Related papers: Taylor series for the Askey-Wilson operator and cl…
This article gives a brief introduction to $q$-special functions, i.e., $q$-analogues of the classical special functions. Here $q$ is a deformation parameter, usually $0<q<1$, where $q=1$ is the classical case. The main topics to be treated…
In the present paper, we propose to give an extension to the context of Dunkl theory of the notion of translation and in connection with this a corresponding extension of Taylor's formula. More precisely, we prove some properties and…
We describe the utility of integral representations for sums of basic hypergeometric functions. In particular we use these to derive an infinite sequence of transformations for symmetrizations over certain variables which the functions…
By employing certain extended classical summation theorems, several surprising \pi and other formulae are displayed.
Automorphisms of the infinite dimensional Onsager algebra are introduced. Certain quotients of the Onsager algebra are formulated using a polynomial in these automorphisms. In the simplest case, the quotient coincides with the classical…
We prove a new Bailey-type transformation relating WP-Bailey pairs. We then use this transformation to derive a number of new 3- and 4-term transformation formulae between basic hypergeometric series.
We propose a new integral based on Taylor measures, study its properties extensively, and we illustrate that it includes many concepts from mathematics as special cases. In particular, the new integral emerges as a generalization of the…
In this paper, we introduce the so-called elliptic Askey-Wilson polynomials which are homogeneous polynomials in two special theta functions. With regard to the significance of polynomials of such kind, we establish some general elliptic…
We present some elementary derivations of summation and transformation formulas for q-series, which are different from, and in several cases simpler or shorter than, those presented in the Gasper and Bahman [1990] "Basic Hypergeometric…
We prove a factorization formula for the Taylor series coefficients of a zero of a polynomial as a function of the polynomial's coefficients. This result extends to more general functions which we call "complex-exponent polynomials". To…
We study inverse factorial series and their relation to Stirling numbers of the first kind. We prove a special representation of the polylogarithm function in terms of series with such numbers. Using various identities for Stirling numbers…
In this paper the double-sided Talor's approximations are used to obtain generalisations and improvements of some trigonometric inequalities.
A master formula of transformation formulas for bilinear sums of basic hypergeometric series is proposed. It is obtained from the author's previous results on a transformation formula for Milne's multivariate generalization of basic…
This paper investigates summability principles for multilinear summing operators. The main result presents a novel inclusion theorem for a class of summing operators, which generalizes several classical results. As applications, we derive…
A generalization of the classical Lipschitz summation formula is proposed. It involves new polylogarithmic rational functions constructed via the Fourier expansion of certain sequences of Bernoulli--type polynomials. Related families of…
A Taylor variety consists of all fixed order Taylor polynomials of rational functions, where the number of variables and degrees of numerators and denominators are fixed. In one variable, Taylor varieties are given by rank constraints on…
We perform certain alternating binomial summations with parameters that occur in the analysis of algorithms. A combination of integral and special function and special number representations is used. The results are sufficiently general to…
We take advantage of the combinatorial interpretations of many sequences of polynomials of binomial type to define a sequence of symmetric functions corresponding to each sequence of polynomials of binomial type. We derive many of the…
We unify the recently developed abstract theories of universal series and extended universal series to include sums of the form $\sum_{k=0}^n a_k x_{n,k}$ for given sequences of vectors $(x_{n,k})_{n\geq k\geq 0}$ in a topological vector…
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…