# Bilateral basic hypergeometric series

I continue to marvel at the high level at which you are working. Very nice work. It might take a while to be fully appreciated but I think quality eventually will win out. It seems relevant to what you're doing Post a Comment. Wednesday, March 17, Hypergeneralization.

The generalized hypergeometric function can itself be generalized endlessly. I have recently implemented three such extensions in mpmath: bilateral series, two-dimensional hypergeometric series, and q -analog or "basic" hypergeometric series.

Anyway, the convergent case is the interesting one. There are three other Appell functions: F2, F3, F4. The Horn functions are 34 distinct functions of order two, containing the Appell functions as special cases. The new hyper2d function in mpmath can evaluate all these named functions, and more general functions still. The trick for speed is to write the series as a nested series, where the inner series is a generalized hypergeometric series that can be evaluated efficiently with hyperand where the outer series has a rational recurrence formula.

This rewriting also permits evaluating the analytic continuation with respect to the inner variable as implemented by hyper. The user specifies the format of the series in quasi-symbolic form, and the rewriting to nested form is done automatically by mpmath.

The Appell F2-F4 functions have also been added explicitly as appellf2appellf3appellf4. Hypergeometric functions of two or more variables have numerous applications, such as solving high-order algebraic equations, expressing various derivatives and integrals in closed form, and solving differential equations, but I have not yet found any simple examples that make good demonstrations except for F1. I have mostly found examples of that take half a page to write down. Any such examples for the documentation would be a welcome contribution!

But of course, nsum is much slower than hyper2d. Hypergeometric q -series Before introducing the q -analog of the hypergeometric series, I should introduce the q -Pochhammer symbolThis itself is a new function in mpmath, implemented as qp a,q,n with two- and one-argument forms qp a,q and qp q also permitted and is the basis for more general computation involving q-analogs.

The q -factorial and q -gamma function have also been added as qfac and qgammabut are not yet documented. The q -analogs have important applications in number theory. Replacing the rising factorials Pochhammer symbols in the generalized hypergeometric series with their q -analogs gives the hypergeometric q -series or basic hypergeometric series This function is implemented as qhyper. If there is interest, such function could be added explicitly to mpmath.

For more information and examples of the functions discussed in this post, see the sections on hypergeometric functions and q -functions a little terse at the moment in the mpmath documentation. Posted by Fredrik Johansson at Wednesday, March 17, Labels: mpmathsagesympy.

Newer Post Older Post Home. Subscribe to: Post Comments Atom. View my complete profile.This paper is motivated by the umbral calculus approach to basic hypergeometric series as initiated by Goldman-Rota, Andrews, and Roman, et al. We develop a method of deriving hypergeometric identities by parameter augmentation, which means that a hypergeometric identity with multiple parameters may be derived from its special case obtained by reducing some parameters to zero.

This process is experimental and the keywords may be updated as the learning algorithm improves. This is a preview of subscription content, log in to check access. Andrews, On a transformation of bilateral series with applications, Proc. Andrews, On the foundations of combinatorial theory V. Eulerian differential operators, Studies in Appl. Askey, ed. Google Scholar. Monthly 8689— Andrews and R. Askey and M. MathSciNet Google Scholar. Monthly 87— Oxford 32 2, — Askey and J.

Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Memoirs. Bailey, Series of hypergeometric type which are infinite in both directions, Quart. CrossRef Google Scholar. Oxford 1 2, — Gasper and M. Goldman and G.

Tutte, ed. Goulden and D. Paule, On identities of the Rogers-Ramanujan type, J. Rogers, Third memoir on the expansion of certain infinite product, Proc. London Math. Roman, More on the umbral calculus, with emphasis on the q -umbral calculs, J. Slater and A.Libraries and resellers, please contact cust-serv ams. See our librarian page for additional eBook ordering options. The theory of partitions, founded by Euler, has led in a natural way to the idea of basic hypergeometric series, also known as Eulerian series. These series were first studied systematically by Heine, but many early results are attributed to Euler, Gauss, and Jacobi.

However, the theory has been developed to such an extent and with such a profusion of powerful and general results that the subject can appear quite formidable to the uninitiated.

By providing a simple approach to basic hypergeometric series, this book provides an excellent elementary introduction to the subject. The author presents an elementary method for using these equations to obtain transformations of the original function. A bilateral series, formed from this function, is summed as an infinite product, thereby providing an elegant and fruitful result which goes back to Ramanujan.

By exploiting a special case, the author is able to evaluate the coefficients of several classes of infinite products in terms of divisor sums. He also touches on general transformation theory for basic series in many variables and the basic multinomial, which is a generalization of a finite sum. These developments lead naturally to the arithmetic domains of partition theory, theorems of Liouville type, and sums of squares.

Contact is also made with the mock theta-functions of Ramanujan, which are linked to the rank of partitions. The author gives a number of examples of modular functions with multiplicative coefficients, along with the beginnings of an elementary constructive approach to the field of modular equations. Requiring only an undergraduate background in mathematics, this book provides a rapid entry into the field. Students of partitions, basic series, theta-functions, and modular equations, as well as research mathematicians interested in an elementary approach to these areas, will find this book useful and enlightening. Because of the simplicity of its approach and its accessibility, this work may prove useful as a textbook.

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Rich with examples and results from the theory of partitions, the study of Ramanujan's mock theta functions, and modular equations.

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Advanced search. Author s Product display : Nathan Fine.Enter your mobile number or email address below and we'll send you a link to download the free Kindle App. Then you can start reading Kindle books on your smartphone, tablet, or computer - no Kindle device required.

To get the free app, enter your mobile phone number. Would you like to tell us about a lower price? If you are a seller for this product, would you like to suggest updates through seller support? The theory of partitions, founded by Euler, has led in a natural way to the idea of basic hypergeometric series, also known as Eulerian series.

These series were first studied systematically by Heine, but many early results are attributed to Euler, Gauss, and Jacobi. However, the theory has been developed to such an extent and with such a profusion of powerful and general results that the subject can appear quite formidable to the uninitiated.

By providing a simple approach to basic hypergeometric series, this book provides an excellent elementary introduction to the subject.

The author presents an elementary method for using these equations to obtain transformations of the original function. A bilateral series, formed from this function, is summed as an infinite product, thereby providing an elegant and fruitful result which goes back to Ramanujan.

By exploiting a special case, the author is able to evaluate the coefficients of several classes of infinite products in terms of divisor sums. He also touches on general transformation theory for basic series in many variables and the basic multinomial, which is a generalization of a finite sum.

These developments lead naturally to the arithmetic domains of partition theory, theorems of Liouville type, and sums of squares.

### Bilateral hypergeometric series

Contact is also made with the mock theta-functions of Ramanujan, which are linked to the rank of partitions. The author gives a number of examples of modular functions with multiplicative coefficients, along with the beginnings of an elementary constructive approach to the field of modular equations.

Requiring only an undergraduate background in mathematics, this book provides a rapid entry into the field. Students of partitions, basic series, theta-functions, and modular equations, as well as research mathematicians interested in an elementary approach to these areas, will find this book useful and enlightening.

Because of the simplicity of its approach and its accessibility, this work may prove useful as a textbook. Read more Read less. Beyond your wildest dreams. Listen free with trial. Kindle Cloud Reader Read instantly in your browser. Register a free business account. Review Rich with examples and results from the theory of partitions, the study of Ramanujan's mock theta functions, and modular equations.

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The model takes into account factors including the age of a rating, whether the ratings are from verified purchasers, and factors that establish reviewer trustworthiness. Top Reviews Most recent Top Reviews. There was a problem filtering reviews right now.The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition. I was in graduate school when the first edition of this book appeared in I knew some of the subject matter from Dick Askey's special functions courses and seminars, and I had seen a preliminary version of the book at a conference at Ohio State inand I remember very well how excited I was when my copy arrived.

I want to write "It did not disappoint," but actually that's not quite true; I was a dumb kid, and I was hoping it would have more on the combinatorial aspects of q-series than it does. Nevertheless I immediately set about reading it and doing problems, and I believe I was one of the first to send in a list of errata. I still have George Gasper's reply of 27 Augustthanking me for my list and commiserating with me on my struggles with exercise 5. The book really was a major event in one part of mathematics, and it was very well received.

George Andrews American Mathematical Monthlyvol. It is great! More recently, Steven Milne has emphasized the importance of the book in both his research and his teaching on p. Andrews and Wimp both stressed the pedagogical qualities of the book, particularly the large collection of exercises, and both noted that this was what set it apart from other books on q-series that appeared in the s.

Gasper and Rahman tried to write a useful book, rather than a beautiful one, and, as Andrews said, "they have masterfully pulled it off. Let me attempt a brief description of the subject first. The word "Basic", though standard, is apt to cause confusion. In the most interesting cases there are as many shifted factorials in the numerator as in the denominator, except that the denominator always has an n!

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A 1-F-0 is the binomial series 1-x -a. Since there would typically be as many of these in the numerator as in the denominator, there are some factors of 1-q n that could be cancelled, so one might as well look at quotients of products of the form. This then is a basic or q- hypergeometric series — a power series where the coefficient c n of x n is a quotient of products of these q-shifted factorials a;q n ; q is the "base", and typically for convergence q n in the denominator analogous to the n!

Thus a trigonometric analogue of the shifted factorial would be something like So this idea, while intriguing at first sight, seems on reflection not to be very interesting. It really is interesting, though, because it hints at a further generalization. There are three new chapters in this edition, of which the last is by far the most interesting. It is about so-called elliptic hypergeometric functions, power series whose coefficients are quotients of products of shifted factorials built up as above from "elliptic numbers", which are themselves quotients of theta functions, and therefore far-reaching generalizations of the trigonometric numbers.

Alternatively, one could build these "elliptic" shifted factorials directly out of theta functions. These elliptic hypergeometric functions arose for the first time in the mids, in work by Frenkel and Turaev on a model in statistical mechanics, and this was rather like the discovery of a new Pacific island. The preface to the second edition lists 28 people who have done important work on these functions in the last 10 years.

This is the most exciting development in q-series in that time, and the authors say that "a new edition could be justified only if we included a chapter" on them. Some new material e. Chapters are largely unchanged. Several sections of chapter 2 are slightly improved by using the idea of a very-well-poised-balanced series. There are about 4 new problems in each chapter more in chapter 8, less in chapter 4added at the end so that all the original problems still have the same numbers.

One of the strongest points of the book is the references, which have been updated they now occupy pagesas have the bibliographical notes at the end of each chapter. To my mind, this also amply justifies a new edition. The result of exercise 1. This is V. Lebesgue, in a paper in Liouville's Journal innot Henri Lebesgue, then aged It appears without proof in Jacobi's epoch-making Fundamenta Nova offormula 8 in the 66th and final section. As Lebesgue and Jacobi both point out, it generalizes an identity of Gauss from Gauss's identity is also a special case of Jacobi's triple product identity, the subject of section 64 of the Fundamenta Nova.The q -derivative operator approach is illustrated by reviewing several typical summation formulae of terminating basic hypergeometric series.

Al-Salam, A. Verma: On quadratic transformations of basic series. SIAM J. Andrews: On q -analogues of the Watson and Whipple summations. Andrews: Connection coefficient problems and partitions. Pure Math. Bailey: Generalized Hypergeometric Series. Cambridge University Press, Cambridge Bailey: A note on certain q -identities. Oxford 12 — Oxford 1 — Bressoud: Almost poised basic hypergeometric series. Indian Acad.

## Parameter Augmentation for Basic Hypergeometric Series, I

Carlitz: Some formulas of F. Carlitz: Some q -expansion formulas. Glasnik Mat. Chu: Inversion techniques and combinatorial identities: Basic hypergeometric identities. Debrecen 44 3—4 — Chu: Inversion techniques and combinatorial identities: Strange evaluations of basic hypergeometric series.

Chu: Partial fractions and bilateral summations. Chu: Partial-fraction expansions and well-poised bilateral series. Acta Sci. Chu: q -Derivative operators and basic hypergeometric series. Results Math. In: S. Elaydi et al. World Scientific Publishers — Chu: Divided differences and generalized Taylor series. Forum Math. Discrete Math. Chu: Elementary proofs for convolution identities of Abel and Hagen-Rothe. Chu, C. Wang: Bilateral inversions and terminating basic hypergeometric series identities.These series were first studied systematically by Heine, but many early results are attributed to Euler, Gauss, and Jacobi. However, the theory has been developed to such an extent and with such a profusion of powerful and general results that the subject can appear quite formidable to the uninitiated.

By providing a simple approach to basic hypergeometric series, this book provides an excellent elementary introduction to the subject. The author presents an elementary method for using these equations to obtain transformations of the original function.

A bilateral series, formed from this function, is summed as an infinite product, thereby providing an elegant and fruitful result which goes back to Ramanujan. By exploiting a special case, the author is able to evaluate the coefficients of several classes of infinite products in terms of divisor sums.

He also touches on general transformation theory for basic series in many variables and the basic multinomial, which is a generalization of a finite sum. These developments lead naturally to the arithmetic domains of partition theory, theorems of Liouville type, and sums of squares. Contact is also made with the mock theta-functions of Ramanujan, which are linked to the rank of partitions. The author gives a number of examples of modular functions with multiplicative coefficients, along with the beginnings of an elementary constructive approach to the field of modular equations.

Requiring only an undergraduate background in mathematics, this book provides a rapid entry into the field. Students of partitions, basic series, theta-functions, and modular equations, as well as research mathematicians interested in an elementary approach to these areas, will find this book useful and enlightening. Because of the simplicity of its approach and its accessibility, this work may prove useful as a textbook.

Author by : George Gasper Languange : en Publisher by : Cambridge University Press Format Available : PDF, ePub, Mobi Total Read : 14 Total Download : File Size : 43,7 Mb Description : This revised and expanded new edition will continue to meet the needs for an authoritative, up-to-date, self contained, and comprehensive account of the rapidly growing field of basic hypergeometric series, or q-series.

Simplicity, clarity, deductive proofs, thoughtfully designed exercises, and useful appendices are among its strengths.

Hypergeometric equation and its solution. (MATH)

The first five chapters cover basic hypergeometric series and integrals, whilst the next five are devoted to applications in various areas including Askey-Wilson integrals and orthogonal polynomials, partitions in number theory, multiple series, orthogonal polynomials in several variables, and generating functions.

Chapters are new for the second edition, the final chapter containing a simplified version of the main elements of the theta and elliptic hypergeometric series as a natural extension of the single-base q-series. Some sections and exercises have been added to reflect recent developments, and the Bibliography has been revised to maintain its comprehensiveness. Author by : Kevin W.

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Contributions were made by Gauss, Jacobi, and Cauchy, but the first attempt at a systematic development, especially from the point of view of studying series with the products in the summands, was made by E. Heine in In the latter part of the nineteenth and in the early part of the twentieth centuries, two English mathematicians, L. Rogers and F. Jackson, made fundamental contributions. InG. This is now one of the fundamental theorems of the subject.

This event gathered mathematicians from the world over to lecture and discuss their research. This volume presents nineteen of the papers presented at the conference. It will be of interest to researchers and graduate students that would like to learn of recent developments in the theory of q-series and modular and how it relates to number theory, combinatorics and special functions.

Many of the latest advances in the field were inspired by the works of R. Askey and colleagues on basic hypergeometric series and I. Macdonald on orthogonal polynomials related to root systems. Significant progress was made by the use of algebraic techniques involving quantum groups, Hecke algebras, and combinatorial methods.

The CRM organized a workshop for key researchers in the field to present an overview of current trends.