Big q-Legendre polynomials

In mathematics, the big q-Legendre polynomials are an orthogonal family of polynomials defined in terms of Heine's basic hypergeometric series as

$\displaystyle P_{n}(x;c;q)={}_{3}\phi _{2}(q^{-n},q^{n+1},x;q,cq;q,q)$.

They obey the orthogonality relation

$\int _{cq}^{q}P_{m}(x;c;q)P_{n}(x;c;q)\,dx=q(1-c){\frac {1-q}{1-q^{2n+1}}}{\frac {(c^{-1}q;q)_{n}}{(cq;q)_{n}}}(-cq^{2})^{n}q^{n \choose 2}\delta _{mn}$

and have the limiting behavior

$\displaystyle \lim _{q\to 1}P_{n}(x;0;q)=P_{n}(2x-1)$

where $P_{n}$ is the $n$th Legendre polynomial.