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 HahnâÂÂExton q-Bessel function.
The three Jackson q-Bessel functions are given in terms of the q-Pochhammer symbol and the basic hypergeometric function by
They can be reduced to the Bessel function by the continuous limit:<br>
There is a connection formula between the first and second Jackson q-Bessel function ():
For integer order, the q-Bessel functions satisfy
By using the relations ():
we obtain
Hahn mentioned that has infinitely many real zeros (). Ismail proved that for all non-zero roots of are real ().
The function is a completely monotonic function ().
The first and second Jackson q-Bessel function have the following recurrence relations (see and ):
When , the second Jackson q-Bessel function satisfies:
(see .)
For ,
(see .)
The following formulas are the q-analog of the generating function for the Bessel function (see ):<br>
is the q-exponential function.
The second Jackson q-Bessel function has the following integral representations (see and ):
where is the q-Pochhammer symbol. This representation reduces to the integral representation of the Bessel function in the limit .
The second Jackson q-Bessel function has the following hypergeometric representations (see , ):
An asymptotic expansion can be obtained as an immediate consequence of the second formula.
For other hypergeometric representations, see .
The q-analog of the modified Bessel functions are defined with the Jackson q-Bessel function ( and ):
There is a connection formula between the modified q-Bessel functions:
For statistical applications, see .
By the recurrence relation of Jackson q-Bessel functions and the definition of modified q-Bessel functions, the following recurrence relation can be obtained ( also satisfies the same relation) ():
For other recurrence relations, see .
The ratio of modified q-Bessel functions form a continued fraction ():
The function has the following representation ():
The modified q-Bessel functions have the following integral representations ():