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A generalization of z!

Published online by Cambridge University Press:  09 April 2009

W. B. White-Smith  and
V. T. Buchwald
W. B. White-Smith
Affiliation:
Departments of Mathematics, University of Sydney.
V. T. Buchwald
Affiliation:
Departments of Mathematics, University of Sydney.
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Summary

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A generalised factorial function (z: k)! is defined as an infinite product similar to the Euler product for z!, but with the sequences of integers replaced by the roots of F(z) = sin πz+kπz. It is proved that, apart from poles in (z) < 0, (z: k)! is analytic in both variables, and that F(z) may be expressed in the form F(z) = πz/(z: k)!(—z: k)!

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 4 , Issue 3 , August 1964 , pp. 327 - 341
Copyright
Copyright © Australian Mathematical Society 1964

References

[1]Koiter, W. T., Kon. Ned. Acad. Wet. Proc. B, 59 (1956), 558.Google Scholar
[2]Noble, B., Methods Based on the Wiener-Hopf Technique, Pergamon Press, London (1958), 162.Google Scholar
[3]Titchmarsh, E. C., Theory of Functions, Second Edition, Oxford (1939), Chapter 8.Google Scholar