By Thomas Ernst

ISBN-10: 303480430X

ISBN-13: 9783034804301

To date, the theoretical improvement of q-calculus has rested on a non-uniform foundation. mostly, the cumbersome Gasper-Rahman notation was once used, however the released works on q-calculus regarded varied counting on the place and via whom they have been written. This confusion of tongues not just complex the theoretical improvement but in addition contributed to q-calculus last a ignored mathematical box. This e-book overcomes those difficulties by means of introducing a brand new and fascinating notation for q-calculus in accordance with logarithms.For example, q-hypergeometric capabilities at the moment are visually transparent and simple to track again to their hypergeometric mom and dad. With this new notation it's also effortless to determine the relationship among q-hypergeometric services and the q-gamma functionality, whatever that beforehand has been overlooked.

The ebook covers many themes on q-calculus, together with exact services, combinatorics, and q-difference equations. except an intensive assessment of the ancient improvement of q-calculus, this e-book additionally offers the domain names of recent physics for which q-calculus is appropriate, reminiscent of particle physics and supersymmetry, to call only a few.

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**Example text**

The USA and the Commonwealth, for two main reasons: Immediately after the first world war, ca. 1920 until 1925, the German and Austrian mathematicians were barred from participation in the big mathematical conferences; they were simply deemed political pariahs by the French, unwanted, and the communications with the English speaking q-analysts were for a period limited, as was the exchange of ideas. The second and more important reason has to do with languages. Most of the mathematicians of the Austrian School wrote in either German or French and, as regards the oldest, namely Euler, Gauß and Jacobi, in Latin.

1 kg, which is the unit for mass, can be compared to the q-factorial. 31) 1 + q a+m . 32) m=0 n−1 (a; q)n ≡ m=0 n−1 a; q n ≡ m=0 A q-analogue of a complex number is also a complex number. 34) {k}q , {0}q ! ≡ 1, q ∈ C\{0, 1}. 35) k=1 n {n}q ! ≡ k=1 20 1 Introduction Definition 3 The q-Pochhammer-symbol {a}n,q is defined by n−1 {a}n,q ≡ {a + m}q . 36) m=0 An equivalent symbol is defined in Exton [194, p. 18] and is used throughout that book. This quantity can be very useful in some cases where we are looking for q-analogues and it is included in the new notation.

Euler has very much contributed to the Proceedings of the St. Petersburg Academy. After his death, his successors could not keep the high level from before. Euler himself died in St. Petersburg, and it is a well-established fact that both his direct influence and also his unpublished papers and work remained in Russia, which explains in part the high level of mathematics in Russia. Many Russian mathematicians from the nineteenth century did excellent work in the area of Bernoulli numbers and umbral calculus, among them Grigoriew, Chistiakov and Imchenetsky.

### A Comprehensive Treatment of q-Calculus by Thomas Ernst

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