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qDifferential Operator and Its Applications
Author: FangJianPing
Tutor: LiuZhiGuo
School: East China Normal University
Course: Basic mathematics
Keywords: Basic hypergeometric series qDifferential operator General hypergeometric series Heine’s transformation formula Sears’ transformation formula Rogers Fine identity
CLC: O175.3
Type: PhD thesis
Year: 2008
Downloads: 96
Quote: 2
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Abstract
In 1893,Rogers[122]constructed the two qexponential operators by the qdifferential operator.Then he used them to study some properties of the qseries.One hundred years later,ChenLiu[46,47]discovered them independently.And they first gave the two identities of the qoperators.Then they used the two identities to give a system ways of studying the theory of qseries.Inspired by the two operator identities given by Chen Liu,in this paper,we first extend the two qexponential operators defined by Rogers.Then we give some applications of them.Using the operators extended by us,we not only obtain all of formulas proven by the old but can get some natural extensions.At the same time,applying them to study the theory of qseries,some properties are more excellent than the old.For example,we can straightforward to derive the Heine’s transformation formula and the terminating and nonterminating Sears’ _{3}Φ_{2} transformation formula from the properties of the qexponential operators which cannot need the other qseries formulas.Applying the qexponential operators extended by us,an extension formula of qPfaffSaalsch（??）tz summation formula and an extension formula of Sears’ terminating balanced _{4}Φ_{3} series transformation formula and an extension involving multiple sum about finite Heine’s _{2}Φ_{1} transformation formula are given.We obtain some extensions involving multiple sum about qChuVandermonde’s identities by using the extended operator identities.From these extensions,we get an extension of Dilcher’s identity and an extension of FuLascoux’s formula.An interesting extension involving multiple sum about the finite RogersFine identity is obtained by applying our operators.In addition,we also use it to give some interesting properties of the homogeneous AlSalam and Carlitz polynomials defined by us.Such as,the generating function and the qMehler formula of this polynomials.As applications,we also give some formal extension formulas of the basic hypergeometric series.Such as,formal extension of Jackson’s _{2}Φ_{2} transformation formula, formal extension of three terms of Sears’ _{3}Φ_{2} transformation formula and formal extension of AskeyRoy integral formula and so on.At last,in this paper,we also give some other interesting applications of the qdifferential operator.

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CLC: > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Differential equations, integral equations > Differential operator theory
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