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Unformatted text preview: Gegenbauer polynomials 1 Definition Generating function: 1 (1- 2 xt + t 2 ) = summationdisplay n =0 C ( ) n ( x ) t n for negationslash = 0. The Gegenbauer polynomials C ( ) 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, = 1 / 2 gives the Legendre polynomials, and = 1 gives the type II Chebyshev polynomials. The case = 0 gives the type I Chebyshev polynomials, though this case must be handled differently than negationslash = 0. Specifically, one defines C (0) n ( x ) = lim C ( ) n and the relation to type I Chebyshev polynomials is discussed in Arfken-Weber section 13.3. The first few Gegenbauer polynomials are C ( ) ( x ) = 1 C ( ) 1 ( x ) = 2 x C ( ) 2 ( x ) =- + 2 (1 + ) x 2 C ( ) 3 ( x ) =- 2 (1 + ) x + 4 3 (1 + )(2 + ) x 3 By comparison, note that the first few Legendre polynomials (...
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