Jackson q-Bessel function

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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 (1903, 1903b, 1905, 1905b). The third Jackson q-Bessel function is the same as the Hahn–Exton q-Bessel function.


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

They 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 (Gasper & Rahman (2004)):

For integer order, the q-Bessel functions satisfy


Negative Integer Order[edit]

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

we obtain


Hahn mentioned that has infinitely many real zeros (Hahn (1949)). Ismail proved that for all non-zero roots of are real (Ismail (1982)).

Ratio of q-Bessel Functions[edit]

The function has complete monotonicity (Ismail (1982)).

Recurrence Relations[edit]

The first and second Jackson q-Bessel function has the following recurrence relations (see Ismail (1982) and Gasper & Rahman (2004)):


When , the second Jackson q-Bessel function satisfies: (see Zhang (2006).)
For , (see Koelink (1993).)

Generating Function[edit]

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

is the q-exponential function.

Alternative Representations[edit]

Integral Representations[edit]

The second Jackson q-Bessel functions has the following integral representations (see Rahman (1987) and Ismail and Zhang (2016)):

is the q-Pochhammer symbol, this representation reduces to the integral representation of the Bessel function with the limit .

Hypergeometric Representations[edit]

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

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[edit]

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

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

Recurrence Relations[edit]

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) (Ismail (1981)):

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

Continued Fraction Representation[edit]

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

Alternative Representations[edit]

Hypergeometric Representations[edit]

The function has the following representation(Ismail & Zhang (2015)):

Integral Representations[edit]

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

See also[edit]