盖根鲍尔多项式 盖根鲍尔多项式
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{\displaystyle C_{n}^{(\alpha )}}
超球多项式 ，是定义在区间
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{\displaystyle [-1,1]}
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{\displaystyle (1-x^{2})^{\alpha -1/2}}
正交多项式 。它是勒让德多项式 和切比雪夫多项式 的推广，又是雅可比多项式 的特殊情况。它以奥地利数学家Leopold Gegenbauer 命名。
盖根鲍尔多项式具有若干性质：
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{\displaystyle {\frac {1}{(1-2xt+t^{2})^{\alpha }}}=\sum _{n=0}^{\infty }C_{n}^{(\alpha )}(x)t^{n}.}
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{\displaystyle {\begin{aligned}C_{0}^{\alpha }(x)&=1\\C_{1}^{\alpha }(x)&=2\alpha x\\C_{n}^{\alpha }(x)&={\frac {1}{n}}[2x(n+\alpha -1)C_{n-1}^{\alpha }(x)-(n+2\alpha -2)C_{n-2}^{\alpha }(x)].\end{aligned}}}
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{\displaystyle (1-x^{2})y''-(2\alpha +1)xy'+n(n+2\alpha )y=0.\,}
当 α = 1/2, 方程约化为勒让德方程, 盖根鲍尔多项式约化为勒让德多项式 .
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{\displaystyle C_{n}^{(\alpha )}(z)={\frac {(2\alpha )_{n}}{n!}}\,_{2}F_{1}\left(-n,2\alpha +n;\alpha +{\frac {1}{2}};{\frac {1-z}{2}}\right).}
(Abramowitz & Stegun p. 561 （页面存档备份 ，存于互联网档案馆 ）). 其中(2α)n 上升阶乘幂 . 具体来说,
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{\displaystyle C_{n}^{(\alpha )}(z)=\sum _{k=0}^{\lfloor n/2\rfloor }(-1)^{k}{\frac {\Gamma (n-k+\alpha )}{\Gamma (\alpha )k!(n-2k)!}}(2z)^{n-2k}.}
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{\displaystyle C_{n}^{(\alpha )}(x)={\frac {(2\alpha )_{n}}{(\alpha +{\frac {1}{2}})_{n}}}P_{n}^{(\alpha -1/2,\alpha -1/2)}(x).}
因而满足罗德里格公式
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{\displaystyle C_{n}^{(\alpha )}(x)={\frac {(-2)^{n}}{n!}}{\frac {\Gamma (n+\alpha )\Gamma (n+2\alpha )}{\Gamma (\alpha )\Gamma (2n+2\alpha )}}(1-x^{2})^{-\alpha +1/2}{\frac {d^{n}}{dx^{n}}}\left[(1-x^{2})^{n+\alpha -1/2}\right].}
当n ≠ m 时，对于固定的α 和权函数
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{\displaystyle w(z)=\left(1-z^{2}\right)^{\alpha -{\frac {1}{2}}}.}
盖根鲍尔多项式在区间[−1, 1]上加权正交 (Abramowitz & Stegun p. 774 （页面存档备份 ，存于互联网档案馆 ）)
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{\displaystyle \int _{-1}^{1}C_{n}^{(\alpha )}(x)C_{m}^{(\alpha )}(x)(1-x^{2})^{\alpha -{\frac {1}{2}}}\,dx=0.}
归一性：
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{\displaystyle \int _{-1}^{1}\left[C_{n}^{(\alpha )}(x)\right]^{2}(1-x^{2})^{\alpha -{\frac {1}{2}}}\,dx={\frac {\pi 2^{1-2\alpha }\Gamma (n+2\alpha )}{n!(n+\alpha )[\Gamma (\alpha )]^{2}}}.}
盖根鲍尔多项式作为勒让德多项式的扩展经常出现在势理论 和谱分析 中. R n 牛顿势 可以在α = (n − 2)/2情况下展开为盖根鲍尔多项式,
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{\displaystyle {\frac {1}{|\mathbf {x} -\mathbf {y} |^{n-2}}}=\sum _{k=0}^{\infty }{\frac {|\mathbf {x} |^{k}}{|\mathbf {y} |^{k+n-2}}}C_{n,k}^{(\alpha )}(\mathbf {x} \cdot \mathbf {y} ).}
当n = 3, 可以得到引力势 的勒让德展开。类似的表达式还有球中泊松核 的展开(Stein & Weiss 1971 ).
当只考虑x 时，
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{\displaystyle C_{n,k}^{((n-2)/2)}(\mathbf {x} \cdot \mathbf {y} )}
球谐函数 。
盖根鲍尔多项式在正定函数 理论中亦有涉及。
Abramowitz, Milton; Stegun, Irene Ann (编). Chapter 22 . Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series 55 Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first. Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. 1983: 773. ISBN 978-0-486-61272-0LCCN 64-60036 MR 0167642 Bayin, S.S., Mathematical Methods in Science and Engineering, Wiley, 2006 Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F., Orthogonal Polynomials , Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (编), NIST Handbook of Mathematical Functions , Cambridge University Press, 2010, ISBN 978-0521192255MR 2723248 Stein, Elias ; Weiss, Guido , Introduction to Fourier Analysis on Euclidean Spaces, Princeton, N.J.: Princeton University Press, 1971, ISBN 978-0-691-08078-9 Suetin, P.K., Ultraspherical polynomials , Hazewinkel, Michiel (编), 数学百科全书 , Springer , 2001, ISBN 978-1-55608-010-4 
![{\displaystyle [-1,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51e3b7f14a6f70e614728c583409a0b9a8b9de01)
![{\displaystyle {\begin{aligned}C_{0}^{\alpha }(x)&=1\\C_{1}^{\alpha }(x)&=2\alpha x\\C_{n}^{\alpha }(x)&={\frac {1}{n}}[2x(n+\alpha -1)C_{n-1}^{\alpha }(x)-(n+2\alpha -2)C_{n-2}^{\alpha }(x)].\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7f7e040cb401c72d41eab0bda662f115229feed)
![{\displaystyle C_{n}^{(\alpha )}(x)={\frac {(-2)^{n}}{n!}}{\frac {\Gamma (n+\alpha )\Gamma (n+2\alpha )}{\Gamma (\alpha )\Gamma (2n+2\alpha )}}(1-x^{2})^{-\alpha +1/2}{\frac {d^{n}}{dx^{n}}}\left[(1-x^{2})^{n+\alpha -1/2}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07eff10ed90eb66bf06c2b329b1d266114dff433)
![{\displaystyle \int _{-1}^{1}\left[C_{n}^{(\alpha )}(x)\right]^{2}(1-x^{2})^{\alpha -{\frac {1}{2}}}\,dx={\frac {\pi 2^{1-2\alpha }\Gamma (n+2\alpha )}{n!(n+\alpha )[\Gamma (\alpha )]^{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9b5696990f291fed55949f3b9e2f1669b9f4c83)
