Jackson q-Bessel function
Encyclopedia
In mathematics, a Jackson q-Bessel function (or basic Bessel function) is one of the three q-analog
s of the Bessel function
introduced by . The third Jackson q-Bessel function is the same as the Hahn–Exton q-Bessel function
.
and the basic hypergeometric function φ by
Q-analog
Roughly speaking, in mathematics, specifically in the areas of combinatorics and special functions, a q-analog of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as q → 1...
s of the Bessel function
Bessel function
In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...
introduced by . The third Jackson q-Bessel function is the same as the Hahn–Exton q-Bessel function
Hahn–Exton q-Bessel function
In mathematics, the Hahn–Exton q-Bessel function or the third Jackson q-Bessel function is a q-analog of the Bessel function, introduced by in a special case and by in general....
.
Definition
The three Jackson q-Bessel functions are given in terms of the Pochhammer symbolPochhammer symbol
In mathematics, the Pochhammer symbol introduced by Leo August Pochhammer is the notation ', where is a non-negative integer. Depending on the context the Pochhammer symbol may represent either the rising factorial or the falling factorial as defined below. Care needs to be taken to check which...
and the basic hypergeometric function φ by