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.
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 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)).
The function
is a completely monotonic function (Ismail (1982)).
The first and second Jackson q-Bessel function have the following recurrence relations (see Ismail (1982) and Gasper & Rahman (2004)): When
, the second Jackson q-Bessel function satisfies:
exp
log
2 log q
(see Zhang (2006).)
(see Koelink (1993).)
The following formulas are the q-analog of the generating function for the Bessel function (see Gasper & Rahman (2004)):
is the q-exponential function.
The second Jackson q-Bessel function has the following integral representations (see Rahman (1987) and Ismail & Zhang (2018a)): 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 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).
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: For statistical applications, see Kemp (1997).
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).
The ratio of modified q-Bessel functions form a continued fraction (Ismail (1981)): The function
has the following representation (Ismail & Zhang (2018b)): The modified q-Bessel functions have the following integral representations (Ismail (1981)):