蓋根鮑爾多項式

蓋根鮑爾多項式${\displaystyle C_{n}^{(\alpha )}}$又稱超球多項式，是定義在區間${\displaystyle [-1,1]}$上、權函數為${\displaystyle (1-x^{2})^{\alpha -1/2}}$的正交多項式。它是勒壤得多項式和柴比雪夫多項式的推廣，又是雅可比多項式的特殊情況。它以奧地利數學家Leopold Gegenbauer命名。

蓋根鮑爾多項式

性質

蓋根鮑爾多項式具有若干性質：

• 蓋根鮑爾多項式可由其母函數表示 (Stein & Weiss 1971，§IV.2):

${\displaystyle {\frac {1}{(1-2xt+t^{2})^{\alpha }}}=\sum _{n=0}^{\infty }C_{n}^{(\alpha )}(x)t^{n}.}$

• 蓋根鮑爾多項式滿足遞迴關係 (Suetin 2001):

${\displaystyle {\begin{aligned}C_{0}^{\alpha }(x)&=1\\C_{1}^{\alpha }(x)&=2\alpha x\\C_{n}^{\alpha }(x)&={\frac {1}{n}}[2x(n+\alpha -1)C_{n-1}^{\alpha }(x)-(n+2\alpha -2)C_{n-2}^{\alpha }(x)].\end{aligned}}}$

• 蓋根鮑爾多項式是蓋根鮑爾微分方程式的特解 (Suetin 2001):

${\displaystyle (1-x^{2})y''-(2\alpha +1)xy'+n(n+2\alpha )y=0.\,}$

當 α = 1/2, 方程式約化為勒壤得方程式, 蓋根鮑爾多項式約化為勒壤得多項式.

• 可由高斯超幾何級數表示:

${\displaystyle C_{n}^{(\alpha )}(z)={\frac {(2\alpha )_{n}}{n!}}\,_{2}F_{1}\left(-n,2\alpha +n;\alpha +{\frac {1}{2}};{\frac {1-z}{2}}\right).}$

(Abramowitz & Stegun p. 561 （頁面存檔備份，存於網際網路檔案館）). 其中(2α)_n 為上升階乘冪. 具體來說,

${\displaystyle C_{n}^{(\alpha )}(z)=\sum _{k=0}^{\lfloor n/2\rfloor }(-1)^{k}{\frac {\Gamma (n-k+\alpha )}{\Gamma (\alpha )k!(n-2k)!}}(2z)^{n-2k}.}$

• 它是雅可比多項式的特例 (Suetin 2001):

${\displaystyle C_{n}^{(\alpha )}(x)={\frac {(2\alpha )_{n}}{(\alpha +{\frac {1}{2}})_{n}}}P_{n}^{(\alpha -1/2,\alpha -1/2)}(x).}$

因而滿足羅德里格公式

${\displaystyle C_{n}^{(\alpha )}(x)={\frac {(-2)^{n}}{n!}}{\frac {\Gamma (n+\alpha )\Gamma (n+2\alpha )}{\Gamma (\alpha )\Gamma (2n+2\alpha )}}(1-x^{2})^{-\alpha +1/2}{\frac {d^{n}}{dx^{n}}}\left[(1-x^{2})^{n+\alpha -1/2}\right].}$

正交歸一性

當n ≠ m時，對於固定的α和權函數

${\displaystyle w(z)=\left(1-z^{2}\right)^{\alpha -{\frac {1}{2}}}.}$,

蓋根鮑爾多項式在區間[−1, 1]上加權正交 (Abramowitz & Stegun p. 774 （頁面存檔備份，存於網際網路檔案館）)

${\displaystyle \int _{-1}^{1}C_{n}^{(\alpha )}(x)C_{m}^{(\alpha )}(x)(1-x^{2})^{\alpha -{\frac {1}{2}}}\,dx=0.}$

歸一性：

${\displaystyle \int _{-1}^{1}\left[C_{n}^{(\alpha )}(x)\right]^{2}(1-x^{2})^{\alpha -{\frac {1}{2}}}\,dx={\frac {\pi 2^{1-2\alpha }\Gamma (n+2\alpha )}{n!(n+\alpha )[\Gamma (\alpha )]^{2}}}.}$

應用

蓋根鮑爾多項式作為勒壤得多項式的擴展經常出現在勢理論和譜分析中. Rⁿ空間中的牛頓勢可以在α = (n − 2)/2情況下展開為蓋根鮑爾多項式,

${\displaystyle {\frac {1}{|\mathbf {x} -\mathbf {y} |^{n-2}}}=\sum _{k=0}^{\infty }{\frac {|\mathbf {x} |^{k}}{|\mathbf {y} |^{k+n-2}}}C_{n,k}^{(\alpha )}(\mathbf {x} \cdot \mathbf {y} ).}$

當n = 3, 可以得到引力勢的勒壤得展開。類似的表達式還有球中卜瓦松核的展開(Stein & Weiss 1971).

當只考慮x時，${\displaystyle C_{n,k}^{((n-2)/2)}(\mathbf {x} \cdot \mathbf {y} )}$ 為球諧函數。

蓋根鮑爾多項式在正定函數理論中亦有涉及。

另見

• 柴比雪夫多項式

參考文獻

• Abramowitz, Milton; Stegun, Irene Ann (編). 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 978-0-486-61272-0. LCCN 64-60036. MR 0167642.
  LCCN 65-12253
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• Bayin, S.S., Mathematical Methods in Science and Engineering, Wiley, 2006, Chapter 5.
• 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. (編), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248
• 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 (編), 数学百科全书, Springer, 2001, ISBN 978-1-55608-010-4.