By Thomas Ernst

ISBN-10: 303480430X

ISBN-13: 9783034804301

To date, the theoretical improvement of q-calculus has rested on a non-uniform foundation. commonly, the cumbersome Gasper-Rahman notation used to be used, however the released works on q-calculus seemed diverse reckoning on the place and by means of whom they have been written. This confusion of tongues not just complex the theoretical improvement but additionally contributed to q-calculus closing a ignored mathematical box. This e-book overcomes those difficulties through introducing a brand new and fascinating notation for q-calculus in keeping with logarithms.For example, q-hypergeometric capabilities at the moment are visually transparent and straightforward to track again to their hypergeometric mom and dad. With this new notation it's also effortless to work out the relationship among q-hypergeometric capabilities and the q-gamma functionality, whatever that previously has been overlooked.

The e-book covers many themes on q-calculus, together with unique features, combinatorics, and q-difference equations. except a radical assessment of the old improvement of q-calculus, this ebook additionally provides the domain names of contemporary physics for which q-calculus is appropriate, corresponding to particle physics and supersymmetry, to call only a few.

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**Additional info for A Comprehensive Treatment of q-Calculus**

**Example text**

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. Since products of q-Pochhammer symbols and q-shifted factorials occur so often, to simplify them we shall frequently use the following more compact notation. Let (a) = (a1 , . . , aA ) be a vector with A elements. Then A (a) n,q ≡ {a1 , . . 37) j =1 A (a); q ≡ a1 , . . , aA ; q n n≡ aj ; q n . 38) j =1 Let the Gaussian q-binomial coefficients [225] be defined by n k ≡ q 1; q 1; q n k 1; q , n−k k = 0, 1, .

Their short correspondence in 1654 [66, p. 107] founded the theory of probability.

Cayley defined a group as a set of objects with a multiplication 38 2 The different languages of q-calculus and a symbolic equation θ n = 1. Blissard was obviously influenced by Cayley’s work, at least in what concerns his notation. Another French entry came from E. Lucas, who invented a modern notation for umbral calculus. The Lucas umbral calculus was widespread in Russia, for example one finds the defining formula for the Bernoulli numbers in Chistiakov 1895 [127, p. 105]. The Blissard umbral calculus has attracted attention in Russia; in [127, p.

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

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