By Thomas Ernst
To date, the theoretical improvement of q-calculus has rested on a non-uniform foundation. more often than not, the cumbersome Gasper-Rahman notation was once used, however the released works on q-calculus seemed various looking on the place and by way of whom they have been written. This confusion of tongues not just advanced the theoretical improvement but additionally contributed to q-calculus ultimate a ignored mathematical box. This e-book overcomes those difficulties through introducing a brand new and fascinating notation for q-calculus according to logarithms.For example, q-hypergeometric features are actually visually transparent and simple 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 services and the q-gamma functionality, anything that previously has been overlooked.
The e-book covers many themes on q-calculus, together with designated features, combinatorics, and q-difference equations. except an intensive overview of the old improvement of q-calculus, this ebook additionally provides the domain names of recent physics for which q-calculus is appropriate, comparable to particle physics and supersymmetry, to call only a few.
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Extra info for A Comprehensive Treatment of q-Calculus
60) q Definition 13 Let a and b be elements of a commutative multiplicative semigroup. Then the NWA q-addition is given by n (a ⊕q b)n ≡ k=0 n k a k bn−k , q n = 0, 1, 2, . . 7 The first q-functions 25 Furthermore, we will put n (a n q b) ≡ k=0 n k a k (−b)n−k , n = 0, 1, 2, . . 60). 59) is valid. 60). Example 1 We display the NWA as a function of the first n = 1, . . 005. 99 (a ⊕q b)n n=1 n=2 n=3 n=4 n=5 n=6 n=7 n=8 n=9 1. 2. 63) first grows and then decreases and thus has an absolute maximum.
Like Blissard, Jackson worked as a priest his whole life; both of them had studied in Cambridge. To honour Jackson, we will use his notation for Eq (x). Steffensen , Jordan  and Milne-Thomson  wrote books about finite differences intended both for mathematicians and statisticians. Johann Cigler wrote an excellent book  on finite differences with a view to umbral calculus. 5 Elliptic and Theta Schools and notations, the oldest roots–the q-forerunners Just as the nicest equations in mechanics are connected with the torque (and the angular momentum), the theta functions and the elliptic functions give the most beautiful equations in calculus.
Elementary trigonometric formulas were given, and these trigonometric functions were used to treat the physics involved. This was natural since many earlier mathematicians, like the Bernoullis, were both physicians and physicists. John Bernoulli (1) has in fact even written a memoir about mathematical medicine. We will start with a brief history about trigonometry and its relationship to spherical triangles and astronomy. Probably Aristarchus of Samos, Greece (−260) [392, p. 108] used ratios similar to tangens.
A Comprehensive Treatment of q-Calculus by Thomas Ernst