Q貝塞爾多項式

q-貝塞爾多項式，也稱q-貝索函數，其通過基本超幾何函數定義：

qBessel function 2D plot

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

正交關係

$\sum _{k=0}^{\infty }({\frac {a^{k}}{(q;q)_{n}}}*q^{k+1 \choose 2}*y_{m}*(q^{k};a;q)*y_{n}*(q^{k};a;q)=(q;q)_{n}*(-aq^{n};q)_{\infty }{\frac {a^{n}*q^{n+1 \choose 2}}{1+aq^{2n}}}\delta _{mn}$

圖集

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

參考文獻

• 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, Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (編), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248