Q贝塞尔多项式

qBessel function 2D plot

q-贝塞尔多项式，也称q-贝塞尔函数，其通过基本超几何函数定义[1]：


y n ( x ; a ; q ) = 2 ϕ 1 ( q − N − a q n 0 ; q , q x ) {\displaystyle y_{n}(x;a;q)=\;_{2}\phi _{1}\left({\begin{matrix}q^{-N}&-aq^{n}\\0\end{matrix}};q,qx\right)}

正交关系

∑ k = 0 ∞ ( a k ( q ; q ) n ∗ q ( k + 1 2 ) ∗ y m ∗ ( q k ; a ; q ) ∗ y n ∗ ( q k ; a ; q ) = ( q ; q ) n ∗ ( − a q n ; q ) ∞ a n ∗ q ( n + 1 2 ) 1 + a q 2 n δ m n {\displaystyle \sum _{k=0}^{\infty }({\frac {a^{k}}{(q;q)_{n}}}*q^{k+1 \choose 2}*y_{m}*(q^{k};a;q)*y_{n}*(q^{k};a;q)=(q;q)_{n}*(-aq^{n};q)_{\infty }{\frac {a^{n}*q^{n+1 \choose 2}}{1+aq^{2n}}}\delta _{mn}} [2]

图集

QBessel function abs complex 3D Maple plot
QBessel function Im complex 3D Maple plot
QBessel function Re complex 3D Maple plot
QBessel function abs density Maple plot
QBessel function Im density Maple plot
QBessel function Re density Maple plot

参考文献

  1. ^ Roelof Koekoek, Peter Lesky Rene Swarttouw,Hypergeometric Orthogonal Polynomials and their q-Analogues, p526 Springer 2010
  2. ^ Roelof p527
  • Gasper, George; Rahman, Mizan, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications 96 2nd, Cambridge University Press, 2004, ISBN 978-0-521-83357-8, MR 2128719, doi:10.2277/0521833574 
  • Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F., Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, 2010, ISBN 978-3-642-05013-8, MR 2656096, doi:10.1007/978-3-642-05014-5 
  • Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F., http://dlmf.nist.gov/18 |contribution-url=缺少标题 (帮助), Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (编), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248