uu.seUppsala University Publications

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A new method *q*-calculusPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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2002 (English)Doctoral thesis, monograph (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Uppsala: Acta Universitatis Upsaliensis , 2002. , p. 116
##### Series

Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 25
##### Keywords [en]

Mathematics
##### Keywords [sv]

MATEMATIK
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

URN: urn:nbn:se:uu:diva-2698ISBN: 91-506-1628-5 (print)OAI: oai:DiVA.org:uu-2698DiVA, id: diva2:162070
##### Public defence

2002-11-14, Room 2146, Mathematics and Information Technology Center, Polacksbacken, Uppsala, 13:15
##### Opponent

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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt446",{id:"formSmash:j_idt446",widgetVar:"widget_formSmash_j_idt446",multiple:true}); Available from: 2002-10-17 Created: 2002-10-17Bibliographically approved

In *q-*calculus we are looking for *q-*analogues of mathematical objects, which have the original object as limits when *q* tends to 1. *q*-Calculus has wide-ranging applications in analytic number theory and theoretical physics. The main topic of the thesis is the invention of the tilde operator and the renaissance of the *q-*addition. There are two types of *q-*addition, the Ward-AlSalam *q-*addition and the Hahn *q-*addition. The first is both commutative and associative, while the second is neither. This is one of the reasons why sometimes more than one *q-*analogue exist. These two operators form the basis of the method which unites hypergeometric series and *q-*hypergeometric series and which gives many formulas of *q-*calculus a natural form reminding directly of their classical origin. This method is reminiscent of Heine, who mentioned the case where one parameter in a *q-*hypergeometric series is plus infinity. The *q-*addition is the natural way to extend addition to the *q-*case as is shown when restating addition formulas for *q-*trigonometric functions.

We give a more lucid definition of the *q-*difference operator. A new notation for powers of *q* reminding of the exponential function is given. A *q-*Taylor formula with remainder term expressed as *q-*integral is proved.

We present a new expression for generalized Vandermonde determinants, and thus for the Schur function. We also obtain an equivalence relation on the set of all generalized Vandermonde determinants. We find a more general expression for the Vandermonde determinant. We show the connection to a determinant of Flowe and Harris and to the solution of difference and *q-*difference equations with constant coefficients.

isbn
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