小q雅可比多項式

小q-雅可比多項式(Little q-Jacobi polynomials)是一個以基本超幾何函數定義的正交多項式，定義如下^[1].

${\displaystyle \displaystyle p_{n}(x;a,b;q)={}_{2}\phi _{1}(q^{-n},abq^{n+1};aq;q,xq)}$

例子

k=3,

${\displaystyle p=1+q*x/((1-a*q)*(1-q))-q^{2}*x*a*b*q^{n}/((1-a*q)*(1-q))-q*x/((1-a*q)*(1-q)*q^{n})+q^{2}*x*a*b/((1-a*q)*(1-q))}$ ${\displaystyle (1-q^{(}-n))*(1-q^{(}-n)*q)*(1-a*b*q^{(}n+1))*(1-a*b*q^{(}n+1)*q)*q^{2}*x^{2}/((1-a*q)*(1-a*q^{2})*(1-q)*(1-q^{2}))}$ ${\displaystyle +(1-q^{(}-n))*(1-q^{(}-n)*q)*(1-q^{(}-n)*q^{2})*(1-a*b*q^{(}n+1))*(1-a*b*q^{(}n+1)*q)*(1-a*b*q^{(}n+1)*q^{2})*q^{3}*x^{3}/((1-a*q)*(1-a*q^{2})*(1-a*q^{3})*(1-q)*(1-q^{2})*(1-q^{3}))}$

圖集

LITTLE Q-JACOBI POLYNOMIALS ABS COMPLEX 3D MAPLE PLOT

LITTLE Q-JACOBI POLYNOMIALS IM COMPLEX 3D MAPLE PLOT

LITTLE Q-JACOBI POLYNOMIALS RE COMPLEX 3D MAPLE PLOT

LITTLE Q-JACOBI POLYNOMIALS ABS DENSITY MAPLE PLOT

LITTLE Q-JACOBI POLYNOMIALS IM DENSITY MAPLE PLOT

LITTLE Q-JACOBI POLYNOMIALS RE DENSITY MAPLE PLOT

參考文獻

1. Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin