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Unformatted text preview: Gegenbauer polynomials 1 Definition Generating function: 1 (1- 2 xt + t 2 ) α = ∞ summationdisplay n =0 C ( α ) n ( x ) t n for α negationslash = 0. The Gegenbauer polynomials C ( α ) n ( x ) are also known as ultraspherical polynomials (see Arfken-Weber end of section 12.1). The Gegenbauer polynomials include a number of polynomials we have seen previously as special cases: for example, α = 1 / 2 gives the Legendre polynomials, and α = 1 gives the type II Chebyshev polynomials. The case α = 0 gives the type I Chebyshev polynomials, though this case must be handled differently than α negationslash = 0. Specifically, one defines C (0) n ( x ) = lim α → C ( α ) n α and the relation to type I Chebyshev polynomials is discussed in Arfken-Weber section 13.3. The first few Gegenbauer polynomials are C ( α ) ( x ) = 1 C ( α ) 1 ( x ) = 2 αx C ( α ) 2 ( x ) =- α + 2 α (1 + α ) x 2 C ( α ) 3 ( x ) =- 2 α (1 + α ) x + 4 3 α (1 + α )(2 + α ) x 3 By comparison, note that the first few Legendre polynomials (...
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- Spring '11
- Legendre polynomials, Special hypergeometric functions, Orthogonal polynomials, chebyshev polynomials, Gegenbauer polynomials