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The Stokes phenomenon and generalized contiguous transformations of some generalized hypergeometric functions

Published online by Cambridge University Press:  24 October 2008

J. Heading
J. Heading
Affiliation:
University College of Wales, Aberystwyth
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Abstract

Previous investigations by the author into the Stokes phenomenon pertaining to solutions of the differential equation dnu/dzn = (–1)nzmu are extended in order to find when different equations have the same set of Stokes multipliers, with perhaps a series of zeros being additionally allowed. The reason for periodic cycles to exist (with n fixed and m varying), with the same Stokes multipliers regained after a complete cycle, is traced to certain transformation properties of the equations. Within the first cycle (with n fixed and m varying) further remarkable identities exist between the Stokes multipliers, and this also is traced to special transformations between the equations. Relations are found toexist when values of m are chosen so that the highest common factors of the two integers n and n/(n + m) are identical. Finally, a transformation of the independent variable is deduced whereby the set of Stokes multipliers for an equation of order n is identical (apart from the additional zeros) to that for an equation of lower order. A hierarchy of equations is thrown up, whereby certain basic equations are transformed to yield more advanced equations of higher order and different m.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 76 , Issue 2 , September 1974 , pp. 423 - 442
Copyright
Copyright © Cambridge Philosophical Society 1974

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References

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