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Does anyone know of a monograph/survey on the modern history of (basic or elliptic) hypergeometric functions and their applications?

I haven't had much time to search the literature, and because it is summer it is hard to reach professors or specialists, which is why I am asking the question here. It is also likely that there are obvious choices out there that I am unaware of because of my ignorance in the field. I would appreciate it a lot if along with the suggestions you could give a quick description of what the book/article treats.

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3 Answers 3

up vote 7 down vote accepted

Gjergji, there are remarkable articles by Richard Askey:

(1) "Ramanujan and hypergeometric and basic hypergeometric series" in
Russian Math. Surveys 45:1 (1990) 37--86; reprinted in Ramanujan: essays and surveys, Hist. Math. 22 Amer. Math. Soc., Providence, RI, 2001, pp. 277--324;

(2) "A look at the Bateman project" in The mathematical legacy of Wilhelm Magnus: groups, geometry and special functions (Brooklyn, NY, 1992), 29--43, Contemp. Math. 169, Amer. Math. Soc., Providence, RI, 1994.

(I asked Dick exactly this question, maybe without accenting on "modern theory", some years ago.) The modern theory is mostly multiple hypergeometric functions related to root systems; for a nice survey on the roots of these functions, the Selberg integral, see

(3) P. Forrester and S.O. Warnaar, "The importance of the Selberg integral", Bull. Amer. Math. Soc. (N.S.) 45:4 (2008) 489--534.

Elliptic functions are hypergeometric functions of the 21st century:

(4) V.P. Spiridonov, "Essays on the theory of elliptic hypergeometric functions", Russian Math. Surveys 63:3 (2008) 405--472.

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I am familiar with Spiridonov's article you mention, but the rest is all new to me. Thank you! –  Gjergji Zaimi Jul 17 '10 at 11:01
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"Basic Hypergeometric Series" 2nd Edition, George Gasper and Mizan Rahman, ISBN: 0521833574 would be a good place to start. Chapters 9-11 of the second edition are new and deal with:

  1. Linear and bilinear generating functions for basic orthogonal polynomials;
  2. q-series in two or more variables;
  3. Elliptic, modular, and theta hypergeometric series
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That does sound like a great start, thanks! I've never looked at the second edition. –  Gjergji Zaimi Jul 17 '10 at 9:09
    
The $q$-Bible definitely covers some history. Slater's book also does... –  Wadim Zudilin Jul 17 '10 at 9:35
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Hyper Geometric Functions, My Love: Modular Interpretations of Configuration Spaces (Aspects of Mathematics) by Masaaki Yoshida

Discriminants, Resultants, and Multidimensional Determinants Israel M. Gelfand, Mikhail Kapranov, Andrei Zelevinsky

I heard a talk by Rivoal a decade back when he had just had his breakthrough concerning irrationality of zeta values. He quoted some very classical but not well-known results and got asked where he learned them. He recommended this book:

Confluent Hypergeometric Functions L. J. Slater

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