Big q-Legendre polynomials

Last updated March 13, 2024

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

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

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References

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

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

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