Diagram Technique for Hypergeometric Series of Several Variables

A convenient graphical algorithm for searching and constructing explicit analytical representations of all hypergeometric series belonging to any given class is proposed. The algorithm is based on a set of rules that define a one-to-one correspondence between analytic representations of series and diagrams. The algorithm can be easily implemented in the form of a computer program.     

Computer-Aided Analysis of Transformation Formulas for Appel and Horn Functions

Simultaneous use of general and special linear transformations allows 36 transformations for the Appel function F ₄ to be obtained. This result is important because F ₄ is widely used in physics and mathematics. The transformations express F ₄ in terms of the Appel functions F ₁, F ₂, and F ₃; the Horn function H ₂; and non-Hornian series G, K, and . Until now, it was not possible to obtain linear multiplets having such a large dimensionality. A program based on Maple V4 was developed for formula generation. Details of program operation and possible applications of the results obtained are discussed.     

New algorithms in the theory of hypergeometric series and Burchnall-Chaundy expansions

It is shown that the Burchnall-Chaundy expansions, which are of fundamental importance in the theory of Appell's functions, can easily be implemented and generalized by means of the operator factorization method, which provides a simple and universal base, both for a new theory of hypergeometric series and for the development of effective new algorithms for computer-aided symbolic transformations of these series. Five new generalized expansions are derived, including 44 Burchnall-Chaundy expansions, as well as many new expansions, some of which are related to the Horn series.     

New algorithms for deriving reduction formulas for hypergeometric series of multiple variables

A Maple-implementation of algorithms for the automatic derivation of hypergeometric series reduction formulas of the most general form is considered. The list of elementary reductions is presented.