Continuous q-Hahn polynomials

In mathematics, the continuous q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.

Definition

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The polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol by [1]

 

 

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CONTINUOUS q hahn ABS COMPLEX3D Maple PLOT
 
CONTINUOUS q hahn IIM COMPLEX3D Maple PLOT
 
CONTINUOUS q hahn RE COMPLEX3D Maple PLOT
 
CONTINUOUS q hahn ABS density Maple PLOT
 
CONTINUOUS q hahn im density Maple PLOT
 
CONTINUOUS q hahn RE density Maple PLOT

References

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  1. ^ Roelof p433, Springer 2010
  • 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
  • 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.