拉卡多项式拉卡多项式(Racah polynomials)是数学中以Guilio Racah命名的正交多项式，由下列广义超几何函数定义[1] 拉卡多项式 p n ( x ( x + γ + δ + 1 ) ) = 4 F 3 [ − n n + α + β + 1 − x x + γ + δ + 1 α + 1 γ + 1 β + δ + 1 ; 1 ] . {\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].} 拉卡多项式的前数条是 h y p e r g e o m ( [ − 1 , − x , 2 + a + b , x + c + d + 1 ] , [ a + 1 , c + 1 , b + d + 1 ] , 1 ) h y p e r g e o m ( [ − 2 , − x , 3 + a + b , x + c + d + 1 ] , [ a + 1 , c + 1 , b + d + 1 ] , 1 ) h y p e r g e o m ( [ − 3 , − x , 4 + a + b , x + c + d + 1 ] , [ a + 1 , c + 1 , b + d + 1 ] , 1 ) h y p e r g e o m ( [ − 4 , − x , 5 + a + b , x + c + d + 1 ] , [ a + 1 , c + 1 , b + d + 1 ] , 1 ) h y p e r g e o m ( [ − 5 , − x , 6 + a + b , x + c + d + 1 ] , [ a + 1 , c + 1 , b + d + 1 ] , 1 ) h y p e r g e o m ( [ − 6 , − x , 7 + a + b , x + c + d + 1 ] , [ a + 1 , c + 1 , b + d + 1 ] , 1 ) . {\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}}} 极限关系 拉卡多项式→哈恩多项式 lim δ → ∞ R n ( λ ( x ) ; − N − 1 , δ ) = Q n ( x ; α , β , N ) {\displaystyle \lim _{\delta \to \infty }R_{n}(\lambda (x);-N-1,\delta )=Q_{n}(x;\alpha ,\beta ,N)} 拉卡多项式→双重哈恩多项式 lim β → ∞ R n ( λ ( x ) ; − N − 1 , β , γ , δ ) = R n ( λ ( x ) ; γ , δ , N ) {\displaystyle \lim _{\beta \to \infty }R_{n}(\lambda (x);-N-1,\beta ,\gamma ,\delta )=R_{n}(\lambda (x);\gamma ,\delta ,N)} 参考文献 ^ 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].}](/media/math_img/2159/2f8669e91de2c035849fc87d8ce46aa12600695e.svg)
![{\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}}}](/media/math_img/2159/c9f5bd393d24a60ce7d64dbb817a4463b8e8b971.svg)
