Taylor Series for the Askey-wilson Operator and Classical Summation Formulas
نویسندگان
چکیده
Abstract. An analogue of Taylor’s formula, which arises by substituting the classical derivative by a divided difference operator of Askey-Wilson type, is developed here. We study the convergence of the associated Taylor series. Our results complement a recent work by Ismail and Stanton. Quite surprisingly, in some cases the Taylor polynomials converge to a function which differs from the original one. We provide explicit expressions for the integral remainder. As application, we obtain some summation formulas for basic hypergeometric series. As far as we know, one of them is new. We conclude by studying the different forms of the binomial theorem in this context.
منابع مشابه
Taylor Series for the Askey-wilson Operator and Classical Summation Formulas
Abstract. An analog of Taylor’s formula, which arises by substituting the classical derivative by a divided difference operator of Askey-Wilson type, is developed here. As an application, a generalization of the binomial theorem is obtained. Besides, this method becomes quite useful to obtain summation formulas of basic hypergeometric series. New proofs of several well-known summation formulas ...
متن کاملThe Cauchy Operator for Basic Hypergeometric Series
We introduce the Cauchy augmentation operator for basic hypergeometric series. Heine’s 2φ1 transformation formula and Sears’ 3φ2 transformation formula can be easily obtained by the symmetric property of some parameters in operator identities. The Cauchy operator involves two parameters, and it can be considered as a generalization of the operator T (bDq). Using this operator, we obtain extensi...
متن کاملA New Approach to the Theory of Classical Hypergeometric Polynomials
In this paper we present a unified approach to the spectral analysis of an hypergeometric type operator whose eigenfunctions include the classical orthogonal polynomials. We write the eigenfunctions of this operator by means of a new Taylor formula for operators of Askey-Wilson type. This gives rise to some expressions for the eigenfunctions, which are unknown in such a general setting. Our met...
متن کاملAn Expansion Formula for the Askey-Wilson Function
The Askey-Wilson function transform is a q-analogue of the Jacobi function transform with kernel given by an explicit non-polynomial eigenfunction of the Askey-Wilson second order q-difference operator. The kernel is called the Askey-Wilson function. In this paper an explicit expansion formula for the Askey-Wilson function in terms of Askey-Wilson polynomials is proven. With this expansion form...
متن کاملBilinear Summation Formulas from Quantum Algebra Representations
The tensor product of a positive and a negative discrete series representation of the quantum algebra Uq ( 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 li...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2005