scholarly journals Middle multiplicative convolution and hypergeometric equations

2021 ◽  
Vol 23 ◽  
Author(s):  
Nicolas Martin
Keyword(s):  
Multiplicative Convolution ◽  
Hypergeometric Equations

scholarly journals Crystallization of Random Matrix Orbits

2018 ◽  
Vol 2020 (3) ◽  
pp. 883-913 ◽  
Author(s):  
Vadim Gorin ◽  
Adam W Marcus
Keyword(s):  
Random Matrix ◽  
Special Functions ◽  
Free Field ◽  
Gaussian Free Field ◽  
Complex Quaternion ◽  
Cutting Out ◽  
Multiplicative Convolution ◽  
Global Asymptotics ◽  
Associated Special Functions ◽  
Derivatives Of

Abstract Three operations on eigenvalues of real/complex/quaternion (corresponding to $\beta =1,2,4$) matrices, obtained from cutting out principal corners, adding, and multiplying matrices, can be extrapolated to general values of $\beta>0$ through associated special functions. We show that the $\beta \to \infty $ limit for these operations leads to the finite free projection, additive convolution, and multiplicative convolution, respectively. The limit is the most transparent for cutting out the corners, where the joint distribution of the eigenvalues of principal corners of a uniformly-random general $\beta $ self-adjoint matrix with fixed eigenvalues is known as the $\beta $-corners process. We show that as $\beta \to \infty $ these eigenvalues crystallize on an irregular lattice consisting of the roots of derivatives of a single polynomial. In the second order, we observe a version of the discrete Gaussian Free Field put on top of this lattice, which provides a new explanation as to why the (continuous) Gaussian Free Field governs the global asymptotics of random matrix ensembles.

scholarly journals k-Hypergeometric Series Solutions to One Type of Non-Homogeneous k-Hypergeometric Equations

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 262 ◽  
Author(s):  
Shengfeng Li ◽  
Yi Dong
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Hypergeometric Series ◽  
Second Order ◽  
Series Solutions ◽  
Frobenius Method ◽  
General Solutions ◽  
Hypergeometric Differential Equation ◽  
Polynomial Term ◽  
Hypergeometric Equations

In this paper, we expound on the hypergeometric series solutions for the second-order non-homogeneous k-hypergeometric differential equation with the polynomial term. The general solutions of this equation are obtained in the form of k-hypergeometric series based on the Frobenius method. Lastly, we employ the result of the theorem to find the solutions of several non-homogeneous k-hypergeometric differential equations.

scholarly journals Voros coefficients for the hypergeometric differential equations and Eynard–Orantin’s topological recursion: Part II: For confluent family of hypergeometric equations

2019 ◽  
Vol 4 (1) ◽  
Author(s):  
Kohei Iwaki ◽  
Tatsuya Koike ◽  
Yumiko Takei
Keyword(s):  
Free Energy ◽  
Differential Equations ◽  
Classical Limit ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Spectral Curves ◽  
Topological Recursion ◽  
Hypergeometric Equations ◽  
Explicit Expressions

Abstract We show that each member of the confluent family of the Gauss hypergeometric equations is realized as quantum curves for appropriate spectral curves. As an application, relations between the Voros coefficients of those equations and the free energy of their classical limit computed by the topological recursion are established. We will also find explicit expressions of the free energy and the Voros coefficients in terms of the Bernoulli numbers and Bernoulli polynomials. Communicated by: Youjin Zhang

Multiplicative Convolution on the Algebra of Block-Symmetric Analytic Functions

2020 ◽  
Vol 246 (2) ◽  
pp. 245-255
Author(s):  
А. V. Zagorodnyuk ◽  
V. V. Kravtsiv
Keyword(s):  
Analytic Functions ◽  
Multiplicative Convolution

POLYNOMIALS ASSOCIATED WITH THE HYPERGEOMETRIC FUNCTIONS WITH FINITE MONODROMY GROUPS

2004 ◽  
Vol 15 (07) ◽  
pp. 629-649 ◽  
Author(s):  
HIROYUKI OCHIAI ◽  
MASAAKI YOSHIDA
Keyword(s):  
Hypergeometric Functions ◽  
Integral Parameter ◽  
Monodromy Groups ◽  
Hypergeometric Equations

The hypergeometric equations with polyhedral monodromy groups derive 3-integral-parameter families of polynomials.

Confluent hypergeometric equations and related solvable potentials in quantum mechanics

2000 ◽  
Vol 41 (12) ◽  
pp. 7964-7996 ◽  
Author(s):  
J. Negro ◽  
L. M. Nieto ◽  
O. Rosas-Ortiz
Keyword(s):  
Quantum Mechanics ◽  
Solvable Potentials ◽  
Hypergeometric Equations

scholarly journals Hypergeometric equations and weighted projective spaces

2011 ◽  
Vol 54 (8) ◽  
pp. 1577-1590 ◽  
Author(s):  
Alessio Corti ◽  
Vasily Golyshev
Keyword(s):  
Projective Spaces ◽  
Hypergeometric Equations
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