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

where ${\displaystyle P_{n))$ is the $n$ th Legendre polynomial.^{[citation needed]}

References

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