Luo, M.-J.; Raina, R.K. New results involving a class of generalized Hurwitz-Lerch zeta functions and their applications. Turk. J. Anal. Number Theory 2013, 1, 26–35. [Google Scholar] [CrossRef]
  43. Bayad, A.; Chikhi, J. Reduction and duality of the generalized Hurwitz-Lerch zetas. Fixed Point Theory Appl. 2013, 2013, 82. [Google Scholar] [CrossRef]
  44. Erdelyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F.G. Higher Transcendental Functions; McGraw-Hill Book Company: New York, NY, USA; Toronto, ON, Canada; London, UK, 1953. [Google Scholar]
  45. Srivastava, H.M.; Chaudhry, M.A.; Qadir, A.; Tassaddiq, A. Some extensions of the Fermi-Dirac and Bose-Einstein functions with applications to zeta and related functions. Russ. J. Math. Phys. 2011, 18, 107–121. [Google Scholar] [CrossRef]
  46. Chaudhry, M.A.; Zubair, S.M. On a Class of Incomplete Gamma Functions with Applications; Chapman and Hall (CRC Press Company): Boca Raton, FL, USA; London, UK; New York, NY, USA; Washington, DC, USA, 2001. [Google Scholar]
  47. Choi, J. Explicit formulas for Bernoulli polynomials of order n. Indian J. Pure Appl. Math. 1996, 27, 667–674. [Google Scholar]
  48. Lippert, R.A. A probabilistic interpretation of the Hurwitz zeta function. Adv. Math. 1993, 97, 278–284. [Google Scholar]
  49. Saxena, R.K.; Pogany, T.K.; Saxena, R.; Jankov, D. On generalized Hurwitz-Lerch Zeta distributions occurring in statistical inference. Acta Univ. Sapientiae Math. 2011, 3, 43–59. [Google Scholar]
  50. Zayed, A.I. Handbook of Functions and Generalized Function Transforms; CRC Press: Boca Raton, FL, USA, 1996. [Google Scholar]
Table 1. Different special cases of λ-generalized Hurwitz–Lerch zeta functions [18].
Table 1. Different special cases of λ-generalized Hurwitz–Lerch zeta functions [18].
min [ ( a ) , ( s ) ] > 0 ; ( b ) 0 ; λ 0 ;
ρ = ρ 1 , , ρ p ; σ = σ 1 , , σ q   ;   λ * = λ 1 , , λ p ; μ = μ 1 , , , μ q
( p 1 = q = 0 ;   λ 1 = μ ;   ρ 1 = 1 ) ( p 1 = q = 0 ;   λ 1 = μ ;   ρ 1 = 1 )
λ = 1 μ = 1   λ = μ = 1   b = 0   b = 0   μ = 1 ; b = 0  
λ-Generalized Hurwitz–Lerch Zeta Functions Φ   λ * , μ ( ρ ; σ ) ( ± z , s , a ; b , λ )
[1], (p. 1487), Equation (1.14)
Θ μ λ ( z , s , a ; b )
[41], (p. 90), Equation (6) and [42]
Φ μ * ( ± z , s , a , b ) Φ b ( ± z , s , a , λ ) Φ b ( ± z , s , a ) Φ   λ * , μ ( ρ ; σ ) ( ± z , s , a )
([1], p. 1486, Equation (1.11)) & [2]
Φ μ * ( ± z , s , a )
([43], p. 100, Equation (1.5))
Φ ( ± z , s , a ) ([44], p. 27, Equation (1.11))
Φ   λ * , μ ( ρ ; σ ) ( ± e x , s , a ; b , λ ) λ-Generalized Extended Fermi–Dirac and Extended Bose–Einstein Functions Θ   λ * , μ ( ρ ; σ ) ( x , s , a ; b , λ ) Θ μ λ ( x , s , a ; b ) Θ μ * ( x , s , a , b ) Θ b ( x , s , a ; λ ) Θ b ( x , s , a ) Θ   λ * , μ ( ρ ; σ ) ( x , s , a ) Θ μ * ( x , s , a )
([43], p. 12, Equation (45))
Θ a ( x ; s ) ([45], p. 9, Equation (3.14))
Ψ   λ * , μ ( ρ ; σ ) ( x , s , a ; b , λ ) Ψ μ λ ( x , s , a ; b ) Ψ μ * ( x , s , a , b ) Ψ b ( x , s , a , λ ) Ψ b ( x , s , a ) Ψ   λ * , μ ( ρ ; σ ) ( x , s , a ) Ψ μ * ( x , s , a )
([43], p. 12, Equation (45))
Ψ a ( x ; s ) [45], p. 115, Equation (4.4)
Φ   λ * , μ ( ρ ; σ ) ( ± z , s , 1 ; b , λ )
λ-Generalized Polylogarithm Functions
Li   λ * , μ ( ρ ; σ ) ( ± z , s ; b , λ ) Li μ λ ( z , s , a ; b ) Li μ * ( z , s , b ) Li b ( z , s , λ ) Li b ( z , s ) Li   λ * , μ ( ρ ; σ ) ( z , s ) Li μ * ( z , s )
([43], p. 12, Equation (47))
Li s ( z )
[44], (Chapter 1)
Φ   λ * , μ ( ρ ; σ ) ( ± e x , s + 1 , 1 ; b , λ )
λ-Generalized Fermi–Dirac and Bose–Einstein Functions
F   λ * , μ ( ρ ; σ ) ( x , s ; b , λ ) F μ λ ( x , s , a ; b ) F μ * ( x , s , b ) F b ( x , s , λ ) F b ( x , s ) F   λ * , μ ( ρ ; σ ) ( x , s ) F μ * ( x , s )
([43], p. 12, Equation (47))
F s ( x ) ([45], p. 109, Equation (1.12)
B   λ * , μ ( ρ ; σ ) ( x , s ; b , λ ) B μ λ ( x , s , a ; b ) B μ * ( x , s , b ) B b ( x , s , λ ) B b ( x , s ) B   λ * , μ ( ρ ; σ ) ( x , s ) B μ * ( x , s )
([43], p. 12, Equation (45))
B s ( x ) ([45], p. 109, Equation (1.12))
Φ   λ * , μ ( ρ ; σ ) ( ± 1 , s , a ; b , λ )
λ-Generalized Hurwitz zeta Functions
ζ   λ * , μ ( ρ ; σ ) ( s , a ; b , λ ) ζ μ λ ( s , a ; b ) ζ μ * ( s , a , b ) ζ b ( s , a , λ ) ζ b ( s , a )
[46], p. 308
ζ   λ * , μ ( ρ ; σ ) ( s , a ) ζ μ * ( s , a ) [43] ζ ( s , a )
[44], (Chapter 1)
Φ   λ * , μ ( ρ ; σ ) ( ± 1 , s , 1 ; b , λ )
λ-Generalized Riemann Zeta Functions
ζ   λ * , μ ( ρ ; σ ) ( s ) ζ μ λ s ; b ) ζ μ * ( s , b ) ζ b ( s , λ ) ζ b ( s )
[46], p. 308
ζ   λ * , μ ( ρ ; σ ) ( s ) ζ μ * ( s ) [43] ζ ( s )
[44], (Chapter 1)
Table 2. Computation of 0 t s 1 e 3 t ( 1 e t ) 3 d t .
Table 2. Computation of 0 t s 1 e 3 t ( 1 e t ) 3 d t .
sDirect Evaluation by MathematicaUsing difference Equation (34)
4 0.610229 0.610229
30 4.29669 × 10 16 4.29669 × 10 16
40 1.67783 × 10 27 1.67783 × 10 27
45 8.998 × 10 32 8.998 × 10 32
46 1.3497 × 10 34 1.3497 × 10 34
48 3.24227 × 10 36 3.24227 × 10 36
48.5 1.29353 × 10 37 1.29353 × 10 37
48.9 3.92781 × 10 37 3.92781 × 10 37
49 5.18762 × 10 37 5.18762 × 10 37
52 2.40071 × 10 41 2.40071 × 10 41
56 2.426 × 10 46 2.426 × 10 46
160Unable to compute 1.349   × 10 206
220Unable to compute 1.121 × 10 314
400Unable to compute 2.269   × 10 675
Table 3. Computation of 0 t s 1 e 4 t ( 1 e t ) 4 d t .
Table 3. Computation of 0 t s 1 e 4 t ( 1 e t ) 4 d t .
sDirect Evaluation by MathematicaBy using difference Equation (40)
90 1.07719 × 10 82 1.07719 × 10 82
100 5.8077 × 10 95 5.8077 × 10 95
140Unable to compute 1.78191 × 10 156
160Unable to compute 1.37955 × 10 186

Share and Cite

MDPI and ACS Style

Tassaddiq, A. Some Difference Equations for Srivastava’s λ-Generalized Hurwitz–Lerch Zeta Functions with Applications. Symmetry 2019, 11, 311. https://doi.org/10.3390/sym11030311

AMA Style

Tassaddiq A. Some Difference Equations for Srivastava’s λ-Generalized Hurwitz–Lerch Zeta Functions with Applications. Symmetry. 2019; 11(3):311. https://doi.org/10.3390/sym11030311

Chicago/Turabian Style

Tassaddiq, Asifa. 2019. "Some Difference Equations for Srivastava’s λ-Generalized Hurwitz–Lerch Zeta Functions with Applications" Symmetry 11, no. 3: 311. https://doi.org/10.3390/sym11030311

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop