I'm looking for a book, set of tables, or other reference which contains a comprehensive list of hypergeometric identities; that is, something which allows a hypergeometric fucntion to be expressed in terms of elementary or other known special functions.

For example, from Wikipedia:


$(1-z)^{-a} = \,_2F_1(a,b;b;z)$

$\arcsin z = z \,_2F_1\left(\tfrac{1}{2}, \tfrac{1}{2}; \tfrac{3}{2};z^2\right)$

Now I should mention that Wolfram Alpha and presumably mathematica can also do this; Here's an example. However, I would like a written reference, preferably in table form, and either with its own proofs of identities, or with references to sources containing proofs.

Does anyone know of any such "dictionaries" for Hypergeometric functions?

  • 1
    $\begingroup$ To save anyone else with similar suspicions to me the effort of checking, Concrete Mathematics doesn't have such a table. $\endgroup$ – Peter Taylor May 29 '11 at 21:30

Have a look here; in particular, see this subsection (Special Cases). Note that you can find more information by pressing the $i$ symbols on the right.

  • $\begingroup$ Note also the link to Prudnikov et al. (1990, pp. 468–488), for an extensive list of elementary representations. $\endgroup$ – Shai Covo May 29 '11 at 11:46

I'm not sure of a table of such results apart from those in the NIST Digital Library of Mathematical Functions that Shai linked to above and those in the wolfram.functions site.

I do know that Mathematica internally makes the translations to and from hypergeometric, Meijer G functions and other special functions. But this behaviour is not exposed to the normal end user.

There is also a GSoC project for SymPy that is looking at doing a large class of integrals using hypergeometric and Meijer G function identities. From it I learned that there is a general procedure for finding such identities (see, e.g., this paper and similar work by Kelly Roach). This project obviously needs to be able to write special functions in terms of hypergeometrics and vice versa - i.e. implement the same functionality that's hidden inside Mathematica.


Here are a few more sources:

K. Oldham, J. Myland, & J. Spanier, $An \ Atlas \ of \ Functions$, Ch. 60, Springer.

A. Erdelyi, $Higher \ Transcendental \ Functions$, Vol. 1 (and particularly, Sec. 2.8), Krieger Publishing.

I.S. Gradshteyn & I.M. Ryzhik, $Tables \ of \ Integrals, \ Series, \ and \ Products$, Sec. 9.1, Academic Press (various editions are available).

Good luck!


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