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 $\varphi$ by

$J_{\nu }^{(1)}(x;q)=\frac{(q^{\nu +1};q{)}_{\mathrm{\infty }}}{(q;q{)}_{\mathrm{\infty }}}(x/2{)}^{\nu }{}_2{\varphi }_1(0,0;q^{\nu +1};q,-x^2/4),|x|<2,$

$J_{\nu }^{(2)}(x;q)=\frac{(q^{\nu +1};q{)}_{\mathrm{\infty }}}{(q;q{)}_{\mathrm{\infty }}}(x/2{)}^{\nu }{}_0{\varphi }_1(;q^{\nu +1};q,-x^2{q}^{\nu +1}/4),x\in ℂ,$

$J_{\nu }^{(3)}(x;q)=\frac{(q^{\nu +1};q{)}_{\mathrm{\infty }}}{(q;q{)}_{\mathrm{\infty }}}(x/2{)}^{\nu }{}_1{\varphi }_1(0;q^{\nu +1};q,q{x}^2/4),x\in ℂ.$

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

$\underset{q\to 1}{\lim }{J}_{\nu }^{(k)}(x(1-q);q)=J_{\nu }(x),k=1,2,3.$

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

$J_{\nu }^{(2)}(x;q)=(-x^2/4;q{)}_{\mathrm{\infty }}{J}_{\nu }^{(1)}(x;q),|x|<2.$

For integer order, the q-Bessel functions satisfy

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

Properties

Negative Integer Order

By using the relations (Gasper & Rahman (2004)):

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

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

we obtain

$J_{-n}^{(k)}(x;q)=(-1{)}^n{J}_n^{(k)}(x;q),k=1,2.$

Zeros

Hahn mentioned that $J_{\nu }^{(2)}(x;q)$ has infinitely many real zeros (Hahn (1949)). Ismail proved that for $\nu >-1$ all non-zero roots of $J_{\nu }^{(2)}(x;q)$ are real (Ismail (1982)).

Ratio of q-Bessel Functions

The function $-i{x}^{-1/2}{J}_{\nu +1}^{(2)}(i{x}^{1/2};q)/J_{\nu }^{(2)}(i{x}^{1/2};q)$ 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)):

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

$J_{\nu }^{(1)}(x\sqrt{q};q)=q^{\pm \nu /2}(J_{\nu }^{(1)}(x;q)\pm \frac{x}{2}{J}_{\nu \pm 1}^{(1)}(x;q)).$

Inequalities

When $\nu >-1$, the second Jackson q-Bessel function satisfies: $|J_{\nu }^{(2)}(z;q)|\le \frac{(-\sqrt{q};q{)}_{\mathrm{\infty }}}{(q;q{)}_{\mathrm{\infty }}}{\left(\frac{|z|}{2}\right)}^{\nu }\exp \left\{\frac{\log (|z{|}^2{q}^{\nu }/4)}{2\log q}\right\}.$ (see Zhang (2006).)

For $n\in \mathbb{Z}$, $|J_n^{(2)}(z;q)|\le \frac{(-q^{n+1};q{)}_{\mathrm{\infty }}}{(q;q{)}_{\mathrm{\infty }}}{\left(\frac{|z|}{2}\right)}^n(-|z{|}^2;q{)}_{\mathrm{\infty }}.$ (see Koelink (1993).)

Generating Function

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

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

$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{t}^n{J}_n^{(3)}(x;q)=e_q(xt/2)E_q(-qx/2t).$

$e_q$ is the q-exponential function.

Alternative Representations

Integral Representations

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

$J_{\nu }^{(2)}(x;q)=\frac{(q^{2\nu };q{)}_{\mathrm{\infty }}}{2\pi (q^{\nu };q{)}_{\mathrm{\infty }}}(x/2{)}^{\nu }\cdot \underset{0}{\overset{\pi }{\int }}\frac{{(e^{2i\theta },e^{-2i\theta },-\frac{ix{q}^{(\nu +1)/2}}{2}{e}^{i\theta },-\frac{ix{q}^{(\nu +1)/2}}{2}{e}^{-i\theta };q)}_{\mathrm{\infty }}}{(e^{2i\theta }{q}^{\nu },e^{-2i\theta }{q}^{\nu };q{)}_{\mathrm{\infty }}}d\theta ,$

$(a_1,a_2,\cdots ,a_n;q{)}_{\mathrm{\infty }}:=(a_1;q{)}_{\mathrm{\infty }}(a_2;q{)}_{\mathrm{\infty }}\cdots (a_n;q{)}_{\mathrm{\infty }},\mathrm{\Re }\nu >0,$

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

$J_{\nu }^{(2)}(z;q)=\frac{(z/2{)}^{\nu }}{\sqrt{2\pi \log q^{-1}}}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{{(\frac{q^{\nu +1/2}{z}^2{e}^{ix}}{4};q)}_{\mathrm{\infty }}\exp \left(\frac{x^2}{\log q^2}\right)}{(q,-q^{\nu +1/2}{e}^{ix};q{)}_{\mathrm{\infty }}}dx.$

Hypergeometric Representations

The second Jackson q-Bessel function has the following hypergeometric representations (see Koelink (1993), Chen, Ismail, and Muttalib (1994)):

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

$J_{\nu }^{(2)}(x;q)=\frac{(x/2{)}^{\nu }(\sqrt{q};q{)}_{\mathrm{\infty }}}{2(q;q{)}_{\mathrm{\infty }}}[f(x/2,q^{(\nu +1/2)/2};q)+f(-x/2,q^{(\nu +1/2)/2};q)],f(x,a;q):=(iax;\sqrt{q}{)}_{\mathrm{\infty }}{}_3{\varphi }_2(\begin{array}{ccc}a,& -a,& 0\\ -\sqrt{q},& iax\end{array};\sqrt{q},\sqrt{q}).$

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)):

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

$K_{\nu }^{(j)}(x;q)=\frac{\pi }{2\sin (\pi \nu )}\{I_{-\nu }^{(j)}(x;q)-I_{\nu }^{(j)}(x;q)\},j=1,2,\nu \in ℂ-\mathbb{Z},$

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

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

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

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 ($K_{\nu }^{(j)}(x;q)$ also satisfies the same relation) (Ismail (1981)):

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

For other recurrence relations, see Olshanetsky & Rogov (1995).

Continued Fraction Representation

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

$\frac{I_{\nu }^{(2)}(z;q)}{I_{\nu -1}^{(2)}(z;q)}=\frac{1}{2(1-q^{\nu })/z+\frac{q^{\nu }}{2(1-q^{\nu +1})/z+\frac{q^{\nu +1}}{2(1-q^{\nu +2})/z+\ddots }}}.$

Alternative Representations

Hypergeometric Representations

The function $I_{\nu }^{(2)}(z;q)$ has the following representation (Ismail & Zhang (2018b)):

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

Integral Representations

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

$I_{\nu }^{(2)}(z;q)={(z^2/4;q)}_{\mathrm{\infty }}(\frac{1}{\pi }\underset{0}{\overset{\pi }{\int }}\frac{\cos \nu \theta d\theta }{{(e^{i\theta }z/2;q)}_{\mathrm{\infty }}{(e^{-i\theta }z/2;q)}_{\mathrm{\infty }}}-\frac{\sin \nu \pi }{\pi }\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{e^{-\nu t}dt}{{(-e^{t}z/2;q)}_{\mathrm{\infty }}{(-e^{-t}z/2;q)}_{\mathrm{\infty }}}),$

$K_{\nu }^{(1)}(z;q)=\frac{1}{2}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{e^{-\nu t}dt}{{(-e^{t/2}z/2;q)}_{\mathrm{\infty }}{(-e^{-t/2}z/2;q)}_{\mathrm{\infty }}},|\arg z|<\pi /2,$

$K_{\nu }^{(1)}(z;q)=\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\cosh \nu dt}{{(-e^{t/2}z/2;q)}_{\mathrm{\infty }}{(-e^{-t/2}z/2;q)}_{\mathrm{\infty }}}.$

See also

• q-Bessel polynomials