Product formulas and Nicholson-type integrals for Jacobi functions. I. Summary of results. [Gegenbauer, Laguerre, Hermite functions]
Journal Article
·
· SIAM J. Math. Anal.; (United States)
Nicholson's formula gives a generalization of the relation sin/sup 2/ x + cos/sup 2/ x = 1 to the case of Bessel functions. A similar result is presented which relates the sum of squares of the Jacobi functions p/sub n//sup (..cap alpha..,..beta..)/ and Q/sub n//sup (..cap alpha..,..beta..)/(x) to an integral over a single Jacobi function of the second kind, with the integrand positive. The Nicholson-type formula is a special case of a general product formula for two Jacobi functions of the second kind with different arguments, Q/sub n//sup (..cap alpha..,..beta..)/(z/sub 1/)Q/sub n//sup (..cap alpha..,..beta..)/(z/sub 2/). Various confluent limits of these expressions give Nicholson-type integrals and product formulas for general Gegenbauer, Laguerre, Bessel, and Hermite functions. These results are summarized in the present paper. Derivations and applications will be given elsewhere.
- Research Organization:
- Univ. of Wisconsin, Madison
- OSTI ID:
- 6424771
- Journal Information:
- SIAM J. Math. Anal.; (United States), Journal Name: SIAM J. Math. Anal.; (United States) Vol. 9:1; ISSN SJMAA
- Country of Publication:
- United States
- Language:
- English
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