- Source: Jackson q-Bessel function
In mathematics, a Jackson q-Bessel function (or basic Bessel function) is one of the three q-analogs of the Bessel function introduced by Jackson (1906a, 1906b, 1905a, 1905b). The third Jackson q-Bessel function is the same as the Hahn–Exton q-Bessel function.
Definition
The three Jackson q-Bessel functions are given in terms of the q-Pochhammer symbol and the basic hypergeometric function
ϕ
{\displaystyle \phi }
by
J
ν
(
1
)
(
x
;
q
)
=
(
q
ν
+
1
;
q
)
∞
(
q
;
q
)
∞
(
x
/
2
)
ν
2
ϕ
1
(
0
,
0
;
q
ν
+
1
;
q
,
−
x
2
/
4
)
,
|
x
|
<
2
,
{\displaystyle J_{\nu }^{(1)}(x;q)={\frac {(q^{\nu +1};q)_{\infty }}{(q;q)_{\infty }}}(x/2)^{\nu }{}_{2}\phi _{1}(0,0;q^{\nu +1};q,-x^{2}/4),\quad |x|<2,}
J
ν
(
2
)
(
x
;
q
)
=
(
q
ν
+
1
;
q
)
∞
(
q
;
q
)
∞
(
x
/
2
)
ν
0
ϕ
1
(
;
q
ν
+
1
;
q
,
−
x
2
q
ν
+
1
/
4
)
,
x
∈
C
,
{\displaystyle J_{\nu }^{(2)}(x;q)={\frac {(q^{\nu +1};q)_{\infty }}{(q;q)_{\infty }}}(x/2)^{\nu }{}_{0}\phi _{1}(;q^{\nu +1};q,-x^{2}q^{\nu +1}/4),\quad x\in \mathbb {C} ,}
J
ν
(
3
)
(
x
;
q
)
=
(
q
ν
+
1
;
q
)
∞
(
q
;
q
)
∞
(
x
/
2
)
ν
1
ϕ
1
(
0
;
q
ν
+
1
;
q
,
q
x
2
/
4
)
,
x
∈
C
.
{\displaystyle J_{\nu }^{(3)}(x;q)={\frac {(q^{\nu +1};q)_{\infty }}{(q;q)_{\infty }}}(x/2)^{\nu }{}_{1}\phi _{1}(0;q^{\nu +1};q,qx^{2}/4),\quad x\in \mathbb {C} .}
They can be reduced to the Bessel function by the continuous limit:
lim
q
→
1
J
ν
(
k
)
(
x
(
1
−
q
)
;
q
)
=
J
ν
(
x
)
,
k
=
1
,
2
,
3.
{\displaystyle \lim _{q\to 1}J_{\nu }^{(k)}(x(1-q);q)=J_{\nu }(x),\ k=1,2,3.}
There is a connection formula between the first and second Jackson q-Bessel function (Gasper & Rahman (2004)):
J
ν
(
2
)
(
x
;
q
)
=
(
−
x
2
/
4
;
q
)
∞
J
ν
(
1
)
(
x
;
q
)
,
|
x
|
<
2.
{\displaystyle J_{\nu }^{(2)}(x;q)=(-x^{2}/4;q)_{\infty }J_{\nu }^{(1)}(x;q),\ |x|<2.}
For integer order, the q-Bessel functions satisfy
J
n
(
k
)
(
−
x
;
q
)
=
(
−
1
)
n
J
n
(
k
)
(
x
;
q
)
,
n
∈
Z
,
k
=
1
,
2
,
3.
{\displaystyle J_{n}^{(k)}(-x;q)=(-1)^{n}J_{n}^{(k)}(x;q),\ n\in \mathbb {Z} ,\ k=1,2,3.}
Properties
= Negative Integer Order
=By using the relations (Gasper & Rahman (2004)):
(
q
m
+
1
;
q
)
∞
=
(
q
m
+
n
+
1
;
q
)
∞
(
q
m
+
1
;
q
)
n
,
{\displaystyle (q^{m+1};q)_{\infty }=(q^{m+n+1};q)_{\infty }(q^{m+1};q)_{n},}
(
q
;
q
)
m
+
n
=
(
q
;
q
)
m
(
q
m
+
1
;
q
)
n
,
m
,
n
∈
Z
,
{\displaystyle (q;q)_{m+n}=(q;q)_{m}(q^{m+1};q)_{n},\ m,n\in \mathbb {Z} ,}
we obtain
J
−
n
(
k
)
(
x
;
q
)
=
(
−
1
)
n
J
n
(
k
)
(
x
;
q
)
,
k
=
1
,
2.
{\displaystyle J_{-n}^{(k)}(x;q)=(-1)^{n}J_{n}^{(k)}(x;q),\ k=1,2.}
= Zeros
=Hahn mentioned that
J
ν
(
2
)
(
x
;
q
)
{\displaystyle J_{\nu }^{(2)}(x;q)}
has infinitely many real zeros (Hahn (1949)). Ismail proved that for
ν
>
−
1
{\displaystyle \nu >-1}
all non-zero roots of
J
ν
(
2
)
(
x
;
q
)
{\displaystyle J_{\nu }^{(2)}(x;q)}
are real (Ismail (1982)).
= Ratio of q-Bessel Functions
=The function
−
i
x
−
1
/
2
J
ν
+
1
(
2
)
(
i
x
1
/
2
;
q
)
/
J
ν
(
2
)
(
i
x
1
/
2
;
q
)
{\displaystyle -ix^{-1/2}J_{\nu +1}^{(2)}(ix^{1/2};q)/J_{\nu }^{(2)}(ix^{1/2};q)}
is a completely monotonic function (Ismail (1982)).
= Recurrence Relations
=The first and second Jackson q-Bessel function have the following recurrence relations (see Ismail (1982) and Gasper & Rahman (2004)):
q
ν
J
ν
+
1
(
k
)
(
x
;
q
)
=
2
(
1
−
q
ν
)
x
J
ν
(
k
)
(
x
;
q
)
−
J
ν
−
1
(
k
)
(
x
;
q
)
,
k
=
1
,
2.
{\displaystyle q^{\nu }J_{\nu +1}^{(k)}(x;q)={\frac {2(1-q^{\nu })}{x}}J_{\nu }^{(k)}(x;q)-J_{\nu -1}^{(k)}(x;q),\ k=1,2.}
J
ν
(
1
)
(
x
q
;
q
)
=
q
±
ν
/
2
(
J
ν
(
1
)
(
x
;
q
)
±
x
2
J
ν
±
1
(
1
)
(
x
;
q
)
)
.
{\displaystyle J_{\nu }^{(1)}(x{\sqrt {q}};q)=q^{\pm \nu /2}\left(J_{\nu }^{(1)}(x;q)\pm {\frac {x}{2}}J_{\nu \pm 1}^{(1)}(x;q)\right).}
= Inequalities
=When
ν
>
−
1
{\displaystyle \nu >-1}
, the second Jackson q-Bessel function satisfies:
|
J
ν
(
2
)
(
z
;
q
)
|
≤
(
−
q
;
q
)
∞
(
q
;
q
)
∞
(
|
z
|
2
)
ν
exp
{
log
(
|
z
|
2
q
ν
/
4
)
2
log
q
}
.
{\displaystyle \left|J_{\nu }^{(2)}(z;q)\right|\leq {\frac {(-{\sqrt {q}};q)_{\infty }}{(q;q)_{\infty }}}\left({\frac {|z|}{2}}\right)^{\nu }\exp \left\{{\frac {\log \left(|z|^{2}q^{\nu }/4\right)}{2\log q}}\right\}.}
(see Zhang (2006).)
For
n
∈
Z
{\displaystyle n\in \mathbb {Z} }
,
|
J
n
(
2
)
(
z
;
q
)
|
≤
(
−
q
n
+
1
;
q
)
∞
(
q
;
q
)
∞
(
|
z
|
2
)
n
(
−
|
z
|
2
;
q
)
∞
.
{\displaystyle \left|J_{n}^{(2)}(z;q)\right|\leq {\frac {(-q^{n+1};q)_{\infty }}{(q;q)_{\infty }}}\left({\frac {|z|}{2}}\right)^{n}(-|z|^{2};q)_{\infty }.}
(see Koelink (1993).)
= Generating Function
=The following formulas are the q-analog of the generating function for the Bessel function (see Gasper & Rahman (2004)):
∑
n
=
−
∞
∞
t
n
J
n
(
2
)
(
x
;
q
)
=
(
−
x
2
/
4
;
q
)
∞
e
q
(
x
t
/
2
)
e
q
(
−
x
/
2
t
)
,
{\displaystyle \sum _{n=-\infty }^{\infty }t^{n}J_{n}^{(2)}(x;q)=(-x^{2}/4;q)_{\infty }e_{q}(xt/2)e_{q}(-x/2t),}
∑
n
=
−
∞
∞
t
n
J
n
(
3
)
(
x
;
q
)
=
e
q
(
x
t
/
2
)
E
q
(
−
q
x
/
2
t
)
.
{\displaystyle \sum _{n=-\infty }^{\infty }t^{n}J_{n}^{(3)}(x;q)=e_{q}(xt/2)E_{q}(-qx/2t).}
e
q
{\displaystyle e_{q}}
is the q-exponential function.
Alternative Representations
= Integral Representations
=The second Jackson q-Bessel function has the following integral representations (see Rahman (1987) and Ismail & Zhang (2018a)):
J
ν
(
2
)
(
x
;
q
)
=
(
q
2
ν
;
q
)
∞
2
π
(
q
ν
;
q
)
∞
(
x
/
2
)
ν
⋅
∫
0
π
(
e
2
i
θ
,
e
−
2
i
θ
,
−
i
x
q
(
ν
+
1
)
/
2
2
e
i
θ
,
−
i
x
q
(
ν
+
1
)
/
2
2
e
−
i
θ
;
q
)
∞
(
e
2
i
θ
q
ν
,
e
−
2
i
θ
q
ν
;
q
)
∞
d
θ
,
{\displaystyle J_{\nu }^{(2)}(x;q)={\frac {(q^{2\nu };q)_{\infty }}{2\pi (q^{\nu };q)_{\infty }}}(x/2)^{\nu }\cdot \int _{0}^{\pi }{\frac {\left(e^{2i\theta },e^{-2i\theta },-{\frac {ixq^{(\nu +1)/2}}{2}}e^{i\theta },-{\frac {ixq^{(\nu +1)/2}}{2}}e^{-i\theta };q\right)_{\infty }}{(e^{2i\theta }q^{\nu },e^{-2i\theta }q^{\nu };q)_{\infty }}}\,d\theta ,}
(
a
1
,
a
2
,
⋯
,
a
n
;
q
)
∞
:=
(
a
1
;
q
)
∞
(
a
2
;
q
)
∞
⋯
(
a
n
;
q
)
∞
,
ℜ
ν
>
0
,
{\displaystyle (a_{1},a_{2},\cdots ,a_{n};q)_{\infty }:=(a_{1};q)_{\infty }(a_{2};q)_{\infty }\cdots (a_{n};q)_{\infty },\ \Re \nu >0,}
where
(
a
;
q
)
∞
{\displaystyle (a;q)_{\infty }}
is the q-Pochhammer symbol. This representation reduces to the integral representation of the Bessel function in the limit
q
→
1
{\displaystyle q\to 1}
.
J
ν
(
2
)
(
z
;
q
)
=
(
z
/
2
)
ν
2
π
log
q
−
1
∫
−
∞
∞
(
q
ν
+
1
/
2
z
2
e
i
x
4
;
q
)
∞
exp
(
x
2
log
q
2
)
(
q
,
−
q
ν
+
1
/
2
e
i
x
;
q
)
∞
d
x
.
{\displaystyle J_{\nu }^{(2)}(z;q)={\frac {(z/2)^{\nu }}{\sqrt {2\pi \log q^{-1}}}}\int _{-\infty }^{\infty }{\frac {\left({\frac {q^{\nu +1/2}z^{2}e^{ix}}{4}};q\right)_{\infty }\exp \left({\frac {x^{2}}{\log q^{2}}}\right)}{(q,-q^{\nu +1/2}e^{ix};q)_{\infty }}}\,dx.}
= Hypergeometric Representations
=The second Jackson q-Bessel function has the following hypergeometric representations (see Koelink (1993), Chen, Ismail, and Muttalib (1994)):
J
ν
(
2
)
(
x
;
q
)
=
(
x
/
2
)
ν
(
q
;
q
)
∞
1
ϕ
1
(
−
x
2
/
4
;
0
;
q
,
q
ν
+
1
)
,
{\displaystyle J_{\nu }^{(2)}(x;q)={\frac {(x/2)^{\nu }}{(q;q)_{\infty }}}\ _{1}\phi _{1}(-x^{2}/4;0;q,q^{\nu +1}),}
J
ν
(
2
)
(
x
;
q
)
=
(
x
/
2
)
ν
(
q
;
q
)
∞
2
(
q
;
q
)
∞
[
f
(
x
/
2
,
q
(
ν
+
1
/
2
)
/
2
;
q
)
+
f
(
−
x
/
2
,
q
(
ν
+
1
/
2
)
/
2
;
q
)
]
,
f
(
x
,
a
;
q
)
:=
(
i
a
x
;
q
)
∞
3
ϕ
2
(
a
,
−
a
,
0
−
q
,
i
a
x
;
q
,
q
)
.
{\displaystyle J_{\nu }^{(2)}(x;q)={\frac {(x/2)^{\nu }({\sqrt {q}};q)_{\infty }}{2(q;q)_{\infty }}}[f(x/2,q^{(\nu +1/2)/2};q)+f(-x/2,q^{(\nu +1/2)/2};q)],\ f(x,a;q):=(iax;{\sqrt {q}})_{\infty }\ _{3}\phi _{2}\left({\begin{matrix}a,&-a,&0\\-{\sqrt {q}},&iax\end{matrix}};{\sqrt {q}},{\sqrt {q}}\right).}
An asymptotic expansion can be obtained as an immediate consequence of the second formula.
For other hypergeometric representations, see Rahman (1987).
Modified q-Bessel Functions
The q-analog of the modified Bessel functions are defined with the Jackson q-Bessel function (Ismail (1981) and Olshanetsky & Rogov (1995)):
I
ν
(
j
)
(
x
;
q
)
=
e
i
ν
π
/
2
J
ν
(
j
)
(
x
;
q
)
,
j
=
1
,
2.
{\displaystyle I_{\nu }^{(j)}(x;q)=e^{i\nu \pi /2}J_{\nu }^{(j)}(x;q),\ j=1,2.}
K
ν
(
j
)
(
x
;
q
)
=
π
2
sin
(
π
ν
)
{
I
−
ν
(
j
)
(
x
;
q
)
−
I
ν
(
j
)
(
x
;
q
)
}
,
j
=
1
,
2
,
ν
∈
C
−
Z
,
{\displaystyle K_{\nu }^{(j)}(x;q)={\frac {\pi }{2\sin(\pi \nu )}}\left\{I_{-\nu }^{(j)}(x;q)-I_{\nu }^{(j)}(x;q)\right\},\ j=1,2,\ \nu \in \mathbb {C} -\mathbb {Z} ,}
K
n
(
j
)
(
x
;
q
)
=
lim
ν
→
n
K
ν
(
j
)
(
x
;
q
)
,
n
∈
Z
.
{\displaystyle K_{n}^{(j)}(x;q)=\lim _{\nu \to n}K_{\nu }^{(j)}(x;q),\ n\in \mathbb {Z} .}
There is a connection formula between the modified q-Bessel functions:
I
ν
(
2
)
(
x
;
q
)
=
(
−
x
2
/
4
;
q
)
∞
I
ν
(
1
)
(
x
;
q
)
.
{\displaystyle I_{\nu }^{(2)}(x;q)=(-x^{2}/4;q)_{\infty }I_{\nu }^{(1)}(x;q).}
For statistical applications, see Kemp (1997).
= Recurrence Relations
=By the recurrence relation of Jackson q-Bessel functions and the definition of modified q-Bessel functions, the following recurrence relation can be obtained (
K
ν
(
j
)
(
x
;
q
)
{\displaystyle K_{\nu }^{(j)}(x;q)}
also satisfies the same relation) (Ismail (1981)):
q
ν
I
ν
+
1
(
j
)
(
x
;
q
)
=
2
z
(
1
−
q
ν
)
I
ν
(
j
)
(
x
;
q
)
+
I
ν
−
1
(
j
)
(
x
;
q
)
,
j
=
1
,
2.
{\displaystyle q^{\nu }I_{\nu +1}^{(j)}(x;q)={\frac {2}{z}}(1-q^{\nu })I_{\nu }^{(j)}(x;q)+I_{\nu -1}^{(j)}(x;q),\ j=1,2.}
For other recurrence relations, see Olshanetsky & Rogov (1995).
= Continued Fraction Representation
=The ratio of modified q-Bessel functions form a continued fraction (Ismail (1981)):
I
ν
(
2
)
(
z
;
q
)
I
ν
−
1
(
2
)
(
z
;
q
)
=
1
2
(
1
−
q
ν
)
/
z
+
q
ν
2
(
1
−
q
ν
+
1
)
/
z
+
q
ν
+
1
2
(
1
−
q
ν
+
2
)
/
z
+
⋱
.
{\displaystyle {\frac {I_{\nu }^{(2)}(z;q)}{I_{\nu -1}^{(2)}(z;q)}}={\cfrac {1}{2(1-q^{\nu })/z+{\cfrac {q^{\nu }}{2(1-q^{\nu +1})/z+{\cfrac {q^{\nu +1}}{2(1-q^{\nu +2})/z+\ddots }}}}}}.}
= Alternative Representations
=Hypergeometric Representations
The function
I
ν
(
2
)
(
z
;
q
)
{\displaystyle I_{\nu }^{(2)}(z;q)}
has the following representation (Ismail & Zhang (2018b)):
I
ν
(
2
)
(
z
;
q
)
=
(
z
/
2
)
ν
(
q
,
q
)
∞
1
ϕ
1
(
z
2
/
4
;
0
;
q
,
q
ν
+
1
)
.
{\displaystyle I_{\nu }^{(2)}(z;q)={\frac {(z/2)^{\nu }}{(q,q)_{\infty }}}{}_{1}\phi _{1}(z^{2}/4;0;q,q^{\nu +1}).}
Integral Representations
The modified q-Bessel functions have the following integral representations (Ismail (1981)):
I
ν
(
2
)
(
z
;
q
)
=
(
z
2
/
4
;
q
)
∞
(
1
π
∫
0
π
cos
ν
θ
d
θ
(
e
i
θ
z
/
2
;
q
)
∞
(
e
−
i
θ
z
/
2
;
q
)
∞
−
sin
ν
π
π
∫
0
∞
e
−
ν
t
d
t
(
−
e
t
z
/
2
;
q
)
∞
(
−
e
−
t
z
/
2
;
q
)
∞
)
,
{\displaystyle I_{\nu }^{(2)}(z;q)=\left(z^{2}/4;q\right)_{\infty }\left({\frac {1}{\pi }}\int _{0}^{\pi }{\frac {\cos \nu \theta \,d\theta }{\left(e^{i\theta }z/2;q\right)_{\infty }\left(e^{-i\theta }z/2;q\right)_{\infty }}}-{\frac {\sin \nu \pi }{\pi }}\int _{0}^{\infty }{\frac {e^{-\nu t}\,dt}{\left(-e^{t}z/2;q\right)_{\infty }\left(-e^{-t}z/2;q\right)_{\infty }}}\right),}
K
ν
(
1
)
(
z
;
q
)
=
1
2
∫
0
∞
e
−
ν
t
d
t
(
−
e
t
/
2
z
/
2
;
q
)
∞
(
−
e
−
t
/
2
z
/
2
;
q
)
∞
,
|
arg
z
|
<
π
/
2
,
{\displaystyle K_{\nu }^{(1)}(z;q)={\frac {1}{2}}\int _{0}^{\infty }{\frac {e^{-\nu t}\,dt}{\left(-e^{t/2}z/2;q\right)_{\infty }\left(-e^{-t/2}z/2;q\right)_{\infty }}},\ |\arg z|<\pi /2,}
K
ν
(
1
)
(
z
;
q
)
=
∫
0
∞
cosh
ν
d
t
(
−
e
t
/
2
z
/
2
;
q
)
∞
(
−
e
−
t
/
2
z
/
2
;
q
)
∞
.
{\displaystyle K_{\nu }^{(1)}(z;q)=\int _{0}^{\infty }{\frac {\cosh \nu \,dt}{\left(-e^{t/2}z/2;q\right)_{\infty }\left(-e^{-t/2}z/2;q\right)_{\infty }}}.}
See also
q-Bessel polynomials
References
Chen, Yang; Ismail, Mourad E. H.; Muttalib, K.A. (1994), "Asymptotics of basic Bessel functions and q-Laguerre polynomials", Journal of Computational and Applied Mathematics, 54 (3): 263–272, doi:10.1016/0377-0427(92)00128-v
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Kata Kunci Pencarian:
- Jackson q-Bessel function
- Bessel function
- Hahn–Exton q-Bessel function
- List of q-analogs
- Green's function
- Q-gamma function
- List of eponyms of special functions
- F. H. Jackson
- Green's function for the three-variable Laplace equation
- Q factor
