Jackson q-Bessel function
In mathematics, a Jackson q-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 byThey can be reduced to the Bessel function by the continuous limit:
There is a connection formula between the first and second Jackson q-Bessel function :
For integer order, the q-Bessel functions satisfy
Properties
Negative Integer Order
By using the relations :we obtain
Zeros
Hahn mentioned that has infinitely many real zeros. Ismail proved that for all non-zero roots of are real.Ratio of ''q''-Bessel Functions
The function is a completely monotonic function.Recurrence Relations
The first and second Jackson q-Bessel function have the following recurrence relations :Inequalities
When, the second Jackson q-Bessel function satisfies:For,
Generating Function
The following formulas are the q-analog of the generating function for the Bessel function :is the q-exponential function.
Alternative Representations
Integral Representations
The second Jackson q-Bessel function has the following integral representations :where is the q-Pochhammer symbol. This representation reduces to the integral representation of the Bessel function in the limit.
Hypergeometric Representations
The second Jackson q-Bessel function has the following hypergeometric representations :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 :There is a connection formula between the modified q-Bessel functions:
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 :For other recurrence relations, see.