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In mathematics, the q -Bessel polynomialsorthogonal 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 by [ 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).}
K
(
x
;
a
;
q
)
.
{\displaystyle K(x;a;q).}
Orthogonality
∑
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 }\left({\frac {a^{k}}{(q;q)_{n}}}*q^{k+1 \choose 2}*y_{m}*(q^{k};a;q)*y_{n}*(q^{k};a;q)\right)=(q;q)_{n}*(-aq^{n};q)_{\infty }{\frac {a^{n}*q^{n+1 \choose 2}}{1+aq^{2n}}}\delta _{mn}}
[ 2] where
(
q
;
q
)
n
and
(
−
a
q
n
;
q
)
∞
{\displaystyle (q;q)_{n}{\text{ and }}(-aq^{n};q)_{\infty }}
q-Pochhammer symbols .
Gallery
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
References
^ Roelof Koekoek, Peter Lesky Rene Swarttouw, Hypergeometric Orthogonal Polynomials and their q-Analogues, p526 Springer 2010
^ Roelof p527
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), "Orthogonal Polynomials" , in Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions ISBN 978-0-521-19225-5 MR 2723248 
