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Kurokawa and Wakayama (Ramanujan J. 10:23–41, 2005) studied a family of elliptic functions defined by certain q-series. This family, in particular, contains the Weierstrass ℘-function. In this paper, we prove that elliptic functions in this family satisfy certain non-linear algebraic differential equations whose coefficients are essentially given by rational functions of the first few Eisenstein series of the modular group.

From the irreducible decompositions' point of view, the structure of the cyclic $GL_n$-module generated by the $\alpha$-determinant degenerates when $\alpha=\pm \frac1k (1\leq k\leq n-1)$. In this paper, we show that $-\frac1k$-determinant shares similar properties which the ordinary determinant possesses. From this fact, one can define a new (relative) invariant called a wreath determinant. Using $(GL_m, GL_n)$-duality in the sense of Howe, we obtain an expression of a wreath determinant by a certain linear combination of the corresponding ordinary minor determinants labeled by suitable rectangular shape tableaux. Also we study a wreath determinant analogue of the Vandermonde determinant, and then, investigate symmetric functions such as Schur functions in the framework of wreath determinants. Moreover, we examine coefficients which we call $(n,k)$-sign appeared at the linear expression of the wreath determinant in relation with a zonal spherical function of a Young subgroup of the symmetric group $S_{nk}$., Comment: 26 pages

金沢大学大学院自然科学研究科, The spectral zeta function for the so-called noncommutative harmonic oscillator is able to be meromorphically extended to the whole complex plane, having only one simple pole at the same point s = 1 where Riemann's zeta function ζ(s) has, and possesses a trivial zero at each nonpositive even integer. The essential part of its proof is sketched. A new result is also given on the lower and upper bounds of the eigenvalues of the noncommutative harmonic oscillator. © 2007 Polish Scientific Publishers PWN, Warszawa.

We give a Jacksonq-integral analogue of Euler’s logarithmic sine integral established in 1769 from several points of view, specifically from the one relating to the Hurwitz zeta function.

We study gamma and sine functions attached to a finite dimensional linear representation of a Lie group including their multiple versions via zeta regularizations. These definitions allow us to understand several multiple gamma and sine functions known so far, such as Barnes' gamma functions, Shintani's sine functions, and their q-analogues, in a unified way. Further, we discuss period integrals for such gamma functions.

We introduce certain families of elliptic functions involving the Weierstrass ℘-function via q-series systematically and investigate their analytic and algebraic properties. Especially, we give examples of non-linear algebraic ordinary differential equations satisfied by some such elliptic functions.

We introduce a Selberg type zeta function of two variables which interpolates several higher Selberg zeta functions. The analytic continuation, the functional equation and the determinant expression of this function via the Laplacian on a Riemann surface are obtained., Comment: 14 pages

This paper investigates the spectral zeta function of the non-commutative harmonic oscillator studied in [PW1, 2]. It is shown, as one of the basic analytic properties, that the spectral zeta function is extended to a meromorphic function in the whole complex plane with a simple pole at s=1, and further that it has a zero at all non-positive even integers, i.e. at s=0 and at those negative even integers where the Riemann zeta function has the so-called trivial zeros. As a by-product of the study, both the upper and the lower bounds are also given for the first eigenvalue of the non-commutative harmonic oscillator.

We discuss zeta extensions in the sense of Kurokawa and Wakayama, Proc. Japan Acad. 2002, for constructing new zeta functions from a given zeta function. This notion appeared when we introduced higher zeta functions such as higher Riemann zeta functions in Kurokawa et al., Kyushu Univ. Preprint, 2003, and a higher Selberg zeta functions in Kurokawa and Wakayama, Comm. Math. Phys., 2004. In this article, we first recall some explicit examples of such zeta extensions and give a conjecture about functional equations satisfied by higher zeta functions. We devote the second part to making a detailed study of the double sine functions which are treated in a framework of the zeta extensions.

The differential algebraicity of the multiple sine functions \(S_{r} (x, \underline{\omega})\) is investigated. The goal of the paper is to show the differential algebraicity of \(S_{r} (x, \underline{\omega})\) when the period \(\underline{\omega})\) is “rational” by using the classical theory due to Ostrowski.

We introduce various Ruelle type zeta functions ζL(s) according to a choice of homogeneous "length functions" for a lattice L in [Formula: see text] via Euler products. The logarithm of each ζL(s) yields naturally a certain arithmetic function. We study the asymptotic distribution of averages of such arithmetic functions. Asymptotic behavior of the zeta functions at the origin s=0 are also investigated.

In the paper [KW2] we introduced a new type of Selberg zeta function for establishing a certain identity among the non-trivial zeroes of the Selberg zeta function and of the Riemann zeta function. We shall call this zeta function a higher Selberg zeta function. The purpose of this paper is to study the analytic properties of the higher Selberg zeta function z Γ(s), especially to obtain the functional equation. We also describe the gamma factor of z Γ(s) in terms of the triple sine function explicitly and, further, determine the complete higher Selberg zeta function with having a discussion of a certain generalized zeta regularization.

We study analytic properties of spectral zeta functions associated to actions of the quantum group SUq(2) such as Z(s, SUq(2)), the zeta function corresponding to the regular representation introduced in [15]. As an application, we show the special value ζ(3) of the Riemann zeta function ζ(s) is given in terms of the classical limit of Z(s, SUq(2)). We further discuss a spectral zeta function [Formula: see text] associated with the so-called model of the representations of [Formula: see text] and show a presence of its series of "trivial" zeros, which is noteworthy.

We study certain higher order Euler products whose Euler factors are given by the polylogarithm.

We define a q-analogue of the basic multiple gamma function [Formula: see text] introduced in [17] which differs from the one defined by Barnes [1] via the zeta regularized product. We call it a q-basic multiple gamma function. Using this q-basic multiple Gamma function we introduce a q-analogue of the multiple sine function of order m + 1. We study properties of such functions from the periodicity, differential-difference equations and duplication formulas, etc. points of view.

The Casimir effect whose existence was first predicted by Casimir in 1948 is considered as a manifestation of macroscopic quantum field theory. This force is evaluated theoretically by using the value of the Riemann zeta function at −3. The aim of the present paper is to introduce a similar Casimir energy for a Riemann surface, and to express it by a special value of the Mellin transform of a theta series arising from the heat kernel and also by a weighted integral of the logarithm of the Selberg zeta function.

In this paper we establish a formula involving Pfaffians for a certain weighted sum of minors of an arbitrary given matrix. First we find a formula where the sum ranges for all columns, and secondly we obtain a formula where the sum ranges for both all rows and columns as the application of the first one. The first formula is stated in the framework of quantum matrix algebra (Matq(m,n)). These sums are weighted by "sub-Pfaffians" of any given (q)skew symmetric matrices. In the last section we provide an application of the minor summation formula to the generating functions of shifted tableaux.

A quantum analogue of the dual pair\(\mathfrak{s}\mathfrak{l}_2 ,\mathfrak{o}_n \) is introduced in terms of the oscillator representation of Uq\((\mathfrak{s}\mathfrak{l}_2 )\). Its commutant and the associated identity of Capelli type are discussed.

We calculate the absolute tensor product ζ(s,Fp1)⊗···⊗ζ(s,Fpr) by observing a relation between the multiple sine function of Shintani's type and Appell's O-function, or equivalently, the multiple elliptic gamma function. This allows us to give an Euler product expression for an absolute tensor product.

As a particular one parameter deformation of the quantum determinant, we introduce a quantum α-determinant det q ( α ) and study the U q ( gl n ) -cyclic module generated by it: We show that the multiplicity of each irreducible representation in this cyclic module is determined by a certain polynomial called the q-content discriminant. A part of the present result is a quantum counterpart for the result of Matsumoto and Wakayama [S. Matsumoto, M. Wakayama, Alpha-determinant cyclic modules of gl n ( C ) , J. Lie Theory 16 (2006) 393–405], however, a new distinguished feature arises in our situation. Specifically, we determine the degeneration of the multiplicities for ‘classical’ singular points and give a general conjecture for singular points involving semi-classical and quantum singularities. Moreover, we introduce a quantum α-permanent per q ( α ) and establish another conjecture which describes a ‘reciprocity’ between the multiplicities of the irreducible summands of the cyclic modules generated respectively by det q ( α ) and per q ( α ) .