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A holonomic systems approach to special functions identities. (English) Zbl 0738.33001
It is possible to treat the subject of identities satisfied by his hypergeometric functions from many points of view. Lie groups and algebras are very useful for certain problems, addition theorems being one but far from the only one. The author has used other algebraic methods to develop methods to prove and often to derive identities, such as recurrence relations satisfied by polynomial hypergeometric series, or generating functions for them. The present paper uses J. Bernstein’s work on holonomic systems to set up a powerful machine, which is often effective in theory, and sometimes in practice.
Later, he developed other methods which are not as complete in theory, but are much more practical, so that identities of a certain important type can now be treated by one method. Some of the later work was joint with H. Wilf. Much of this works for basic hypergeometric series, as well as hypergeometric series.

##### MSC:
 33C20 Generalized hypergeometric series, $${}_pF_q$$
##### Keywords:
summation of series; holonomic systems
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