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

${\displaystyle p_{n}(x(x+\gamma +\delta +1))={}_{4}F_{3}\left[{\begin{matrix}-n&n+\alpha +\beta +1&-x&x+\gamma +\delta +1\\\alpha +1&\gamma +1&\beta +\delta +1\\\end{matrix}};1\right].}$

拉卡多项式的前数条是

${\displaystyle {\begin{aligned}hypergeom([-1,-x,2+a+b,x+c+d+1],[a+1,c+1,b+d+1],1)\\hypergeom([-2,-x,3+a+b,x+c+d+1],[a+1,c+1,b+d+1],1)\\hypergeom([-3,-x,4+a+b,x+c+d+1],[a+1,c+1,b+d+1],1)\\hypergeom([-4,-x,5+a+b,x+c+d+1],[a+1,c+1,b+d+1],1)\\hypergeom([-5,-x,6+a+b,x+c+d+1],[a+1,c+1,b+d+1],1)\\hypergeom([-6,-x,7+a+b,x+c+d+1],[a+1,c+1,b+d+1],1).\end{aligned}}}$

极限关系

拉卡多项式→哈恩多项式

${\displaystyle \lim _{\delta \to \infty }R_{n}(\lambda (x);-N-1,\delta )=Q_{n}(x;\alpha ,\beta ,N)}$ 

拉卡多项式→双重哈恩多项式

${\displaystyle \lim _{\beta \to \infty }R_{n}(\lambda (x);-N-1,\beta ,\gamma ,\delta )=R_{n}(\lambda (x);\gamma ,\delta ,N)}$ 

参考文献

1. 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