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[1]

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

They obey the orthogonality relation

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

and have the limiting behavior

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

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