Quantum Symmetric Pairs and Their Zonal Spherical Functions

We study the space of biinvariants and zonal spherical functions associated to quantum symmetric pairs in the maximally split case. Under the obvious restriction map, the space of biinvariants is proved isomorphic to the Weyl group invariants of the character group ring associated to the restricted roots. As a consequence, there is either a unique set, or an (almost) unique two-parameter set of Weyl group invariant quantum zonal spherical functions associated to an irreducible symmetric pair. Included is a complete and explicit list of the generators and relations for the left coideal subalgebras of the quantized enveloping algebra used to form quantum symmetric pairs.     

Bi-axial Gegenbauer Functions of the Second Kind

Bi-axially symmetric monogenic generating functions on ^{p + q} have been used recently to define generalisations of Gegenbauer polynomials. These polynomials are orthogonal on the unit ball in ^p. Generalised Cauchy transforms of these polynomials are used to define corresponding bi-axial Gegenbauer functions of the second kind. It is demonstrated that these functions of the second kind satisfy second order differential equations related to those satisfied by the corresponding bi-axial Gegenbauer polynomials.     

On Stirling Functions of the Second Kind

We continue the study of Stirling functions of the second kind, S(α,k) where α > 0 is real. In particular, we consider integral and fractional derivative representations of the S(α,k), as well as connections with generalized Bernoulli and Stirling polynomials.     

Minkowski空间中定向曲面上的第二类松弛弹性线

在Minkowski空间中,定义了定向曲面上的第二类松弛弹性线,推导了在定向曲面上的第二类松弛弹性线的Euler-Lagrange方程．进一步阐明了,这些曲线是否落在曲率线上,最后给出相关的实例．     

由泛复函生成多项式型空间调和函数和球函数

本文以泛复变函数为工具,成功地构造出多项式型空间调和函数族,通过坐标变换和正交化过程,进而又获得了球函数.     

Discriminants and Functions of the Second Kind of Orthogonal Polynomials

We consider the problem of evaluating discriminants of general orthogonal polynomials. It is shown that for a general class of weight functions, the functions of the second kind and the orthogonal polynomials are linear independent solutions of the same second order differential equation. We derive a linear fourth-order differential equation satisfied by the numerator polynomials and give two additional linear independent solutions.     

Hypergeometric Type Functions and Their Symmetries

The paper is devoted to a systematic and unified discussion of various classes of hypergeometric type equations: the hypergeometric equation, the confluent equation, the F ₁ equation (equivalent to the Bessel equation), the Gegenbauer equation and the Hermite equation. In particular, recurrence relations of their solutions, their integral representations and discrete symmetries are discussed.     

Symmetric Functions, Noncommutative Symmetric Functions, and Quasisymmetric Functions

This paper is concerned with two generalizations of the Hopf algebra of symmetric functions that have more or less recently appeared. The Hopf algebra of noncommutative symmetric functions and its dual, the Hopf algebra of quasisymmetric functions. The focus is on the incredibly rich structure of the Hopf algebra of symmetric functions and the question of which structures and properties have good analogues for the noncommutative symmetric functions and/or the quasisymmetric functions. This paper attempts to survey the ongoing investigations in this topic as dictated by the knowledge and interests of its author. There are many open questions that are discussed.     

The Hypergeometric Equation and Ramanujan Functions

In this paper we give analogues of the Ramanujan functions and nonlinear differential equations for them. Investigating a modular structure of solutions for nonlinear differential systems, we deduce new identities between the Ramanujan and hypergeometric functions. Another result of this paper is a solution of transcendence problems concerning nonlinear systems.     

Geometric Properties of Generalized Hypergeometric Functions

Let denote the generalized hypergeometric function where no denominator parameter can be zero or a negative integer and (a,n) denotes the ascending factorial notation. Ponnusamy and Vuorinen raised the problem of finding conditions on the parameters a_j > 0, b_j > 0 so that the function is univalent in . The main aim of this paper is to discuss this problem in detail for the case q = 2.     

p-adic超几何函数与Dwork超曲面上的有理点

p-adic超几何函数是经典的Gauss超几何函数在有限域上的模拟,与许多数论问题都有联系.设Fq是q元有限域,λ∈Fq,n为正整数.本文研究了Dwork超曲面Dλ^n:x1^n+x2^n+…+xn^n=nλx1x2…xn及其推广形式上的Fq-有理点,并在n与q(q-1)互素时给出了由p-adic超几何函数表示的各种Fq-有理点个数的公式,从而修正和改进了Barman与Goodson等人的结论.     

p-adic 超几何函数与 Dwork超曲面上的有理点

p-adic超几何函数是经典的Gauss超几何函数在有限域上的模拟,与许多数论问题都有联系.设F_q是q元有限域,λ∈F_q,n为正整数.本文研究了Dwork超曲面D_λⁿ：x₁ⁿ+x₂ⁿ+…+x_nⁿ=nλx₁x₂…x_n及其推广形式上的F_q-有理点,并在n与q（q-1）互素时给出了由p-adic超几何函数表示的各种F_q-有理点个数的公式,从而修正和改进了Barman与Goodson等人的结论.     

On Hypergeometric Functions and Generalizations of Legendre's Relation

We prove some convexity properties for a sum of hypergeometric functions and obtain a generalization of Legendre's relation for complete elliptic integrals. We apply these results to prove some inequalities for hypergeometric functions, incomplete beta-functions, and Legendre functions.     

Probability Measures on Dual Objects to Compact Symmetric Spaces and Hypergeometric Identities

We derive, in several different ways, combinatorial identities which are multidimensional analogs of classical Dougall's formula for a bilateral hypergeometric series of the type ₂H₂. These identities have a representation-theoretic meaning. They make it possible to construct concrete examples of spherical functions on inductive limits of symmetric spaces. These spherical functions are of interest to harmonic analysis.     

Geometric Properties of the Shifted Hypergeometric Functions

We will provide sufficient conditions for the shifted hypergeometric function \(z_2F_1(a,b;c;z)\) to be a member of a specific subclass of starlike functions in terms of the complex parameters a, b and c. For example, we study starlikeness of order \(\alpha ,\) \(\lambda \)-spirallikeness of order \(\alpha \) and strong starlikeness of order \(\alpha .\) In particular, those properties lead to univalence of the shifted hypergeometric functions on the unit disk.     

The Lie Theory of Two-Variable Hypergeometric Functions

We develop a Lie-algebraic method that associates with each of the 34 distinct second-order hypergeometric functions in two variables a canonical system of partial differential equations. The special functions arise by partial separation of variables in these simple systems. Some consequences are a demonstration that all such functions appear as solutions of the 4-variable wave equation and a classification of the possible imbeddings. In each case the functions are characterized by first- and second-order operators in the enveloping algebra of the conformal symmetry algebra for the wave equation. In some cases the 3-variable wave and heat equations and the 2-variable Helmholtz equation also arise. This intimate relationship between Horn functions and some fundamental equations of mathematical physics shows that these functions are more interesting than was previously recognized and permits use of the powerful tools of Lie theory and separation of variables to obtain properties of the functions.