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An extension of a theorem of Mehler's on Hermite polynomials

Published online by Cambridge University Press:  24 October 2008

W. F. Kibble
W. F. Kibble
Affiliation:
Madras Christian CollegeTambaramIndia
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Extract

It was shown by Mehler (1866) that

where Hk(x) denotes the Hermite polynomial

(Hermite, 1864a, b), which can be expressed in terms of Weber's parabolic cylinder function (Whittaker, 1903). The series is convergent if | ρ | < 1, and divergent if | ρ | > 1. If ρ = 1 and x = y = 0 the series is divergent, and Hille's work (1938) shows that it will therefore be divergent for all real or complex values, except possibly real positive values, of x and y.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 41 , Issue 1 , June 1945 , pp. 12 - 15
Copyright
Copyright © Cambridge Philosophical Society 1945

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References

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