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The expansion of functions in ultraspherical polynomials

Published online by Cambridge University Press:  09 April 2009

David Elliott
David Elliott
Affiliation:
Mathematics Department, University of Adelaide, Adelaide, S.A.
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The ultraspherical polynomial (x) of degree n and order λ is defined by for n = 0, 1, 2, …. Of these polynomials, the most commonly used are the Chebyshev polynomials Tn(x) of the first kind, corresponding to λ = 0; the Legendre polynomials Pn(x) for which λ = ½; and the Chebyshev polynomials Un(x) of the second kind (λ = 1). In the first case the standardisation is different from that given in equation (1), since.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 1 , Issue 4 , December 1960 , pp. 428 - 438
Copyright
Copyright © Australian Mathematical Society 1960

References

[1] Szegö, G., Orthogonal Polynomials. American Math. Soc. Colloquium Publ., 23 (1939).Google Scholar
[2] Clenshaw, C. W., The Numerical Solution of Linear Differential Equations in Chebyshev Series. Proc. Camb. Phil. Soc., 53 (1957) 134–149.CrossRefGoogle Scholar
[3] Clenshaw, C. W., A Note on the Summation of Chebyshev Series. M.T.A.C., 9 (1955), no. 51, 118–120.Google Scholar
[4] Tables of Chebyshev Polynomials. N.B.S. Applied Mathematics Series, 9 (1952).Google Scholar
[5] Bernstein, S., Les Propriétés Extrémales. Gauthier-Villars, Paris, (1926).Google Scholar
[6] Elliott, D., The Numerical Solution of Integral Equations using Chebyshev Polynomials. This Journal 1 (1960), 344–356.Google Scholar