By Thomas Ernst

ISBN-10: 303480430X

ISBN-13: 9783034804301

To date, the theoretical improvement of q-calculus has rested on a non-uniform foundation. in most cases, the cumbersome Gasper-Rahman notation used to be used, however the released works on q-calculus seemed diversified reckoning on the place and by way of 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 booklet overcomes those difficulties through introducing a brand new and fascinating notation for q-calculus in response to 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 determine the relationship among q-hypergeometric features and the q-gamma functionality, whatever that earlier has been overlooked.

The publication covers many subject matters on q-calculus, together with designated services, combinatorics, and q-difference equations. except an intensive evaluation of the ancient improvement of q-calculus, this e-book additionally provides the domain names of contemporary physics for which q-calculus is acceptable, comparable to particle physics and supersymmetry, to call only a few.

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Other strong representatives of this School are Eugene Wigner (1902–1995) and John von Neumann (1903–1957), who also worked without q, but strongly influenced the development of quantum groups. Their works are of high quality, but rather difficult. Wigner introduced the 3-j coefficients, but his formulas could have been greatly improved by using hypergeometric functions in his formulas. This came only in our time, when Joris Van der Jeugt used multiple hypergeometric functions for 9-j coefficients.

M. Srivastava was born in India, settled in Canada in the early seventies, but travelled frequently to India until 1985. He was a good friend of and therefore also inspired by Gould of the American–Austrian School. Srivastava left India permanently in the 1980s and without its founder the School quickly dwindled. g. Laguerre and/or differential operators. Recently (2009), Srivastava again visited Vijay Gupta in India and perhaps something interesting will arise from this cooperation. H. L. Manocha (Polytechnic Institute of New York University) has written a book about generating functions [484]; one of his graduate students was Vivek Sahai.

1] n E f (x) = n i=0 n i i f (x). 4 [320, p. 97, 10], [138, p. 13], [385, p. 35, 2], [411, p. 19]. n n (f g) = i=0 n i i f n−i i E g. 7) In Chapter 4, we will retain the binomial coefficients in the corresponding qformulas, whereas in Chapter 5, q-binomial coefficients for the corresponding formulas will be used. Karl Weierstraß said that the calculus of finite differences will once play a leading role in mathematics. Two important elements of the calculus of finite differences are the Bernoulli numbers and the function.

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

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