q-Analogues of Some Multivariable Biorthogonal Polynomials

经典特殊函数的扩展方案

An extended scheme of classical special functions

本文提出一个统一的超几何型经典特殊函数扩展方案，扩展了 Askey 方案及其 q-类似，顶层包含由椭圆超几何级数构成的双正交函数，并将复超几何函数纳入其中。

$BC_2$ type multivariable matrix functions and matrix spherical functions

Matrix spherical functions associated to the compact symmetric pair $(\mathrm{SU}(m+2), \mathrm{S}(\mathrm{U}(2)\times \mathrm{U}(m))$, having reduced root system of type $\mathrm{BC}_2$, are studied. We consider an irreducible $K$-representation $(π,V)$ arising from the $\mathrm{U}(2)$-part of $K$, and the induced representation $\mathrm{Ind}_K^G π$ splits multiplicity free. The corresponding spherical functions, i.e. $Φ\colon G \to \mathrm{End}(V)$ satisfying $Φ(k_1gk_2)=π(k_1)Φ(g)π(k_2)$ for all $g\in G$, $k_1,k_2\in K$, are studied by studying certain leading coefficients which involve hyp…

Multivariable generalizations of the Schur class: positive kernel characterization and transfer function realization

The operator-valued Schur-class is defined to be the set of holomorphic functions $S$ mapping the unit disk into the space of contraction operators between two Hilbert spaces. There are a number of alternate characterizations: the operator of multiplication by $S$ defines a contraction operator between two Hardy Hilbert spaces, $S$ satisfies a von Neumann inequality, a certain operator-valued kernel associated with $S$ is positive-definite, and $S$ can be realized as the transfer function of a dissipative (or even conservative) discrete-time linear input/state/output linear system. Various mul…

A second addition formula for continuous q-ultraspherical polynomials

This paper provides the details of Remark 5.4 in the author's paper "Askey-Wilson polynomials as zonal spherical functions on the SU(2) quantum group", SIAM J. Math. Anal. 24 (1993), 795-813. In formula (5.9) of the 1993 paper a two-parameter class of Askey-Wilson polynomials was expanded as a finite Fourier series with a product of two 3phi2's as Fourier coefficients. The proof given there used the quantum group interpretation. Here this identity will be generalized to a 3-parameter class of Askey-Wilson polynomials being expanded in terms of continuous q-ultraspherical polynomials with a pro…

Askey-Wilson多项式连接公式及相关展开

Connection formulas for Askey--Wilson polynomials and related expansions

本文推导了Askey-Wilson多项式的连接公式，并研究了其与相关多项式展开式及极限关系。

Transformations of elliptic hypergometric integrals

We prove a pair of transformations relating elliptic hypergeometric integrals of different dimensions, corresponding to the root systems BC_n and A_n; as a special case, we recover some integral identities conjectured by van Diejen and Spiridonov. For BC_n, we also consider their "Type II" integral. Their proof of that integral, together with our transformation, gives rise to pairs of adjoint integral operators; a different proof gives rise to pairs of adjoint difference operators. These allow us to construct a family of biorthogonal abelian functions generalizing the Koornwinder polynomials, …

对称函数代数中的q-Askey概型元素

Elements of the q-Askey scheme in the algebra of symmetric functions

本研究在对称函数代数中构造多变量q-Racah对称函数，推导其可退化为big q-Jacobi等三类对称函数，证明四类函数均关于无穷点构形空间上的测度正交，且测度间的极限转移与函数退化一致，丰富了q-Askey概型的多变量对称函数理论。