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The multiple gamma functions and the multiple q-gamma functions

We give an asymptotic expansion (the higher Stirling formula) and an infinite product representation (the Weierstrass product representation) of the Vign\'{e}ras multiple gamma functions by considering the classical limit of the multiple q-gamma functions.


Introduction
The multiple gamma function was introduced by Barnes. It is defined to be an infinite product regularized by the multiple Hurwitz zeta functions [2], [3], [4], [5]. After his discovery, many mathematicians have studied this function: Hardy [7], [8] studied this function from his viewpoint of the theory of elliptic functions, and Shintani [20], [21] applied it to the study on the Kronecker limit formula for zeta functions attached to certain algebraic fields.
In the end of 70's, Vignéras [24] redefined the multiple gamma function to be a function satisfying the generalized Bohr-Morellup theorem, Furthermore, Vignéras [24], Voros [25], Vardi [23] and Kurokawa [12], [13], [14], [15] showed that it plays an essential role to express gamma factors of the Selberg zeta functions of compact Riemann surfaces and the determinants of the Laplacians on some Riemannian manifolds.
As we can see from these studies, the multiple gamma functions are fundamental for the analytic number theory: See also [16], [17]. However we do not think that the theory of the multiple gamma functions has been fully explored.
On the other hand, the second author of this paper introduced a q-analogue of the Vignéras multiple gamma functions and showed it to be characterized by a q-analogue of the generalized Bohr-Morellup theorem [24].
2.4. The Vignéras multiple gamma function. As a generalization of the gamma function and the G-function, Vignéras [24] introduced a hierarchy of functions which she called "the multiple gamma functions".
Theorem 2.1. There exists a unique hierarchy of functions which satisfy (2) G n (1) = 1, Applying Dufresnoy and Pisot's results [6], she showed that these functions satisfying the above properties are uniquely determined and that G n (z + 1) has an infinite product representation The Vignéras multiple gamma function can be regarded as a special case of the Barnes multiple gamma function. Namely G n (z) = Γ n (z, (1, 1 · · · , 1)) (−1) n−1 × (the normalization factor).
In this paper, we will use the word " the multiple gamma function" to refer the Vignéras multiple gamma function.
2.5. The q-gamma function. Throughout this paper, we suppose 0 < q < 1. A q-analogue of the gamma function was introduced by Jackson [9], [10].
Inspired by Moak's works [18], the authors [22] derived a representation of the q-gamma function denotes the Gauss symbol i.e. the integral part of t), and the polynomial M n (x) is defined by the recurrence relation Each term of the formula (2.5) converges uniformly as q → 1−0. So we get another proof of the uniformity of the classical limit of log Γ(z; q).
2.6. The multiple q-gamma function. Recently, one of the authors [19] constructed the function G n (z; q) which satisfies a q-analogue of the generalized Bohr-Morellup theorem: There exists a unique hierarchy of functions which satisfy (1) G n (z + 1; q) = G n−1 (z; q)G n (z; q), (2) G n (1; q) = 1, G 0 (z; q) = [z] q . 6 We call it "the multiple q-gamma function". It is given by the following infinite product representation [19] G n (z + 1; q) : for n ≥ 1, where In the next section, we will derive a representation of the multiple q-gamma function like (2.5) and consider its classical limit. This limit formula gives some important properties of the multiple gamma functions.
3. The Euler-MacLaurin expansion of G n (z + 1; q) By means of the Euler-MacLaurin summation formula we give an expansion formula of the multiple q-gamma functions which we call the Euler-MacLaurin expansion [22]. This formula plays an important role in the following sections.
The polynomial G n,j (z) is introduced through Proof. From (2.7) and the definition of G n,j (z), we obtain Applying the Euler-MacLaurin summation formula, we have Let L r (z) and L r be where Li r (z) is Euler's polylogarithm where n S j is the Stirling number of the first kind defined by L r (z) and L r cause divergence as q → 1 − 0, but we can prove that these divergent terms vanish. In order to simplify such terms, let us express L r (z) as the sum of L r and convergent terms.
Proof. By partial integration, we have Then (3.6) reads In order to write L n+1 (z) with L n+1 and T k (z), we use induction on n. It can be seen that which is equivalent to the claim of Lemma 3.2.
Applying (3.4), (3.5) and Lemma 3.2 to (3.1), we obtain We prove that the coefficients of the divergent terms in (3.8) vanish. By the following lemma, we can see that the first term and the second term in (3.8) are canceled out.
Proof. We consider the exponential generating functions of the coefficients of L r+2 in both sides. The generating function of the left hand side is On the other hand, the generating function of the right hand side is Therefore, the coefficients of L r+2 in both sides coincide. Thus, the claim follows.
Next, we prove the coefficient of L 1 vanishes. Since We prove that the right hand side of the above formula is equal to zero. In a formal sense, Imposing the boundary condition at z = 0, we make the both sides above act on z−1 n−1 . Then, it can be seen that Thus we have the formula which shows that the coefficient of L 1 is equal to zero. Hence we have proved that the all coefficients of L r vanish in (3.8).
Finally, we calculate the coefficients of log 1−q z+1 1−q and T r (z + 1). Using the formula (3.9), we have In order to calculate the coefficients of T r (z + 1) in (3.8), we note that Using this identity, we have Substituting (3.10) and (3.11) in (3.8), we obtain Proposition 1.1.

4.
The classical limit of G n (z + 1; q) and the asymptotic expansion of G n (z + 1) In this section, we study the classical limit of G n (z + 1; q) using the Euler-MacLaurin expansion. We will see that this limit formula gives an asymptotic expansion for the multiple gamma functions, which is a generalization of the Stirling formula for the gamma function.
Furthermore this convergence is uniform on any compact set in {z ∈ C|ℜz > −1}.
Proof. Taking Proposition 3.1 into acount, we must show that and further have to show that this convergence is uniform. Here we prove only (4.4). The other formulas can be verified in a similar way. Since in order to show (4.4), it is sufficient to prove that the procedure of taking the classical limit commutes with the integration. Let us introduce polynomials M r (x) through [18]). They satisfy the recurrence relation.
Suppose −1 < y 0 ≤ ℜz ≤ y 1 . By Lemma 4.2, there exists a constant C depending only on y 0 and y 1 such that Since m − j ≥ 2, the right hand side above is integrable over 1 ≤ t < +∞. Therefore, Lebesgue's convergence theorem ensures (4.6). We have thus proved the limit formula (4.4). Next we show that this convergence is uniform on any compact set in the domain {z ∈ C|ℜz > −1}. Put From the consideration above, we see that there exists a constant C depending only on y 0 and y 1 such that dt t 2 for ℜz ∈ [y 0 , y 1 ] and 0 < q ≤ 1.
Hence {Φ(z, q)|0 < q ≤ 1} is a uniformly bounded family of functions and By Vitali's convergence theorem, this convergence is uniform on any compact set in the domain {z ∈ C|ℜz > −1}.
The constant C j in Proposition 4.1 can be expressed in terms of the special value of the Riemann zeta function.
Proof. From the definition of ζ(s), By the Euler-MacLaurin summation formula, we obtain (4.7) can be analytically continued to {z ∈ C|ℜz > −n + 1}. So if we put s = −j, n = j + 2, then we obtain Next we prove that the limit function in Proposition 4.1 coincides with the multiple gamma funciton.
Theorem 4.4. Suppose m > n. Then, as q → 1 − 0, G n (z + 1; q) converges to G n (z + 1) uniformly on any compact set in the domain C\Z <0 and log G n (z + 1) (4.8) Proof. In the domain {z ∈ C|ℜz > −1}, we have already proved the existence of the limit function and uniformity of the convergence. Let us putG Because of the uniformity of the convergence, we have particularly {log G n (z + 1; q)} = d dz n+1 logG n (z + 1) 19 so that, from Theorem 2.2,G n (z+1) satisfies the conditions in Theorem 2.1. Namely Since a hierarchy of such functions is uniquely determined, so G n+1 (z + 1) =G n (z + 1) in {z ∈ C|ℜz > −1}. Thus the claim of the theorem in the case that ℜz > −1 has been proved for {z ∈ C|ℜz > −1}.
Next, we show that in {z ∈ C|ℜz ≤ −1, z = −1}, G n (z + 1; q) → G n (z + 1) as q → 1 − 0 and that the convergence is uniform on any compact set in this domain.
For the proof, we use induction on n.

4.2.
Asymptotic expansion of G n (z + 1). Let us call the expression (4.8) the Euler-MacLaurin expansion of G n (z + 1). We should note that (4.8) is valid for z ∈ C\{ℜz ≤ −1}. We show that it gives an asymptotic expansion of G n (z + 1) as |z| → ∞, i.e. the higher Stirling formula.
Proof. Straightforward calculation shows that Thus, if r > n, then F n,r−1 (z) = O(z −r+n ) as |z| → ∞. So we can see that  Furthermore we can see that Noting that Hence, as |z| → ∞ in the sector.
Let us exhibit some examples of the higher Stirling formula. In the case that n = 1, we obtain log G 1 (z + 1) = log Γ(z + 1) This is the Stirling formula since ζ ′ (0) = − 1 2 log(2π). 22 Furthermore, in the case that n = 2, we obtain log G 2 (z + 1) which coincides with the formula (2.3). In the case that n = 3, 4, and 5, we have the following results.

5.
The Weierstrass product repesentation for the G n (z + 1) By calculating the formula (4.8) in the case of m = n + 1, we derive the Weierstrass product representation for the multiple gamma functions. Main theorem of this section is the following: Proof of this theorem will be carried out through the sections 5.1 ∼ 5.4, and some examples of this representation will be discussed in the section 5.5.

5.1.
Rewriting the Euler-MacLaurin expansion of G n (z + 1). In this section we prove the following proposition.
Furthermore the infinite sum of the last term is absolutely convergent.
Proof. From the definiftion of G n,r (z), we have and hence In order to calculate the coefficient of log(z + 1), we use the following lemma.
Proof. Since the both sides are polynomials in z, it is sufficient to prove that the formula holds for a sufficiently large, arbitrary integer N.
Noting that From (5.2) and Lemma 5.3, it follows that Next we calculate R n,n+1 (z). From the definition of G n,r (z), we have Here we have, for j ≤ n − 1, so that the infinite sum in (5.6) converges absolutely because j−n−1 ≤ −2.
By means of the Euler-MacLaurin summation formula, we have The coefficient of log(z + 1) of the above formula vanishes because of the following lemma.
Proof. The first equality follows from the identity The second equality holds because of the Euler-MacLaurin summation formula for (z + t + k) j on [0, 1].
The next lemma which will be useful in the section 5.3 is deduced from the above proof.
In other words, the polynomial P j (z + k + 1) − P j (z + k) is a convergent factor such that the infinite sum in (5.9) converges absolutely. 29 5.
2. An infinite product representation for the ζ ′ (−j). We derive an infinite product representation for ζ ′ (−j) in the same way as in the section 3.1.
exp (P j (k + 1) − P j (k))    and the infinite product converges absolutely.
A(k, z) is equal to P j (z + k + 1) − P j (z + k).
Proof. By integrating the both side of is uniquely determined and the polynomial P j (z + k + 1) − P j (z + k) satisfies (5.14) by Lemma 5.5. Therefore, A(k, z) = P j (z + k + 1) − P j (z + k).
Using these lemmas, we prove the following proposition.