Inversions of integral operators and elliptic beta integrals on root systems

Expansion formulas for multiple basic hypergeometric series over root systems

We extend expansion formulas of Liu given in 2013 to the context of multiple series over root systems. Liu and others have shown the usefulness of these formulas in Special Functions and number-theoretic contexts. We extend Wang and Ma's generalizations of Liu's work which they obtained using $q$-Lagrange inversion. We use the $A_n$ and $C_n$ Bailey transformation and other summation theorems due to Gustafson, Milne, Milne and Lilly, and others, from the theory of $A_n$, $C_n$ and $D_n$ basic hypergeometric series.

根系统上的复与有理超几何函数

Complex and rational hypergeometric functions on root systems

本文研究了根系统上椭圆超几何积分的新极限，通过准周期退化获得可精确求值的复 beta 积分，证明了非紧自旋链理论中的 Derkachov--Manashov 猜想，并导出了有理函数积分之和的新超几何恒等式。

Rarefied elliptic hypergeometric functions

Two exact evaluation formulae for multiple rarefied elliptic beta integrals related to the simplest lens space are proved. They generalize evaluations of the type I and II elliptic beta integrals attached to the root system $C_n$. In a special $n=1$ case, the simplest $p\to 0$ limit is shown to lead to a new class of $q$-hypergeometric identities. Symmetries of a rarefied elliptic analogue of the Euler-Gauss hypergeometric function are described and the respective generalization of the hypergeometric equation is constructed. Some extensions of the latter function to $C_n$ and $A_n$ root system…

Theta hypergeometric integrals

We define a general class of (multiple) integrals of hypergeometric type associated with the Jacobi theta functions. These integrals are related to theta hypergeometric series through the residue calculus. In the one variable case, we get theta function extensions of the Meijer function. A number of multiple generalizations of the elliptic beta integral [S2] associated with the root systems $A_n$ and $C_n$ is described. Some of the $C_n$-examples were proposed earlier by van Diejen and the author, but other integrals are new. An example of the biorthogonality relations associated with the elli…

Essays on the theory of elliptic hypergeometric functions

We give a brief review of the main results of the theory of elliptic hypergeometric functions -- a new class of special functions of mathematical physics. We prove the most general univariate exact integration formula generalizing Euler's beta integral, which is called the elliptic beta integral. An elliptic analogue of the Gauss hypergeometric function is constructed together with the elliptic hypergeometric equation for it. Biorthogonality relations for this function and its particular subcases are described. We list known elliptic beta integrals on root systems and consider symmetry transfo…

Schwarzian-Newton方法用于求解非线性方程及其应用

The Schwarzian-Newton method for solving nonlinear equations, with applications

本文提出Schwarzian-Newton方法，该方法在常数Schwarzian导数下精确求解非线性方程，具有四阶收敛性，并应用于累积分布函数和椭圆积分的求逆问题。