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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 (1906a, 1906b, 1905a, 1905b). The third Jackson q-Bessel function is the same as the Hahn–Exton q-Bessel function.

Definition

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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:

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

Properties

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Negative Integer Order

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

we obtain

Zeros

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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

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The function is a completely monotonic function (Ismail (1982)).

Recurrence Relations

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

Inequalities

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

For , (see Koelink (1993).)

Generating Function

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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

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Integral Representations

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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 .

Hypergeometric Representations

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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

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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).

Recurrence Relations

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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

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

Alternative Representations

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Hypergeometric Representations

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The function has the following representation (Ismail & Zhang (2018b)):

Integral Representations

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

See also

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References

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