Big q-Legendre polynomials explained

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

They obey the orthogonality relation

\int ^{q} _{cq} P _{m(x;c;q)P} _{n(x;c;q)} dx = q( 1- c) (1- q)/(1- q ^{2n+1}) ((c ^{-1} q;q) _{n})/((cq;q) _{n}) ( - cq ^{2)} ^{n} q ^{n} \delta _{mn}

and have the limiting behavior

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

where

P _{n} is the n th Legendre polynomial.

References

1. Roelof Koekoek, Peter Lesky, Rene Swattouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues, p 443, Springer