Big q-Legendre polynomials

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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]

Pn(x;c;q)=3ϕ2(qn,qn+1,x;q,cq;q,q).

They obey the orthogonality relation

cqqPm(x;c;q)Pn(x;c;q)dx=q(1c)1q1q2n+1(c1q;q)n(cq;q)n(cq2)nq(n2)δmn

and have the limiting behavior

limq1Pn(x;0;q)=Pn(2x1)

where Pn is the nth Legendre polynomial.[citation needed]

References

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