拉卡多項式 拉卡多項式 （Racah polynomials）是數學中以Guilio Racah命名的正交多項式，由下列廣義超幾何函數 定義[1]
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{\displaystyle p_{n}(x(x+\gamma +\delta +1))={}_{4}F_{3}\left[{\begin{matrix}-n&n+\alpha +\beta +1&-x&x+\gamma +\delta +1\\\alpha +1&\gamma +1&\beta +\delta +1\\\end{matrix}};1\right].}
拉卡多項式的前數條是
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{\displaystyle {\begin{aligned}hypergeom([-1,-x,2+a+b,x+c+d+1],[a+1,c+1,b+d+1],1)\\hypergeom([-2,-x,3+a+b,x+c+d+1],[a+1,c+1,b+d+1],1)\\hypergeom([-3,-x,4+a+b,x+c+d+1],[a+1,c+1,b+d+1],1)\\hypergeom([-4,-x,5+a+b,x+c+d+1],[a+1,c+1,b+d+1],1)\\hypergeom([-5,-x,6+a+b,x+c+d+1],[a+1,c+1,b+d+1],1)\\hypergeom([-6,-x,7+a+b,x+c+d+1],[a+1,c+1,b+d+1],1).\end{aligned}}}
極限關係 [ 編輯 ] 拉卡多項式→哈恩多項式
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{\displaystyle \lim _{\delta \to \infty }R_{n}(\lambda (x);-N-1,\delta )=Q_{n}(x;\alpha ,\beta ,N)}
拉卡多項式→雙重哈恩多項式
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{\displaystyle \lim _{\beta \to \infty }R_{n}(\lambda (x);-N-1,\beta ,\gamma ,\delta )=R_{n}(\lambda (x);\gamma ,\delta ,N)}
參考文獻 [ 編輯 ]
^ Askey, Richard; Wilson, James (1979), "A set of orthogonal polynomials that generalize the Racah coefficients or 6-j symbols", SIAM Journal on Mathematical Analysis 10 (5): 1008–1016
![{\displaystyle p_{n}(x(x+\gamma +\delta +1))={}_{4}F_{3}\left[{\begin{matrix}-n&n+\alpha +\beta +1&-x&x+\gamma +\delta +1\\\alpha +1&\gamma +1&\beta +\delta +1\\\end{matrix}};1\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f8669e91de2c035849fc87d8ce46aa12600695e)
![{\displaystyle {\begin{aligned}hypergeom([-1,-x,2+a+b,x+c+d+1],[a+1,c+1,b+d+1],1)\\hypergeom([-2,-x,3+a+b,x+c+d+1],[a+1,c+1,b+d+1],1)\\hypergeom([-3,-x,4+a+b,x+c+d+1],[a+1,c+1,b+d+1],1)\\hypergeom([-4,-x,5+a+b,x+c+d+1],[a+1,c+1,b+d+1],1)\\hypergeom([-5,-x,6+a+b,x+c+d+1],[a+1,c+1,b+d+1],1)\\hypergeom([-6,-x,7+a+b,x+c+d+1],[a+1,c+1,b+d+1],1).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9f5bd393d24a60ce7d64dbb817a4463b8e8b971)
