Abstract

The purpose of this paper is to introduce and investigate a new class of multiple zeta functions of variables. We study its properties, integral representations, differential relation, series expansion and discuss the link with known results.

1. Introduction

The generalized (Hurwitz’s) zeta function is defined by ^{1, 2}

(1.1)

so that when , we have

(1.2)

where is the Riemann zeta function. The function extends (1.1) further, and this generalized Hurwitz-Lerch zeta function [ ¹, p. 316], is defined by

(1.3)

A generalization of (1.3) is the Zeta function which is defined by [ ³, p.100, (1.5)]:

(1.4)

where for denotes the Pochhammer’s symbol and denotes the Gamma function. Evidently, we have

(1.5)

(1.6)

(1.7)

The zeta functions in (1.3) and (1.4) have since been extended and generalized by a number of workers (see e.g. ^{3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}. The present sequel to these earlier papers is motivated largely by the aforementioned works of Matsumoto and Kamano ^{15, 16} in which the zeta function in (1.1) was generalized to the following multiple Hurwitz zeta function

(1.8)

In the present paper we introduce a new class of zeta functions which is defined by

(1.9)

Clearly, we have the following relationship

In the case when we have simply

which implies the next result.

Corollary 1.1. Let

Then

(1.10)

where is the Lauricella function of variables defined by the series (see e.g. ¹⁷ and ¹⁸),

(1.11)

By using the Hankel's contour integral for Gamma function ( for details see e.g. [ ¹⁹, Section 12.12])

(1.12)

we can derive the following interesting formula.

Corollary 1.2. Let Then

(1.13)

Proof. The result follow directly from definition (1.9) and the integral representation (1.11).

2. Integral Representations

First, by using Eulerian integral formula of the second kind (see e.g. ²):

(2.1)

and the formula(see, e.g., [ ²⁰, p.67, Eq (22)]) :

(2.2)

we should proof the following result.

Theorem 2.1. Let and either

Then

(2.3)

Proof. Denote, for convenience, the right-hand side of equation (2.3) by I. Then in view of (2.2), it is easily seen that

Upon using (2.1) and in view of the definition (1.9), we led finally to the left-hand side of the formula (2.3). Next, by using the contour integral formula [ ², p.14 (4)]:

(2.4)

one can derive the following contour integral representation.

Theorem 2.2. Let and

Then

(2.5)

Proof. Denote, for convenience, the right-hand side of equation (2.5) by I. Then in view of (2.2), it is easily seen that

Upon using (2.4) and in view of the definition (1.9) with the reflection formula

for Gamma function, we led finally to the left-hand side of the formula (2.5).

Further, we evaluate some definite integrals involving the function First, we recall the Eulerian integral formula of first kind (cf. e.g ¹²):

(2.6)

From the term-by-term integration, we can derive the following formula.

Theorem 2.3. Let

(2.7)

Proof. Denote for convenience the left-hand side of equation (2.7) by I.

Then in view of definition (1.9), it is easily seen that

Upon using (2.6) and the relation

we are finally led to the right-hand of the relation (2.7).

Finally, we prove the following result.

Theorem 2.4. Let Then

(2.8)

Proof. Setting and

in the Eulerian Beta function formula (see, e.g., [ ²⁰, p.8, Eq. (45)]):

we find that

which, by appealing to the definition (1.9), immediately yields the assertion (2.8).

3. Operational Relations

The multiple Zeta function satisfies some operational relations. Fortunately these properties of be developed directly from the definition (1.9). First, by recalling the familiar derivative formula from calculus in terms of the gamma function ⁸

(3.1)

where we aim now to derive the following differential relations for

Theorem 3.1. Let Then

(3.2)

Proof. By starting from the left-hand side of (3.2) and in view of (1.9) and by using the relation (3.1), we get:

(3.3)

Now, letting in (3.3) using the formula and considering the definition (1.9), we get the right-hand side of formula (3.2).

Theorem 3.2. Let

Then

(3.4)

Proof. By starting from the left-hand side of (3.4) and in view of (1.9) and by using the relation

we get:

Now, considering the definition (1.9), we get the right-hand side of formula (3.4).

Closely associated with the derivative of the gamma function is the digamma function defined by [ ²¹, p.74(2.51)]:

(3.5)

Now, we wish to establish the derivative of the function with respect to the parameters

Theorem 3.3. Let Then

(3.6)

Proof. By starting from the left-hand side of formula (3.6) and by using the relation (3.5), then according to the result by using:

(3.7)

we obtain the right-hand side of formula (3.6).

Next, let us recall the definition of the Weyl fractional derivative of exponential function of order in the form (see [ ²², p.248(7.4)] ):

(3.8)

We now proceed to find the fractional derivative of the function with respect to .

Theorem 3.4. Let Then

(3.9)

Proof. Since we have

The desired result now follows by applying the formula (3.8) to the above identity.

4. Series Expansions

First we derive the following basic sums of series

Theorem 4.1. Let . Then

(4.1)

Proof. If in formula (1.9), we replace by multiply throughout by and then sums up, we get (4.1).

Further, from definition (1.9) we easily have the following interesting series relation.

Theorem 4.2. Let Then

(4.2)

(4.3)

Proof. Since

it is easily seen that :

The desired result (4.2) now follows by using definition (1.9). Also, by right-hand formula (4.3) and using (3.4) we get to left-hand side of formula (4.3).

Theorem 4.3. Let Then

(4.4)

where

(4.5)

where

(4.6)

where and

(4.7)

where are the Lauricella's hypergeometric functions of -variables (see [ ¹⁸, p.60, Eq (1), (2), (3), (4)]).

Proof. We refer to the proof of Theorem 4.2.

Theorem 4.4. Let Then

(4.8)

Proof. By starting from the right-hand side of (4.8) and in view of (1.9) and by using the relation

we get the left-hand side of formula (4.8).

Theorem 4.5. Let and Then

(4.9)

where is the Appell's function of two variables defined by the series ¹⁸

Proof. By starting from the right-hand side of (4.9) and in view of (1.9), we get

Since

Hence, the right-hand side of formula (4.9) follows.

References