Continuous q-Legendre polynomials

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In mathematics, the continuous q-Legendre polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme.Koekoek, Lesky & Swarttouw (2010) give a detailed list of their properties.^[1]

Definition

The polynomials are given in terms of basic hypergeometric functions by

P_{n}(x|q)={}_{4}\phi _{3}\left(q^{-n},q^{n+1},q^{\frac {1}{4}}e^{i\theta },q^{\frac {1}{4}}e^{-i\theta };q,-q^{-{\frac {1}{2}}},-q;q,q\right),\quad x=\cos \,\theta .

References

^ Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin: Springer-Verlag, p. 14, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096

Sadjang, Patrick Njionou. Moments of Classical Orthogonal Polynomials (Ph.D.). Universität Kassel. CiteSeerX 10.1.1.643.3896.

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[1] Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin: Springer-Verlag, p. 14, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096

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