Taylor series for the Askey-Wilson operator and classical summation formulas

具有互补余项的适定基本 q-Taylor 展开与双基核

Well-poised basic q-Taylor expansions with complementary remainders and a two-basis kernel

本文证明了一个非终止的适定基本 q-Taylor 展开，该展开具有互补余项，适用于第二作者在文献中隐含提出的双基无穷乘积核，并利用适定 Cooper 公式、Jackson 终止型求和、Rogers 求和以及 theta 插值完成证明，同时恢复了 Bailey 非终止求和作为推论。

Fourier transforms on the quantum SU(1,1) group

The main goal is to interpret the Askey-Wilson function and the corresponding transform pair on the quantum SU(1,1) group. A weight on the C^*-algebra of continuous functions vanishing at infinity on the quantum SU(1,1) group is studied, which is left and right invariant in a weak sense with respect to a product defined using Wall functions. The Haar weight restricted to certain subalgebras are explicitly determined in terms of an infinitely supported Jackson integral and in terms of an infinitely supported Askey-Wilson type measure. For the evaluation the spectral analysis of explicit unbound…

From Taylor series of analytic functions to their global analysis

We analyze the conditions on the Taylor coefficients of an analytic function to admit global analytic continuation, complementing a recent paper of Breuer and Simon on general conditions for natural boundaries to form. A new summation method is introduced to convert a relatively wide family of infinite sums and local expansions into integrals. The integral representations yield global information such as analytic continuability, position of singularities, asymptotics for large values of the variable and asymptotic location of zeros.

Bilinear summation formulas from quantum algebra representations

The tensor product of a positive and a negative discrete series representation of the quantum algebra U_q(su(1,1)) decomposes as a direct integral over the principal unitary series representations. Discrete terms can appear, and these terms are a finite number of discrete series representations, or one complementary series representation. From the interpretation as overlap coefficients of little q-Jacobi functions and Al-Salam and Chihara polynomials in base q and base q^{-1}, two closely related bilinear summation formulas for the Al-Salam and Chihara polynomials are derived. The formulas inv…

关于Bailey的${}_5ψ_5$恒等式

On ${}_5ψ_5$ identities of Bailey

本文利用Carlitz的${}_5ϕ_4$恒等式证明Bailey的两个${}_5ψ_5$求和公式，并通过极限导出并证明对应的两个${}_3ψ_3$求和公式。

The summation of infinite partial fraction decomposition I: some formulae related to the Hurwitz zeta function

In this paper we establish a new summation method by expanding $\prod_{k}(1-\frac{z}{a_{k}})^{-1}$ with two approaches: the Taylor expansion and the infinite partial fraction decomposition. Here we focus on the case when $a_{k}$ is arithmetic sequence. By this summation we obtain many equalities involve Hurwitz zeta function and Gammma function.

On Taylor coefficients of smooth functions

We study the Borel map, which maps infinitely differentiable functions on an interval to the jets of their Taylor coefficients at a given point in the interval. Our main results include a complete description of the image of the Borel map for Beurling classes of smooth functions and a moment-type summation method which allows one to recover a function from its Taylor jet. A surprising feature of this description is an unexpected threshold at the logarithmic class. Another interesting finding is a "duality" between non-quasianalytic and quasianalytic classes, which reduces the description of th…

q-special functions, basic hypergeometric series and operators

In the lecture notes we start off with an introduction to the $q$-hypergeometric series, or basic hypergeometric series, and we derive some elementary summation and transformation results. Then the $q$-hypergeometric difference equation is studied, and in particular we study solutions given in terms of power series at $0$ and at $\infty$. Factorisations of the corresponding operator are considered in terms of a lowering operator, which is the $q$-derivative, and the related raising operator. Next we consider the $q$-hypergeometric operator in a special case, and we show that there is a natural…

Hypergeometric and basic hypergeometric series and integrals associated with root systems

We give an overview of some of the main results from the theories of hypergeometric and basic hypergeometric series and integrals associated with root systems. In particular, we list a number of summations, transformations and explicit evaluations for such multiple series and integrals. We concentrate on such results which do not directly extend to the elliptic level. This text is a provisional version of a chapter on hypergeometric and basic hypergeometric series and integrals associated with root systems for volume 2 of the new Askey--Bateman project which deals with "Multivariable special f…

发散矩阵级数的求和

Summing divergent matrix series

本研究将经典标量级数求和方法推广为完全非交换的矩阵版本，证明其正则性，给出适配的数值算法，可应用于各类矩阵级数并在多个矩阵分析场景中发挥作用。

从给定通项求任意级数的和

Finding the sum of any series from a given general term

欧拉通过泰勒展开推导欧拉-麦克劳林求和公式，并应用于整数幂和、调和级数部分和及zeta函数近似。