In mathematics, the big q-Legendre polynomials are an orthogonal family of polynomials defined in terms of Heine's basic hypergeometric series as[1]
.
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}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
and have the limiting behavior
![{\displaystyle \displaystyle \lim _{q\to 1}P_{n}(x;0;q)=P_{n}(2x-1)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
where
is the
th Legendre polynomial.[citation needed]
References
- ^ Roelof Koekoek, Peter Lesky, Rene Swattouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues, p 443, Springer