Jackson q-Bessel function

The three Jackson q-Bessel functions are given in terms of the q-Pochhammer symbol and the basic hypergeometric function $\backslash phi$ by

:

:$J\_\backslash nu^\{(2)\}(x;q)\; =\; \backslash frac\{(q^\{\backslash nu+1\};q)\_\backslash infty\}\{(q;q)\_\backslash infty\}\; (x/2)^\backslash nu\; \{\}\_0\backslash phi\_1(;q^\{\backslash nu+1\};q,-x^2q^\{\backslash nu\; +1\}/4),\; \backslash quad\; x\backslash in\backslash mathbb\{C\},$

:$J\_\backslash nu^\{(3)\}(x;q)\; =\; \backslash frac\{(q^\{\backslash nu+1\};q)\_\backslash infty\}\{(q;q)\_\backslash infty\}\; (x/2)^\backslash nu\; \{\}\_1\backslash phi\_1(0;q^\{\backslash nu+1\};q,qx^2/4),\; \backslash quad\; x\backslash in\backslash mathbb\{C\}.$

They can be reduced to the Bessel function by the continuous limit:

:$\backslash lim\_\{q\backslash to1\}J\_\backslash nu^\{(k)\}(x(1-q);q)=J\_\backslash nu(x),\; \backslash \; k=1,2,3.$

There is a connection formula between the first and second Jackson q-Bessel function ({{harvtxt|Gasper|Rahman|2004}}):

:

For integer order, the q-Bessel functions satisfy

:$J\_n^\{(k)\}(-x;q)=(-1)^n\; J\_n^\{(k)\}(x;q),\; \backslash \; n\backslash in\backslash mathbb\{Z\},\; \backslash \; k=1,2,3.$

By using the relations ({{harvtxt|Gasper|Rahman|2004}}):

:$(q^\{m+1\};q)\_\backslash infty=(q^\{m+n+1\};q)\_\backslash infty\; (q^\{m+1\};q)\_n,$

:$(q;q)\_\{m+n\}=(q;q)\_m\; (q^\{m+1\};q)\_n,\backslash \; m,n\backslash in\backslash mathbb\{Z\},$

we obtain

:$J\_\{-n\}^\{(k)\}(x;q)=(-1)^n\; J\_n^\{(k)\}(x;q),\; \backslash \; k=1,2.$

Hahn mentioned that $J\_\backslash nu^\{(2)\}(x;q)$ has infinitely many real zeros ({{harvs|txt|authorlink=Wolfgang Hahn|last=Hahn|year=1949}}). Ismail proved that for $\backslash nu>-1$ all non-zero roots of $J\_\backslash nu^\{(2)\}(x;q)$ are real ({{harvs|txt|authorlink=Mourad E. H. Ismail|last=Ismail|year=1982}}).

The function $-ix^\{-1/2\}J\_\{\backslash nu+1\}^\{(2)\}(ix^\{1/2\};q)/J\_\{\backslash nu\}^\{(2)\}(ix^\{1/2\};q)$ is a completely monotonic function ({{harvs|txt|authorlink=Mourad E. H. Ismail|last=Ismail|year=1982}}).

The first and second Jackson q-Bessel function have the following recurrence relations (see {{harvtxt|Ismail|1982}} and {{harvtxt|Gasper|Rahman|2004}}):

:$q^\backslash nu\; J\_\{\backslash nu+1\}^\{(k)\}(x;q)=\backslash frac\{2(1-q^\backslash nu)\}\{x\}J\_\backslash nu^\{(k)\}(x;q)-J\_\{\backslash nu-1\}^\{(k)\}(x;q),\; \backslash \; k=1,2.$

:$J\_\{\backslash nu\}^\{(1)\}(x\backslash sqrt\{q\};q)=q^\{\backslash pm\backslash nu/2\}\backslash left(J\_\backslash nu^\{(1)\}(x;q)\backslash pm\; \backslash frac\{x\}\{2\}J\_\{\backslash nu\backslash pm1\}^\{(1)\}(x;q)\backslash right).$

When $\backslash nu>-1$, the second Jackson q-Bessel function satisfies:

\left|J_{\nu}^{(2)}(z;q)\right|\leq\frac{(-\sqrt{q};q)_{\infty}}{(q;q)_{\infty}}\left(\frac

(see {{harvs|txt|last=Zhang|year=2006}}.)

For $n\backslash in\backslash mathbb\{Z\}$,

\left|J_{n}^{(2)}(z;q)\right|\leq\frac{(-q^{n+1};q)_{\infty}}{(q;q)_{\infty}}\left(\frac

(see {{harvs|txt|last=Koelink|year=1993}}.)

The following formulas are the q-analog of the generating function for the Bessel function (see {{harvtxt|Gasper|Rahman|2004}}):

:$\backslash sum\_\{n=-\backslash infty\}^\{\backslash infty\}t^nJ\_n^\{(2)\}(x;q)=(-x^2/4;q)\_\{\backslash infty\}e\_q(xt/2)e\_q(-x/2t),$

:$\backslash sum\_\{n=-\backslash infty\}^\{\backslash infty\}t^nJ\_n^\{(3)\}(x;q)=e\_q(xt/2)E\_q(-qx/2t).$

$e\_q$ is the q-exponential function.

The second Jackson q-Bessel function has the following integral representations (see {{harvtxt|Rahman|1987}} and {{harvtxt|Ismail|Zhang|2018a}}):

:$$

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{i x q^{(\nu+1)/2}}{2}e^{i\theta}, -\frac{i x q^{(\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,\cdots,a_n;q)_{\infty}:=(a_1;q)_{\infty}(a_2;q)_{\infty}\cdots(a_n;q)_{\infty}, \ \Re \nu>0,

where $(a;q)\_\{\backslash infty\}$ is the q-Pochhammer symbol. This representation reduces to the integral representation of the Bessel function in the limit $q\backslash to\; 1$.

:$$

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^2e^{ix}}{4};q\right)_{\infty}\exp\left(\frac{x^2}{\log q^2}\right)}{(q,-q^{\nu+1/2}e^{ix};q)_{\infty}}\,dx.

The second Jackson q-Bessel function has the following hypergeometric representations (see {{harvs|txt|last=Koelink|year=1993}}, {{harvs|txt|last1=Chen|authorlink2=Mourad Ismail|last2=Ismail|last3=Muttalib|year=1994}}):

:$$

J_{\nu}^{(2)}(x;q)=\frac{(x/2)^{\nu}}{(q;q)_{\infty}}\ _1\phi_1(-x^2/4;0;q,q^{\nu+1}),

:$$

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 {{harvtxt|Rahman|1987}}.

The q-analog of the modified Bessel functions are defined with the Jackson q-Bessel function ({{harvtxt|Ismail|1981}} and {{harvtxt|Olshanetsky|Rogov|1995}}):

:$I\_\backslash nu^\{(j)\}(x;q)=e^\{i\backslash nu\backslash pi/2\}J\_\{\backslash nu\}^\{(j)\}(x;q),\; \backslash \; j=1,2.$

:$K\_\backslash nu^\{(j)\}(x;q)=\backslash frac\{\backslash pi\}\{2\backslash sin(\backslash pi\backslash nu)\}\backslash left\backslash \{I\_\{-\backslash nu\}^\{(j)\}(x;q)-I\_\backslash nu^\{(j)\}(x;q)\backslash right\backslash \},\; \backslash \; j=1,2,\backslash \; \backslash nu\backslash in\backslash mathbb\{C\}-\backslash mathbb\{Z\},$

:$K\_n^\{(j)\}(x;q)=\backslash lim\_\{\backslash nu\backslash to\; n\}K\_\backslash nu^\{(j)\}(x;q),\backslash \; n\backslash in\backslash mathbb\{Z\}.$

There is a connection formula between the modified q-Bessel functions:

:$I\_\backslash nu^\{(2)\}(x;q)=(-x^2/4;q)\_\backslash infty\; I\_\backslash nu^\{(1)\}(x;q).$

For statistical applications, see {{harvtxt|Kemp|1997}}.

By the recurrence relation of Jackson q-Bessel functions and the definition of modified q-Bessel functions, the following recurrence relation can be obtained ($K\_\backslash nu^\{(j)\}(x;q)$ also satisfies the same relation) ({{harvtxt|Ismail|1981}}):

:$q^\backslash nu\; I\_\{\backslash nu+1\}^\{(j)\}(x;q)=\backslash frac\{2\}\{z\}(1-q^\backslash nu)I\_\backslash nu^\{(j)\}(x;q)+I\_\{\backslash nu-1\}^\{(j)\}(x;q),\; \backslash \; j=1,\; 2.$

For other recurrence relations, see {{harvtxt|Olshanetsky|Rogov|1995}}.

The ratio of modified q-Bessel functions form a continued fraction ({{harvtxt|Ismail|1981}}):

:$\backslash frac\{I\_\backslash nu^\{(2)\}(z;q)\}\{I\_\{\backslash nu-1\}^\{(2)\}(z;q)\}=\backslash cfrac\{1\}\{2(1-q^\backslash nu)/z+\backslash cfrac\{q^\backslash nu\}\{2(1-q^\{\backslash nu+1\})/z+\backslash cfrac\{q^\{\backslash nu+1\}\}\{2(1-q^\{\backslash nu+2\})/z+\backslash ddots\}\}\}.$

The function $I\_\backslash nu^\{(2)\}(z;q)$ has the following representation ({{harvtxt|Ismail|Zhang|2018b}}):

:$$

I_\nu^{(2)}(z;q)=\frac{(z/2)^\nu}{(q,q)_{\infty}} {}_1\phi_1(z^2/4;0;q,q^{\nu+1}).

The modified q-Bessel functions have the following integral representations ({{harvtxt|Ismail|1981}}):

:$I\_\backslash nu^\{(2)\}(z;q)=\backslash left(z^2/4;q\backslash right)\_\backslash infty\backslash left(\backslash frac\{1\}\{\backslash pi\}\backslash int\_0^\backslash pi\backslash frac\{\backslash cos\backslash nu\backslash theta\backslash ,d\backslash theta\}\{\backslash left(e^\{i\backslash theta\}z/2;q\backslash right)\_\backslash infty\backslash left(e^\{-i\backslash theta\}z/2;q\backslash right)\_\backslash infty\}-\backslash frac\{\backslash sin\backslash nu\backslash pi\}\{\backslash pi\}\backslash int\_0^\backslash infty\backslash frac\{e^\{-\backslash nu\; t\}\backslash ,dt\}\{\backslash left(-e^t\; z/2;q\backslash right)\_\backslash infty\backslash left(-e^\{-t\}z/2;q\backslash right)\_\backslash infty\}\backslash right),$

:

:$K\_\backslash nu^\{(1)\}(z;q)=\backslash int\_0^\backslash infty\backslash frac\{\backslash cosh\backslash nu\; \backslash ,dt\}\{\backslash left(-e^\{t/2\}\; z/2;q\backslash right)\_\backslash infty\backslash left(-e^\{-t/2\}z/2;q\backslash right)\_\backslash infty\}.$

• q-Bessel polynomials

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Category:Special functions

Category:Q-analogs