大Q勒让德多项式

大q-勒让德多项式是一个以基本超几何函数定义的正交多项式^[1]：

P_{n} (x; c; q) =_{3} ϕ_{2} (q^{- n}, q^{n + 1}, x; q, c q; q, q)

正交性

大q-勒让德多项式满足下列正交关系

$\int_{c q}^{q} P_{m} (x; c; q) P_{n} (x; c; q) d_{q} x = q (1 - c) \frac{1 - q}{1 - q^{2 n + 1}} \frac{(c^{- 1} q; q)_{n}}{(c q; q)_{n}} (- c q^{2})^{n} q^{(\binom{n}{2})} δ_{m n}$

极限关系

大Q勒让德多项式→勒让德多项式

\lim_{q \to 1} P_{n} (x; 0; q) = P_{n} (2 x - 1)

令大q勒让德项式中的 $c = 0$ ,并且q→1 即得勒让德多项式

验证

将c=0代人7阶(n=7)大q勒让德多项式得：

$q L = P_{n} (x; 0; q) = 1 + \frac{q}{{(1 - q)}^{2}} - \frac{q x}{{(1 - q)}^{2}} - \frac{q^{9}}{{(1 - q)}^{2}} + \frac{x q^{9}}{{(1 - q)}^{2}} - \frac{1}{q^{6} {(1 - q)}^{2}} + \frac{x}{q^{6} {(1 - q)}^{2}} + \frac{q^{2}}{{(1 - q)}^{2}} - \frac{x q^{2}}{{(1 - q)}^{2}} + (1 - q^{- 7}) (1 - q^{- 6}) (1 - q^{8}) (1 - q^{9}) (1 - x) (1 - q x) q^{2} {(1 - q)}^{- 2} {(1 - q^{2})}^{- 2} + (1 - q^{- 7}) (1 - q^{- 6}) (1 - q^{- 5}) (1 - q^{8}) (1 - q^{9}) (1 - q^{10}) (1 - x) (1 - q x) (1 - x q^{2}) q^{3} {(1 - q)}^{- 2} {(1 - q^{2})}^{- 2} {(1 - q^{3})}^{- 2} + (1 - q^{- 7}) (1 - q^{- 6}) (1 - q^{- 5}) (1 - q^{- 4}) (1 - q^{8}) (1 - q^{9}) (1 - q^{10}) (1 - q^{11}) (1 - x) (1 - q x) (1 - x q^{2}) (1 - x q^{3}) q^{4} {(1 - q)}^{- 2} {(1 - q^{2})}^{- 2} {(1 - q^{3})}^{- 2} {(1 - q^{4})}^{- 2} + (1 - q^{- 7}) (1 - q^{- 6}) (1 - q^{- 5}) (1 - q^{- 4}) (1 - q^{- 3}) (1 - q^{8}) (1 - q^{9}) (1 - q^{10}) (1 - q^{11}) (1 - q^{12}) (1 - x) (1 - q x) (1 - x q^{2}) (1 - x q^{3}) (1 - x q^{4}) q^{5} {(1 - q)}^{- 2} {(1 - q^{2})}^{- 2} {(1 - q^{3})}^{- 2} {(1 - q^{4})}^{- 2} {(1 - q^{5})}^{- 2} + (1 - q^{- 7}) (1 - q^{- 6}) (1 - q^{- 5}) (1 - q^{- 4}) (1 - q^{- 3}) (1 - q^{- 2}) (1 - q^{8}) (1 - q^{9}) (1 - q^{10}) (1 - q^{11}) (1 - q^{12}) (1 - q^{13}) (1 - x) (1 - q x) (1 - x q^{2}) (1 - x q^{3}) (1 - x q^{4}) (1 - x q^{5}) q^{6} {(1 - q)}^{- 2} {(1 - q^{2})}^{- 2} {(1 - q^{3})}^{- 2} {(1 - q^{4})}^{- 2} {(1 - q^{5})}^{- 2} {(1 - q^{6})}^{- 2} + (1 - q^{- 7}) (1 - q^{- 6}) (1 - q^{- 5}) (1 - q^{- 4}) (1 - q^{- 3}) (1 - q^{- 2}) (1 - q^{- 1}) (1 - q^{8}) (1 - q^{9}) (1 - q^{10}) (1 - q^{11}) (1 - q^{12}) (1 - q^{13}) (1 - q^{14}) (1 - x) (1 - q x) (1 - x q^{2}) (1 - x q^{3}) (1 - x q^{4}) (1 - x q^{5}) (1 - x q^{6}) q^{7} {(1 - q)}^{- 2} {(1 - q^{2})}^{- 2} {(1 - q^{3})}^{- 2} {(1 - q^{4})}^{- 2} {(1 - q^{5})}^{- 2} {(1 - q^{6})}^{- 2} {(1 - q^{7})}^{- 2}$