Zeilberger, Doron A holonomic systems approach to special functions identities. (English) Zbl 0738.33001 J. Comput. Appl. Math. 32, No. 3, 321-368 (1990). 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. Reviewer: R.Askey (Madison) Cited in 7 ReviewsCited in 225 Documents MathOverflow Questions: D-finiteness of Hilbert series of non-commutative invariant ring under reductive group MSC: 33C20 Generalized hypergeometric series, \({}_pF_q\) Keywords:summation of series; holonomic systems Software:DEtools × Cite Format Result Cite Review PDF Full Text: DOI References: [1] G. Almkvist, Zeilbergers bevismaskin, Elementa, to appear (in Swedish).; G. Almkvist, Zeilbergers bevismaskin, Elementa, to appear (in Swedish). [2] G. Almkvist and D. Zeilberger, The method of differentiation under the integral sign, J. Symbolic Comput., to appear.; G. Almkvist and D. Zeilberger, The method of differentiation under the integral sign, J. Symbolic Comput., to appear. · Zbl 0717.33004 [3] Andrews, G. E., Connection coefficient problems and partitions, (Ray-Chaudhuri, D., AMS Proc. 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Zeilberger, The method of creative telescoping for q-series, in preparation. · Zbl 0738.33002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.