# A good reference to grok hypergeometric functions?

When I was introduced during my degree to special functions, I made friends with a number of nice functions - Laguerre, Legendre, Hermite, Bessel, and whatnot - but I made only a passing acquaintance with the hypergeometric and confluent hypergeometric functions, mostly limited to looking them up in Arfken and recoiling in horror at the dryness of the material and the lack of physical content in the calculations.

I know, of course, that this lack of physical content is also accompanied by an astounding generality. After a while I did get the core of the idea, which I believe is "explore all special functions whose series coefficients are rational functions of $n$", and I do find it appealing, but I've not had the energy nor the motivation to follow that up and see what's interesting about the thing.

However, it appears that the long-delayed moment is here and some pretty hairy integrals (think $\int_0^\infty x^k e^{-\alpha x^2}J_m(\beta x)dx$) have pushed some ugly "${}_1 F_1$" symbols onto my page. So my question is, then: what's a good introduction to hypergeometric and confluent hypergeometric functions? I'd like one where I can get an intuitive understanding of what to expect from them in different circumstances, what nice properties they have, and generally why it really is worth it to deal with them instead of their more specific cases like Laguerre, Legendre, Hermite, Bessel, etc.

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A course of Modern Analysis by Whittaker and Watson. No, I'm not kidding! They have a wonderful chapter that ends with a wonderful classification of the main special functions according to their singularity patterns.

The best modern textbook really is A=B (cited already). The modern approach to all special functions is to realize that they are a very special case of holonomic functions: solutions of linear ordinary differential equations with polynomial coefficients. In fact, the hypergeometrics are special amongst holonomic functions because the coefficients of their series solutions doesn't just satisfy a linear recurrence equation (they all do), they have one of order 1. So that makes them very special indeed, for a structural, mathematical reason, not for socio-cultural reasons (the old view of 'special' functions was that they were special because they had a name that stuck because they seemed to come up often).

Once you get that, A=B will tell you a lot of the simple story. Then you can read things like Kauers' Algorithms for Holonomic Functions for more advanced material.

Then it really depends on where you want to go next. There are papers on fast and provably accurate evaluation, proving identities, or combinatorics [the link between special functions and combinatorics via the theory of Species is fascinating!].

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Whittaker-Watson's wonderful book is freely available at archive.org/details/acoursemodernan00watsgoog, University Press, 1902. –  Papiro Jun 23 '12 at 11:17
I would not be quite so quick to disparage the 'old' view on special functions; this essay by Michael Berry argues eloquently against it. –  Emilio Pisanty Jul 13 '13 at 15:10

Andrews, Askey, Roy "Special Functions" Cambridge (1999) has three chapters totalling 180 pages devoted explicitly to hyper geometric functions and confluent hypergeometric functions, and they arise frequently in other parts. It is a really interesting book.

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A very good place for Hypergeometric Functions is Wolfram Alpha here.

A pre-$\LaTeX$ report from Pincherle, Hypergeometric functions and various related problems, NASA, December 1, 1965, is available here.

1. Chapter 13 Confluent Hypergeometric Functions
2. Chapter 15 Hypergeometric Function

J Pearson, Computation of Hypergeometric Functions, MSc. Dissertation, Oxford, 2009. Matlab code here

Z.X.Wang, D.R. Guo, Special Functions, World Scientific Publishing, 1989

M. Yoshida, Hypergeometric Functions, My Love, Vieweg, 1997

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To avoid some potential misconception that 'report' is considerably older than the answer suggests since it is a translation. The original is late 19th century. –  quid Jun 22 '12 at 19:01
Thanks, quid!! The report is a translation of Delie funzioni ipergeometriche e di varie question! ad esse attinenti, Giomali di Matsmatiche de Battaglini, Vol. 32, pp. 209-291, Naples 1894. –  Papiro Jun 22 '12 at 19:16

A very interesting book is "A=B" Marko Petkovsek, Herbert Wilf and Doron Zeilberger.

This book emphasizes in algorithms for doing the sums symbolically, and is, for instance, the theory behind the function "summation()" in sympy.

The Book "A=B" www.cis.upenn.edu/~wilf/AeqB.html

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Here is a diagram of special function relationships. Hypergeometric functions form the spine of the diagram. It doesn't go into much depth, but it might help establish some mental landmarks.

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Felix Klein's book Über Die Hypergeometrische Function is truly nice. I also second the recommendations to Whittaker and Watson and Andrews, Askey, Roy.

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Physics oriented introduction is given in http://link.springer.com/book/10.1007/978-1-4757-5443-8 (Hypergeometric Functions and Their Applications, by James B. Seaborn).

This review article http://iopscience.iop.org/0036-0279/45/1/R02 (Ramanujan and hypergeometric and basic hypergeometric series, by R. Askey) is also recommended.

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I would also recommend "Special Functions and their Application" by Lebedev and Silverman (http://www.amazon.com/Special-Functions-Their-Applications-Mathematics/dp/0486606244/ref=sr_1_1/179-2267788-6339652?s=books&ie=UTF8&qid=1422720681&sr=1-1) and the series "Higher Transcendental Functions" (part of the Bateman Manuscript Project, edited by Erdelyi) which has the benefit of being free online via Caltech:

http://authors.library.caltech.edu/43491/

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