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Masahiko Ito

Publications and source records attributed to Masahiko Ito.

16 recordsLinked to original sources
arXiv2026

$q$-Selberg extensions of Gasper's two stage $q$-beta integrals

Gasper gave a two-step extension of the Askey-Roy formula for a beta-integral type defined on the unit circle. This involved increasing the number of parameters and adding a balancing condition. We derive a multidimensional generalization of the final formula in Gasper's extension. By considering a two-step degeneration involving parameters our generalization becomes Tarasov-Varchenko's formula -- itself a multidimensional generalization of the original Askey-Roy formula. The proof is done by Aomoto's method. This generalized integration by parts strategy makes use of a class of functions known as the fundamental invariants of type $BC$.

math.CV↗
arXiv2023

Gauss decomposition and $q$-difference equations for Jackson integrals of symmetric Selberg type

We provide explicit expressions for two types of first order $q$-difference systems for the Jackson integral of symmetric Selberg type. One is the $q$-difference system known to be the $q$-KZ equation and the other is the $q$-difference system for parameters different from the $q$-KZ equation. We use a basis of the systems introduced by Matsuo in his study of the $q$-KZ equation. As a result, the similarity of these two systems is discussed by concrete calculations. Intermediate calculations are made use of the Riemann-Hilbelt method for $q$-difference equation from connection matrix established by Aomoto.

math.CV↗
arXiv2020

$q$-Difference Systems for the Jackson Integral of Symmetric Selberg Type

We provide an explicit expression for the first order $q$-difference system for the Jackson integral of symmetric Selberg type. The $q$-difference system gives a generalization of $q$-analog of contiguous relations for the Gauss hypergeometric function. As a basis of the system we use a set of the symmetric polynomials introduced by Matsuo in his study of the $q$-KZ equation. Our main result is an explicit expression for the coefficient matrix of the $q$-difference system in terms of its Gauss matrix decomposition. We introduce a class of symmetric polynomials called interpolation polynomials, which includes Matsuo's polynomials. By repeated use of three-term relations among the interpolation polynomials we compute the coefficient matrix.

math.CA↗
arXiv2020

Elliptic extension of Gustafson's $q$-integral of type $G_2$

The evaluation formula for an elliptic beta integral of type $G_2$ is proved. The integral is expressed by a product of Ruijsenaars' elliptic gamma functions, and the formula includes that of Gustafson's $q$-beta integral of type $G_2$ as a special limiting case as $p\to 0$. The elliptic beta integral of type $BC_1$ by van Diejen and Spiridonov is effectively used in the proof of the evaluation formula.

math.CV↗
arXiv2019

A determinant formula associated with the elliptic hypergeometric integrals of type $BC_n$

We establish a determinant formula for the bilinear form associated with the elliptic hypergeometric integrals of type $BC_n$ by studying the structure of $q$-difference equations to be satisfied by them. The determinant formula is proved by combining the $q$-difference equations of the determinant and its asymptotic analysis along the singularities. The elliptic interpolation functions of type $BC_n$ are essentially used in the study of the $q$-difference equations.

math.CV↗
arXiv2018

Connection Formula for the Jackson Integral of Type $A_n$ and Elliptic Lagrange Interpolation

We investigate the connection problem for the Jackson integral of type $A_n$. Our connection formula implies a Slater type expansion of a bilateral multiple basic hypergeometric series as a linear combination of several specific multiple series. Introducing certain elliptic Lagrange interpolation functions, we determine the explicit form of the connection coefficients. We also use basic properties of the interpolation functions to establish an explicit determinant formula for a fundamental solution matrix of the associated system of $q$-difference equations.

math.CV↗
arXiv2016

A bilateral extension of the $q$-Selberg integral

A multi-dimensional bilateral $q$-series extending the $q$-Selberg integral is studied using concepts of truncation, regularization and connection formulae. Following Aomoto's method, which involves regarding the $q$-series as a solution of a $q$-difference equation fixed by its asymptotic behavior, an infinite product evaluation is obtained. The $q$-difference equation is derived applying the shifted symmetric polynomials introduced by Knop and Sahi. As a special case of the infinite product formula, Askey--Evans's $q$-Selberg integral evaluation and its generalization by Tarasov--Varchenko and Stokman is reclaimed, and an explanation in the context of Aomoto's setting is thus provided.

math.CV↗
arXiv2016

A generalization of the Sears--Slater transformation and elliptic Lagrange interpolation of type $BC_n$

The connection formula for the Jackson integral of type $BC_n$ is obtained in the form of a Sears--Slater type expansion of a bilateral multiple basic hypergeometric series as a linear combination of several specific bilateral multiple series. The coefficients of this expansion are expressed by certain elliptic Lagrange interpolation functions. Analyzing basic properties of the elliptic Lagrange interpolation functions, an explicit determinant formula is provided for a fundamental solution matrix of the associated system of $q$-difference equations.

math.CV↗
arXiv2014

The $q$-Dixon--Anderson integral and multi-dimensional $_1ψ_1$ summations

The Dixon--Anderson integral is a multi-dimensional integral evaluation fundamental to the theory of the Selberg integral. The $_1ψ_1$ summation is a bilateral generalization of the $q$-binomial theorem. It is shown that a $q$-generalization of the Dixon--Anderson integral, due to Evans, and multi-dimensional generalizations of the $_1ψ_1$ summation, due to Milne and Gustafson, can be viewed as having a common origin in the theory of $q$-difference equations as expounded by Aomoto. Each is shown to be determined by a $q$-difference equation of rank one, and a certain asymptotic behavior. In calculating the latter, essential use is made of the concepts of truncation, regularization and connection formulae.

math.CV↗
arXiv2013

Ramanujan's $_1ψ_1$ summation theorem --- perspective, announcement of bilateral $q$-Dixon--Anderson and $q$-Selberg integral extensions, and context

The Ramanujan $_1ψ_1$ summation theorem in studied from the perspective of $q$-Jackson integrals, $q$-difference equations and connection formulas. This is an approach which has previously been shown to yield Bailey's very-well-poised $_6ψ_6$ summation. Bilateral Jackson integral generalizations of the Dixon--Anderson and Selberg integrals relating to the type $A$ root system are identified as natural candidates for multidimensional generalizations of the Ramanujan $_1ψ_1$ summation theorem. New results of this type are announced, and furthermore they are put into context by reviewing from previous literature explicit product formulas for Jackson integrals relating to other roots systems obtained from the same perspective.

math.CV↗
arXiv2012

Crystal structure and superconducting properties of hexagonal lithium-niobium oxynitride

A hexagonal oxynitride (Li0.88_0.12)Nb3.0(O0.13N0.87)4 was synthesized through ammonia nitridation of LiNb3O8. The structural analysis revealed that this oxynitride consists of alternate stacking of octahedral and prismatic layers with different Li/Nb ratios: significant amounts of Li and Nb atoms (Li/Nb = 43/57) coexist in the octahedral layer, while the prismatic site is preferentially occupied by Nb (Li/Nb = 3/97). A metallic behavior was accompanied by an abrupt drop of electrical resistivity at about 3 K. Furthermore, large diamagnetism and specific-heat anomaly were observed below this temperature, suggesting the appearance of superconductivity in the Li-Nb oxynitride.

cond-mat.supr-con↗
arXiv2010

Difference system for Selberg correlation integrals

The Selberg correlation integrals are averages of the products $\prod_{s=1}^m\prod_{l=1}^n (x_s - z_l)^{μ_s}$ with respect to the Selberg density. Our interest is in the case $m=1$, $μ_1 = μ$, when this corresponds to the $μ$-th moment of the corresponding characteristic polynomial. We give the explicit form of a $(n+1) \times (n+1)$ matrix linear difference system in the variable $μ$ which determines the average, and we give the Gauss decomposition of the corresponding $(n+1) \times (n+1)$ matrix. For $μ$ a positive integer the difference system can be used to efficiently compute the power series defined by this average.

math-ph↗
arXiv2009

A First Order $q$-Difference System for the $BC_1$-Type Jackson Integral and Its Applications

We present an explicit expression for the $q$-difference system, which the $BC_1$-type Jackson integral ($q$-series) satisfies, as first order simultaneous $q$-difference equations with a concrete basis. As an application, we give a simple proof for the hypergeometric summation formula introduced by Gustafson and the product formula of the $q$-integral introduced by Nassrallah-Rahman and Gustafson.

math.CA↗