Big q-Jacobi polynomials

In mathematics, the big q-Jacobi polynomialsP_n(x;a,b,c;q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. ^[1]

Definition

The polynomials are given in terms of basic hypergeometric functions by

${\displaystyle \displaystyle P_{n}(x;a,b,c;q)={}_{3}\phi _{2}(q^{-n},abq^{n+1},x;aq,cq;q,q)}$

References

1. Andrews, George E.; Askey, Richard (1985). "Classical orthogonal polynomials". In Brezinski, C.; Draux, A.; Magnus, Alphonse P.; Maroni, Pascal; Ronveaux, A. (eds.). Polynômes orthogonaux et applications. Proceedings of the Laguerre symposium held at Bar-le-Duc, October 15–18, 1984. Lecture Notes in Math. Vol. 1171. Berlin, New York: Springer-Verlag. pp. 36–62. doi:10.1007/BFb0076530. ISBN   978-3-540-16059-5. MR   0838970.

Further reading

• Gasper, George; Rahman, Mizan (2004). Basic hypergeometric series. Encyclopedia of Mathematics and its Applications. Vol. 96 (2nd ed.). Cambridge University Press. ISBN   978-0-521-83357-8. MR   2128719.
• Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010). Hypergeometric orthogonal polynomials and their q-analogues. Springer Monographs in Mathematics. Berlin, New York: Springer-Verlag. doi:10.1007/978-3-642-05014-5. ISBN   978-3-642-05013-8. MR   2656096 ^{[ page needed ]} gives a detailed list of properties.
• Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Chapter 18: Orthogonal Polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions , Cambridge University Press, ISBN   978-0-521-19225-5, MR   2723248 .