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

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

p n ( x ; a , b , c , d | q ) = a − n e − i n u ( a b e 2 i u , a c , a d ; q ) n 4 ϕ 3 ( q − n , a b c d q n − 1 , a e i ( t + 2 u ) , a e − i t ; a b e 2 i u , a c , a d ; q ; q ) {\displaystyle p_{n}(x;a,b,c,d|q)=a^{-n}e^{-inu}(abe^{2iu},ac,ad;q)_{n}{}_{4}\phi _{3}(q^{-n},abcdq^{n-1},ae^{i{(t+2u)}},ae^{-it};abe^{2iu},ac,ad;q;q)}

x = cos ⁡ ( t + u ) {\displaystyle x=\cos(t+u)}

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

  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.