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A new method q-calculus
Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics.
2002 (English)Doctoral thesis, monograph (Other academic)
Abstract [en]

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.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis , 2002. , 116 p.
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 25
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URN: urn:nbn:se:uu:diva-2698ISBN: 91-506-1628-5OAI: oai:DiVA.org:uu-2698DiVA: diva2:162070
Public defence
2002-11-14, Room 2146, Mathematics and Information Technology Center, Polacksbacken, Uppsala, 13:15
Available from: 2002-10-17 Created: 2002-10-17Bibliographically approved

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