Jackson q-Bessel function explained

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

\phi byJ ^{( 1 )} _{\nu} (x;q) = ((q ^{\nu+1} ;q) _{infty})/((q;q) _{infty}) (x/ 2 ) ^{\nu} {} _{2\phi} ^{\nu +1} ^{1 ( 0 , 0 ;q} ;q, - x ^{2/4),} |x|< 2 , J ^{( 2 )} _{\nu} (x;q) = ((q ^{\nu+1} ;q) _{infty})/((q;q) _{infty}) (x/ 2 ) ^{\nu} {} _{0\phi} ^{\nu +1} ^{1 (;q} ;q, - x ^{2q} ^{\nu} / 4 ), x\in C , J ^{( 3 )} _{\nu} (x;q) = ((q ^{\nu+1} ;q) _{infty})/((q;q) _{infty}) (x/ 2 ) ^{\nu} {} _{1\phi} ^{\nu +1} ^{1 ( 0 ;q} ;q,qx ^{2/4),} x\in C . They can be reduced to the Bessel function by the continuous limit:
\lim _{q\to1} J ^{(k)} _{\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 ^{( 2 )} _{\nu} (x;q) = ( - x ^{2 / 4 ;q)} _{inf ty} J ^{( 1 )} _{\nu} (x;q), |x|< 2 . For integer order, the q-Bessel functions satisfyJ ^{(k)} _{n} ( - x;q) = ( -1 ) ^{n} J ^{(k)} _{n} (x;q), n\in Z , k =1 , 2 , 3 .

Properties

Negative Integer Order

By using the relations :

(q ^{m+1} ;q) ^{m + n +1} _{inf ty = (q} ;q) _{infty} (q ^{m+1} ;q) _{n,} (q;q) _{m+n} = (q;q) _{m} (q ^{m+1} ;q) _{n, m,n\inZ,} we obtainJ ^{(k)} _{- n} (x;q) = ( -1 ) ^{n} J ^{(k)} _{n} (x;q), k =1 , 2 .

Zeros

Hahn mentioned that

J ^{( 2 )} _{\nu} (x;q) has infinitely many real zeros . Ismail proved that for \nu> -1 all non-zero roots of J ^{( 2 )} _{\nu} (x;q) are real .

Ratio of q-Bessel Functions

The function

- ix ^{-1/2} J ^{( 2 )} _{\nu +1} (ix ^{1/2} ;q)/J ^{( 2 )} _{\nu} (ix ^{1/2} ;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 ^{(k)} _{\nu +1} (x;q) = (2 ( 1- q ^{\nu)})/(x) J ^{(k)} _{\nu} (x;q) - J ^{(k)} _{\nu -1} (x;q), k =1 , 2 . J ^{( 1 )} _{\nu} (x\sqrt{q};q) = q ^{\pm\nu/2} \left(J ^{( 1 )} _{\nu} (x;q)\pm (x)/(2) J ^{( 1 )} _{\nu\pm 1} (x;q)\right).

Inequalities

When

\nu> -1 , the second Jackson q-Bessel function satisfies:\left|J ^{( 2 )} _{\nu} (z;q)\right|\leq (( - \sqrt{q)/(;q)) _{infty} }\left(\frac\right)^\nu\exp\left\.(see .)

For

n\in Z , \left|J ^{( 2 )} _{n} (z;q)\right|\leq (( - q ^{n+1} ;q) _{infty})/(\left( (q;q) _{infty}) (|z|)/(2) \right) ^{n(-|z|} ^{2 ;q)} ^{inf ty} . (see .)

Generating Function

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

\sum ^{inf ty} _{n =-inf ty} t ^{( 2 )} _{n} (x;q) = ( - x ^{2 / 4 ;q)} _{inf ty} e _{q(xt/2)e} _{q(-x/2t),} \sum ^{inf ty} _{n =-inf ty} t ^{( 3 )} _{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 ^{( 2 )} _{\nu} (x;q) = ((q ^{2\nu} ;q) _{infty})/(2 \pi(q ^{\nu} ;q) _{infty}) (x/ 2 ) ^{\nu} ⋅ \int ^{\pi} _{0} (\left(e ^{2 i\theta} , e ^{-2i\theta} , - (i x q ^{(\nu+1)/2})/(2) e ^{i\theta} , - (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, … ,a} _{n;q)} _{infty} : = (a _{1;q)} _{infty} (a _{2;q)} _{infty} … (a _{n;q)} _{infty} , \Re \nu> 0 , where (a;q) _{infty} is the q-Pochhammer symbol. This representation reduces to the integral representation of the Bessel function in the limit q\to 1 .J ^{( 2 )} _{\nu} (z;q) = ((z/ 2 ) ^{\nu})/(\sqrt{ 2 \pi log q ^{-1}) }\int_^\frac\,dx.

Hypergeometric Representations

The second Jackson q-Bessel function has the following hypergeometric representations (see,):

J ^{( 2 )} _{\nu} (x;q) = ((x/ 2 ) ^{\nu})/((q;q) _{infty}) _{1\phi} ^{2 / 4 ; 0 ;q,q} ^{1 ( - x} ^{\nu+1} ), J ^{( 2 )} _{\nu} (x;q) = ((x/ 2 ) ^{\nu} (\sqrt{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)_\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 ^{(j)} _{\nu} (x;q) = e ^{i\nu\pi/2} J ^{(j)} _{\nu} (x;q), j =1 , 2 . K ^{(j)} _{\nu} (x;q) = (\pi)/(2 \sin(\pi\nu)) \left\{I ^{(j)} _{- \nu} (x;q) - I ^{(j)} _{\nu} (x;q)\right\}, j =1 , 2 , \nu\in C - Z , K ^{(j)} _{n} (x;q) = \lim _{\nu\to} K ^{(j)} _{\nu} (x;q), n\in Z . There is a connection formula between the modified q-Bessel functions:I ^{( 2 )} _{\nu} (x;q) = ( - x ^{2 / 4 ;q)} _{inf ty} I ^{( 1 )} _{\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 ^{(j)} _{\nu} (x;q) also satisfies the same relation) :q ^{\nu} I ^{(j)} _{\nu +1} (x;q) = (2)/(z) ( 1- q ^{(j)} _{\nu} (x;q) + I ^{(j)} _{\nu -1} (x;q), j =1 , 2 . For other recurrence relations, see .

Continued Fraction Representation

The ratio of modified q-Bessel functions form a continued fraction :

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

Alternative Representations

Hypergeometric Representations

The function

I ^{( 2 )} _{\nu} (z;q) has the following representation :I ^{( 2 )} _{\nu} (z;q) = ((z/ 2 ) ^{\nu})/((q,q) _{infty}) {} _{1\phi} ^{2 / 4 ; 0 ;q,q} ^{1 (z} ^{\nu+1} ).

Integral Representations

The modified q-Bessel functions have the following integral representations :

I ^{( 2 )} _{\nu} (z;q) = \left(z ^{2 / 4 ;q\right)} _{inf ty\left( (1)/(\pi)} \int ^{\pi (\cos\nu\thetad\theta)/(\left(e ^{i\theta} z/ 2 ;q\right) ^{- i\theta} _{inf ty\left(e} z/ 2 ;q\right) _{infty})} _{0} - (\sin\nu\pi)/(\pi) \int ^{inf ty (e ^{-\nu} dt)/(\left( - e ^{t} z/ 2 ;q\right) ^{- t} _{inf ty\left( - e} z/ 2 ;q\right) _{infty})} _{0} \right), K ^{( 1 )} _{\nu} (z;q) = (1)/(2) \int ^{inf ty (e ^{-\nu} dt)/(\left( - e ^{t/ 2} z/ 2 ;q\right) ^{- t/ 2} _{inf ty\left( - e} z/ 2 ;q\right) _{infty})} _{0} , |\arg z|<\pi/ 2 , K ^{( 1 )} _{\nu} (z;q) = \int ^{inf ty (\cosh\nu dt)/(\left( - e ^{t/ 2} z/ 2 ;q\right) ^{- t/ 2} _{inf ty\left( - e} z/ 2 ;q\right) _{infty})} _{0} .

See also

• q-Bessel polynomials

References

• • • • • • • • • • • • • • • Olshanetsky . M. A. . Rogov . V. B. . 1995 . The Modified q-Bessel Functions and the q-Bessel-Macdonald Functions . q-alg/9509013 . cs2.
  • Zhang. R.. Plancherel-Rotach Asymptotics for q-Series . 2006 . math/0612216 . cs2.