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

$P_n(x;c;q)={}_3{\varphi }_2(q^{-n},q^{n+1},x;q,cq;q,q)$.

They obey the orthogonality relation

${\int }_{cq}^q{P}_m(x;c;q)P_n(x;c;q)dx=q(1-c)\frac{1-q}{1-q^{2n+1}}\frac{(c^{-1}q;q{)}_n}{(cq;q{)}_n}(-c{q}^2{)}^n{q}^{(\genfrac{}{}{0ex}{}{n}{2})}{\delta }_{mn}$

and have the limiting behavior

$\underset{q\to 1}{lim}{P}_n(x;0;q)=P_n(2x-1)$

where $P_n$ is the $n$th Legendre polynomial.Template:Citation needed

References

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