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 . 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 $\phi$ by

:$J_\nu^(x;q) = \frac (x/2)^\nu _2\phi_1(0,0;q^;q,-x^2/4), \quad , x, <2,$
:$J_\nu^(x;q) = \frac (x/2)^\nu _0\phi_1(;q^;q,-x^2q^/4), \quad x\in\mathbb,$
:$J_\nu^(x;q) = \frac (x/2)^\nu _1\phi_1(0;q^;q,qx^2/4), \quad x\in\mathbb.$
They can be reduced to the Bessel function by the continuous limit:

:$\lim_J_\nu^(x(1-q);q)=J_\nu(x), \ k=1,2,3.$
There is a connection formula between the first and second Jackson ''q''-Bessel function ():
:$J_\nu^(x;q)=(-x^2/4;q)_\infty J_\nu^(x;q), \ , x, <2.$
For integer order, the ''q''-Bessel functions satisfy
:$J_n^(-x;q)=(-1)^n J_n^(x;q), \ n\in\mathbb, \ k=1,2,3.$

Properties

Negative Integer Order

By using the relations ():
:$(q^;q)_\infty=(q^;q)_\infty (q^;q)_n,$
:$(q;q)_=(q;q)_m (q^;q)_n,\ m,n\in\mathbb,$
we obtain
:$J_^(x;q)=(-1)^n J_n^(x;q), \ k=1,2.$

Zeros

Hahn mentioned that $J_\nu^(x;q)$ has infinitely many real zeros (). Ismail proved that for $\nu>-1$ all non-zero roots of $J_\nu^(x;q)$ are real ().

Ratio of ''q''-Bessel Functions

The function $-ix^J_^(ix^;q)/J_^(ix^;q)$ is a completely monotonic function ().

Recurrence Relations

The first and second Jackson ''q''-Bessel function have the following recurrence relations (see and ):
:$q^\nu J_^(x;q)=\fracJ_\nu^(x;q)-J_^(x;q), \ k=1,2.$
:$J_^(x\sqrt;q)=q^\left(J_\nu^(x;q)\pm \fracJ_^(x;q)\right).$

Inequalities

When $\nu>-1$, the second Jackson ''q''-Bessel function satisfies:
$\left, J_^(z;q)\\leq\frac\left(\frac\right)^\nu\exp\left\.$
(see .)

For $n\in\mathbb$,
$\left, J_^(z;q)\\leq\frac\left(\frac\right)^n(-, z, ^2;q)_.$
(see .)

Generating Function

The following formulas are the ''q''-analog of the generating function for the Bessel function (see ):

:$\sum_^t^nJ_n^(x;q)=(-x^2/4;q)_e_q(xt/2)e_q(-x/2t),$
:$\sum_^t^nJ_n^(x;q)=e_q(xt/2)E_q(-qx/2t).$
$e_q$ is the ''q''-exponential function.

Alternative Representations

Integral Representations

The second Jackson ''q''-Bessel function has the following integral representations (see and ):
:$J_^(x;q)=\frac(x/2)^ \cdot\int_0^ \frac\,d\theta,$
:$(a_1,a_2,\cdots,a_n;q)_:=(a_1;q)_(a_2;q)_\cdots(a_n;q)_, \ \Re \nu>0,$
where $(a;q)_$is the ''q''-Pochhammer symbol. This representation reduces to the integral representation of the Bessel function in the limit $q\to 1$.
:$J_^(z;q)=\frac\int_^\frac\,dx.$

Hypergeometric Representations

The second Jackson ''q''-Bessel function has the following hypergeometric representations (see , ):
:$J_^(x;q)=\frac\ _1\phi_1(-x^2/4;0;q,q^),$
:
J_^(x;q)=\frac (x/2,q^;q)+f(-x/2,q^;q) \ f(x,a;q):=(iax;\sqrt)_\infty \ _3\phi_2 \left(\begin
a, & -a, & 0 \\
-\sqrt, & iax \end
; \sqrt,\sqrt \right).

An asymptotic expansion can be obtained as an immediate consequence of the second formula.

For other hypergeometric representations, see .

Modified ''q''-Bessel Functions

The ''q''-analog of the modified Bessel functions are defined with the Jackson ''q''-Bessel function ( and ):
:$I_\nu^(x;q)=e^J_^(x;q), \ j=1,2.$
:$K_\nu^(x;q)=\frac\left\, \ j=1,2,\ \nu\in\mathbb-\mathbb,$
:$K_n^(x;q)=\lim_K_\nu^(x;q),\ n\in\mathbb.$
There is a connection formula between the modified q-Bessel functions:
:$I_\nu^(x;q)=(-x^2/4;q)_\infty I_\nu^(x;q).$
For statistical applications, see .

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 ($K_\nu^(x;q)$ also satisfies the same relation) ():
:$q^\nu I_^(x;q)=\frac(1-q^\nu)I_\nu^(x;q)+I_^(x;q), \ j=1, 2.$
For other recurrence relations, see .

Continued Fraction Representation

The ratio of modified ''q''-Bessel functions form a continued fraction ():
:$\frac=\cfrac.$

Alternative Representations

Hypergeometric Representations

The function $I_\nu^(z;q)$ has the following representation ():
:$I_\nu^(z;q)=\frac _1\phi_1(z^2/4;0;q,q^).$

Integral Representations

The modified ''q''-Bessel functions have the following integral representations ():
:$I_\nu^(z;q)=\left(z^2/4;q\right)_\infty\left(\frac\int_0^\pi\frac-\frac\int_0^\infty\frac\right),$
:$K_\nu^(z;q)=\frac\int_0^\infty\frac,\ , \arg z, <\pi/2,$
:$K_\nu^(z;q)=\int_0^\infty\frac.$

See also

* ''q''-Bessel polynomials

References

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*{{cite arXiv , last1=Zhang, first1=R., title=Plancherel-Rotach Asymptotics for ''q''-Series , year=2006 , eprint=math/0612216 , mode=cs2

Special functions
Q-analogs