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Certain Extensions of the Mehler Formula

Published online by Cambridge University Press:  20 November 2018

J. P. Singhal
J. P. Singhal*
Affiliation:
University of Victoria, Victoria, British Columbia
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For the Hermite polynomials Hn(z) defined by

1

it is easy to see from the Rodrigues formula that

2

where, as usual, Dz=d/dz.

In recent papers ([1], [2]) Carlitz has proved the following formulae:

3

4

where, on the right-hand side of (4), the range of each summation is from i= 1 to i=k(k=1,2,…).

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 1 , March 1972 , pp. 115 - 117
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Carlitz, L., An extension of Mehler's formula, Boll. Un. Mat. Ital. (4) 3 (1970), 43-46.Google Scholar
2. Carlitz, L., Some extensions of the Mehler formula, Collect. Math. 21 (1970), 117-130.Google Scholar
3. Rain, E. D. ville, Special functions, Macmillan, New York, 1960.Google Scholar