FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2002, VOLUME 8, NUMBER 2, PAGES 517-531

A. W. Niukkanen

Abstract

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Using factorization method and introducing canonical forms
of multiple hypergeometric series allows all quadratic transformations
for all series satisfying the corresponding applicability conditions
to be obtained in the form of a small set of general basic relations.
In other words any quadratic transformation for standard series,
for example for the Gauss', Appell's, Horn's, Kamp\'e de F\'eriet's,
Lauricella's, Gelfand's series, etc., as well as for numerous
nameless series, can be obtained as particular cases of the relations
given in the paper. Along with completeness and generality of analysis
the factorization method ensures an essential simplification of the
theory by introducing a natural hierarchical structure into a system
of quadratic connections between nine types of canonical forms.
These results may contribute to development of advanced computer
algebra systems capable to analyze, automatically, important properties
of those multiple series which find large-scale applications
in mathematics, mathematical physics and theoretical chemistry.
```

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Last modified: November 26, 2002