A symbolic approach to multiple Hurwitz zeta values at non - positive integers

: In this article, we give another method to calculate the values of multiple Hurwitz zeta function at non - positive integers by means of Raabe ’ s formula and the Bernoulli numbers and we simplify these values by symbolic computation techniques.


Introduction
for ≤ ≤ k n 1 (see [1]). In recent years, multiple Hurwitz zeta functions have attracted wide attention from many scholars. They are not only important for the general zeta function theory but also appear in different mathematical fields, for example, but not limited, the theory of special functions (see [2]), holomorphic dynamics (see [3]), renormalization theory (see [4]), knot theory (see [5]), and quantum field theory (see [6]).
In [7], Katsurada and Matsumoto proved that for = n 2, the series (1.1) can be continued meromorphically to the whole space 2 . For the general case n, the meromorphic continuation of (1.1) for general r was established by Akiyama and Ishikawa [8].
For ≔ α 1, the series (1.1) becomes the multiple zeta function, which has been studied by many researchers (see [5,9],…). When = n 2, (1.1) is the classical Euler sum, and its analytic continuation was first obtained by Atkinson [10]. In the case > n 1, the analytic continuation of (1.1) was proved by Zhao [11] and proved independently by Akiyama and Ishikawa [8]. For more details on the history of the problem of the series (1.1), the readers are referred, without limitation, to ( [1,12,13],…).
In this article, we give the values at the non-positive integers of a type of the multiple Hurwitz zeta function (see (3.1)) by the use of the "Raabe formula",¹ which expresses an integral in terms of sum. Encouraged by the simplification that symbolic computation may bring to the manipulation of complicated sums, we use here symbols which is simply a scaled version of that introduced by Gessel [16]; these symbols simplify these values and give a general recursion of this series on their depth.

The symbols and 2.1 The generalized Bernoulli symbols
In what follows, we introduce the generalized Bernoulli symbol (see, e.g., [16,17]), with the evaluation rule, for all { } ∈ ⧹ α 0 and for all ∈ n , we have:  where, for all ≤ ≤ + j k 1 1 , the symbol j is defined by The expressions can be expanded to obtain formulas involving only generalized Bernoulli symbols k . For example, the term For more details of these symbols, the reader can see [18].

The main results
For real numbers ∈ α , such that: ≠ − − … α 0, 1, 2, , and complex n-tuples , we define the following type of the multiple Hurwitz zeta function ; , , and the corresponding integral function associated with the multiple Hurwitz zeta function is defined by Remark 3.1. We remark that, if = α 1, then the series (3.1) is corresponding to the multiple zeta function.
We first give the well-known elementary result for the integral function.    So, we have be a point of n , then we have for ∈ + a n ( ) ( )  Finally, by using the Raabe formula we give the main result for the multiple Hurwitz zeta function.  3.1 Application of Theorem 1 in the case = n 1 and = α 1 In this section, we give an application of our main result for = n 1 and = α 1. We have which is the classical zeta function.
For ∈ + a we set Thus, for all ∈ N : Then, Proposition 5.2 of Section 5 shows that where B k is the kth Bernoulli number. Now, we recall the elementary result Finally, we obtain the known result and equations (2.1) and (2.12) give the formula of Theorem 1:

Values of the integral representation of the multiple Hurwitz zeta function
This section is devoted to the proof of Proposition 3.1. Let the integral function If we use the following change of variables: Now, using the following change of variables: . This change gives and we find This integral can be rewritten as follows:  if and only if R( ) − > s 1 0 n . Inductively on n, we find  Therefore, for any point if and only if there exists a

An intermediate estimate
In this section, we show Proposition 3.2.
Let ∈ + a n , such that for all and for all ≤ ≤ i n 1 the relation (4.3) shows that This integral can be written as follows:  Setting We will simplify these values and observe that  which ends the proof of Proposition 3.2.

Raabe formula
In this section, we proceed to the proof of the main Theorem 1. The proof relies on the Raabe formula [14], which expresses the integral in terms of the sum.

Proof.
(1) Let ∈ s n be chosen in such a way that the integral function and the multiple Hurwitz zeta function are absolutely convergent. Thus, for ∈ + t n , we have    More generally, for is a product of Bernoulli numbers.
Proof. It follows from the above lemma, with ( ) . □

Values of the multiple Hurwitz zeta function at non-positive integers
In this section, we give the proof of our main result (Theorem 1). Relation (4.21) shows that for all ∈ + a n ( ) ( )  Bernoulli number. Now, using the generalized Bernoulli symbols, we find a a a n a n a a a a n a a n 1 , Thus, we proved the following theorem. and a a a n a a n , , ,