By Thomas Ernst

To date, the theoretical improvement of q-calculus has rested on a non-uniform foundation. ordinarily, the cumbersome Gasper-Rahman notation used to be used, however the released works on q-calculus seemed varied looking on the place and through whom they have been written. This confusion of tongues not just advanced the theoretical improvement but additionally contributed to q-calculus final a missed mathematical box. This booklet overcomes those difficulties by way of introducing a brand new and engaging notation for q-calculus in accordance with logarithms.For example, q-hypergeometric services 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 services and the q-gamma functionality, whatever that earlier has been overlooked.

The ebook covers many subject matters on q-calculus, together with certain capabilities, combinatorics, and q-difference equations. except a radical evaluate of the ancient improvement of q-calculus, this e-book additionally offers the domain names of recent physics for which q-calculus is acceptable, equivalent to particle physics and supersymmetry, to call only a few.

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1 cd Eq (x) Definition 8 ∞ Eq (z) ≡ k=0 Diagram 1 1 k z . {k}q ! 7 The first q-functions 23 Diagram 2 Diagram 3 The speed can be compared to the q-derivative. 2 ⎧ ϕ(x)−ϕ(qx) ⎪ ⎨ (1−q)x , (Dq ϕ)(x) ≡ dϕ dx (x), ⎪ ⎩ dϕ dx (0), if q ∈ C\{1}, x = 0, if q = 1, if x = 0. 52) 0 < |q| < 1, a ∈ R. 53) f (t, q) dq (t), 0 where a 0 ∞ f (t, q) dq (t) ≡ a(1 − q) f aq n , q q n , n=0 24 1 Introduction The second, which is the unit of measurement for the time, may be compared to the Cigler -operator. This operator is the simplest operator in q-analysis; it is called σ in French literature and η in the papers by Rogers and Jackson and in Hardy and Wright [268, p.

According to Moritz Cantor (1829–1920) [96, p. 800], Descartes and Fermat were the greatest mathematicians mentioned here. 44 2 The different languages of q-calculus At that time Fermat and then also Torricelli, had already generalized the power formula to rational exponents n = −1. Fermat determined areas under curves which he called general parabolas and general hyperbolas. This was equivalent to calculating integrals for fractional powers of x. Vincenzo Viviani (1622–1703), another prominent pupil of Galilei, determined the tangent to the cycloid.

The power function is defined by q a ≡ ea log(q) . 29) means that q a q b = q a+b . 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.