This article has multiple issues. Please help improve it or discuss these issues on the talk page. ( Learn how and when to remove these template messages) This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources:  "Big q-Legendre polynomials" –  news · newspapers · books · scholar · JSTOR (May 2015) ( Learn how and when to remove this message) This article relies largely or entirely on a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources. Find sources:  "Big q-Legendre polynomials" –  news · newspapers · books · scholar · JSTOR (May 2015) ( Learn how and when to remove this message)

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.^{[ citation needed]}

References

This polynomial-related article is a stub. You can help Wikipedia by expanding it.

• v
• t
• e