By Thomas Ernst
To date, the theoretical improvement of q-calculus has rested on a non-uniform foundation. typically, the cumbersome Gasper-Rahman notation used to be used, however the released works on q-calculus regarded diverse 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 missed mathematical box. This publication overcomes those difficulties by way of introducing a brand new and fascinating notation for q-calculus according to logarithms.For example, q-hypergeometric capabilities are actually 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 capabilities and the q-gamma functionality, whatever that earlier has been overlooked.
The booklet covers many issues on q-calculus, together with distinct services, combinatorics, and q-difference equations. except a radical overview of the old improvement of q-calculus, this ebook additionally offers the domain names of recent physics for which q-calculus is appropriate, corresponding to particle physics and supersymmetry, to call only a few.
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Additional resources for A Comprehensive Treatment of q-Calculus
10. 11. 12. 13. the study of elliptic functions from the nineteenth century the development of theta functions additive analytic numbers theory, or theory of partitions the field of hypergeometric functions gamma function theory Bernoulli and Euler polynomials umbral calculus theory of finite differences combinatorics + q-binomial coefficient identities theory of finite fields + primitive roots Mock theta functions multiple hypergeometric functions elliptic integrals and Dedekind eta function The main trunk of the tree is the subject q.
Russian School: Russian mathematicians have greatly influenced the above mentioned Hungarian School through studies of polynomials. Russia has, in general, a strong tradition in mathematics, which dates back to Euler and the 18th century mathematicians who took up this heritage. 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 Euler himself died in St. Petersburg, and it is a well-established fact that both his direct influence and also his unpublished papers and work remained in Russia, which explains in part the high level of mathematics in Russia.
In the footsteps of Nielsen, Niels Erik Nørlund (1885–1981) in the remarkable work  gave the first rigorous treatment of finite differences from the perspective of the mathematician. Nørlund gave lectures on hypergeometric series in Copenhagen until 1955. Nørlund also knew q-calculus; F. Ryde published a thesis on this subject under his supervision in Lund. The next link in the chain is perhaps Per Karlsson (1936–), the expert on multiple hypergeometric functions and friend, among other things, collaborator and model of the author.
A Comprehensive Treatment of q-Calculus by Thomas Ernst