A good reference to grok hypergeometric functions? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T15:17:12Z http://mathoverflow.net/feeds/question/100364 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/100364/a-good-reference-to-grok-hypergeometric-functions A good reference to grok hypergeometric functions? Emilio Pisanty 2012-06-22T14:52:15Z 2012-07-10T16:33:10Z <p>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.</p> <p>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.</p> <p>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: <strong>what's a good introduction to hypergeometric and confluent hypergeometric functions?</strong> 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 <em>is</em> worth it to deal with them instead of their more specific cases like Laguerre, Legendre, Hermite, Bessel, etc.</p> http://mathoverflow.net/questions/100364/a-good-reference-to-grok-hypergeometric-functions/100376#100376 Answer by Kjetil B Halvorsen for A good reference to grok hypergeometric functions? Kjetil B Halvorsen 2012-06-22T16:56:30Z 2012-06-22T16:56:30Z <p>A very interesting book is "A=B" Marko Petkovsek, Herbert Wilf and Doron Zeilberger.</p> <p>This book emphasizes in algorithms for doing the sums symbolically, and is, for instance, the theory behind the function "summation()" in sympy.</p> <p>The Book "A=B" www.cis.upenn.edu/~wilf/AeqB.html</p> http://mathoverflow.net/questions/100364/a-good-reference-to-grok-hypergeometric-functions/100384#100384 Answer by PaPiro for A good reference to grok hypergeometric functions? PaPiro 2012-06-22T18:43:02Z 2012-07-10T16:33:10Z <p>A very good place for Hypergeometric Functions is Wolfram Alpha <a href="http://reference.wolfram.com/mathematica/guide/HypergeometricFunctions.html" rel="nofollow">here</a>.</p> <p>A pre-$\LaTeX$ report from Pincherle, <em>Hypergeometric functions and various related problems</em>, NASA, December 1, 1965, is available <a href="http://archive.org/stream/nasa_techdoc_19660004770/19660004770#page/n0/mode/2up" rel="nofollow">here</a>.</p> <p><strong>ADDED:</strong></p> <p>NIST <a href="http://dlmf.nist.gov/" rel="nofollow">Digital Library of Mathematical Functions:</a></p> <ol> <li>Chapter 13 <a href="http://dlmf.nist.gov/13" rel="nofollow">Confluent Hypergeometric Functions</a></li> <li>Chapter 15 <a href="http://dlmf.nist.gov/15" rel="nofollow">Hypergeometric Function</a></li> </ol> <p>J Pearson, <a href="http://people.maths.ox.ac.uk/porterm/research/pearson_final.pdf" rel="nofollow">Computation of Hypergeometric Functions</a>, MSc. Dissertation, Oxford, 2009. Matlab code <a href="http://people.maths.ox.ac.uk/~porterm/research/hypergeometricpackage.zip" rel="nofollow">here</a></p> <p>Z.X.Wang, D.R. Guo, <a href="http://books.google.com.br/books?id=d4a99Fls3CcC&amp;printsec=frontcover&amp;hl=en&amp;source=gbs_ge_summary_r&amp;redir_esc=y#v=onepage&amp;q&amp;f=false" rel="nofollow">Special Functions</a>, World Scientific Publishing, 1989</p> <p>M. Yoshida, <a href="http://openlibrary.org/books/OL282713M/Hypergeometric_functions_my_love" rel="nofollow">Hypergeometric Functions, My Love</a>, Vieweg, 1997</p> http://mathoverflow.net/questions/100364/a-good-reference-to-grok-hypergeometric-functions/100390#100390 Answer by Chris Godsil for A good reference to grok hypergeometric functions? Chris Godsil 2012-06-22T20:14:53Z 2012-06-23T15:25:51Z <p>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.</p> http://mathoverflow.net/questions/100364/a-good-reference-to-grok-hypergeometric-functions/100409#100409 Answer by John D. Cook for A good reference to grok hypergeometric functions? John D. Cook 2012-06-22T23:21:51Z 2012-06-22T23:27:22Z <p>Here is a <a href="http://www.johndcook.com/special_function_diagram.html" rel="nofollow">diagram</a> 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. </p> http://mathoverflow.net/questions/100364/a-good-reference-to-grok-hypergeometric-functions/100412#100412 Answer by Jacques Carette for A good reference to grok hypergeometric functions? Jacques Carette 2012-06-23T00:26:23Z 2012-06-23T00:26:23Z <p><a href="http://www.amazon.com/Course-Analysis-Cambridge-Mathematical-Library/dp/0521588073" rel="nofollow">A course of Modern Analysis</a> 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.</p> <p>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 <em>holonomic functions</em>: 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 <strong>order 1</strong>. 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).</p> <p>Once you get that, A=B will tell you a lot of the simple story. Then you can read things like Kauers' <a href="http://www.risc.jku.at/people/mkauers/publications/kauers11c.pdf" rel="nofollow">Algorithms for Holonomic Functions</a> for more advanced material.</p> <p>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!].</p> http://mathoverflow.net/questions/100364/a-good-reference-to-grok-hypergeometric-functions/100434#100434 Answer by alvarezpaiva for A good reference to grok hypergeometric functions? alvarezpaiva 2012-06-23T08:54:55Z 2012-06-23T08:54:55Z <p>Felix Klein's book Über Die Hypergeometrische Function is truly nice. I also second the recommendations to Whittaker and Watson and Andrews, Askey, Roy.</p>