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A New Proof of a Watson's Formula

Published online by Cambridge University Press:  20 November 2018

Krzysztof Stempak
Krzysztof Stempak*
Affiliation:
Institute of Mathematics, University of WroclawPL. Grunwaldzki 2/4, 50-384 Wroclaw, Poland(Current Address) Department of Mathematics, The University of GeorgiaAthens, Georgia 30602, USA
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Abstract

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A new proof of a product formula for Laguerre polynomials, due originally to Watson, is given. Considering the commutative Banach algebra of radial functions on the Heisenberg groups Hn, n ≧ 2, we observe that Watson's formula holds for z = 1,2, 3, …. Then, applying a complex function theory argument, we establish the validity of this formula for other complex values of z, i.e. for Re z > - 1/2.

Keywords

Laguerre polynomialsHeisenberg groups33A6533A7543A80
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 31 , Issue 4 , 01 December 1988 , pp. 414 - 418
Copyright
Copyright © Canadian Mathematical Society 1988

References

1. Hulanicki, A. and Ricci, F., A Tauberian theorem and tangential convergence for bounded harmonic functions on balls in Cn Inventiones math. 62 (1980), pp. 325–331.Google Scholar
2. Markett, C., A new proof of Watson's product formula for Laguerre polynomials via a Cauchy problem associated with a singular differential operator SIAM J. Math. Anal. 17 (1986), pp. 1010–1032.Google Scholar
3. Ricci, F., Harmonic analysis on groups of Type H (preprint).Google Scholar
4. Stempak, K., An algebra associated with the generalized sublaplacian Studia Math. 88 (1988), pp. 245–256.Google Scholar
5. Titchmarsh, E. C., The theory of functions Oxford Univ. Press, 1939.Google Scholar
6. Watson, G. N., Another note on Laguerre polynomials J. London Math. Soc. 14 (1939), pp. 19–22.Google Scholar