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The ultraspherical polynomials are solutions
to the Ultraspherical Differential Equation for
Integer
and
. They are generalizations of Legendre Polynomials to
-D space and are proportional to (or, depending on the normalization, equal to) the Gegenbauer
Polynomials
, denoted in Mathematica
(Wolfram Research, Champaign,
IL) GegenbauerC[n,lambda,x]. The ultraspherical polynomials are also Jacobi Polynomials with
. They are given by the Generating Function
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(1) |
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|
(2) |
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(3) |
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(4) |
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(5) |
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(6) |
In terms of the Hypergeometric Functions,
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(7) |
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(8) | |
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(9) |
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(10) |
Derivative identities include
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(11) |
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|
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(12) |
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(13) |
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(14) |
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(15) |
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(16) |
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(17) |
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(18) |
A Recurrence Relation is
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(19) |
Special double- Formulas also exist
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(20) | |||
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(21) | |||
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(22) | |||
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(23) |
Special values are given in the following table.
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Special Polynomial |
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Legendre Polynomial |
1 | Chebyshev Polynomial of the Second Kind |
Koschmieder (1920) gives representations in terms of Elliptic Functions for
and
.
See also Birthday Problem, Chebyshev Polynomial of the Second Kind, Elliptic Function, Hypergeometric Function, Jacobi Polynomial
References
Abramowitz, M. and Stegun, C. A. (Eds.). ``Orthogonal Polynomials.'' Ch. 22 in
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 771-802, 1972.
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 643, 1985.
Iyanaga, S. and Kawada, Y. (Eds.). ``Gegenbauer Polynomials (Gegenbauer Functions).'' Appendix A, Table 20.I in
Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1477-1478, 1980.
Koschmieder, L. ``Über besondere Jacobische Polynome.'' Math. Zeitschrift 8, 123-137, 1920.
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York:
McGraw-Hill, pp. 547-549 and 600-604, 1953.
Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., 1975.
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© 1996-9 Eric W. Weisstein