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]

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

拉卡多項式

拉卡多項式的前數條是

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

極限關係

拉卡多項式→哈恩多項式

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

拉卡多項式→雙重哈恩多項式

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

參考文獻

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