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, and satisfies the Hahn-Exton ''q''-difference equation (). This function was introduced by in a special case and by in general.

The Hahn–Exton ''q''-Bessel function is given by
:$J_\nu^(x;q) = \frac \sum_\frac= \frac x^\nu _1\phi_1(0;q^;q,qx^2).$
$\phi$ is the basic hypergeometric function.

Properties

Zeros

Koelink and Swarttouw proved that $J_\nu^(x;q)$ has infinite number of real zeros.
They also proved that for $\nu>-1$ all non-zero roots of $J_\nu^(x;q)$ are real (). For more details, see . Zeros of the Hahn-Exton ''q''-Bessel function appear in a discrete analog of Daniel Bernoulli's problem about free vibrations of a lump loaded chain (, )

Derivatives

For the (usual) derivative and ''q''-derivative of $J_\nu^(x;q)$, see . The symmetric ''q''-derivative of $J_\nu^(x;q)$ is described on .

Recurrence Relation

The Hahn–Exton ''q''-Bessel function has the following recurrence relation (see ):
:$J_^(x;q)=\left(\frac+x\right)J_\nu^(x;q)-J_^(x;q).$

Alternative Representations

Integral Representation

The Hahn–Exton ''q''-Bessel function has the following integral representation (see ):
:$J_^(z;q)=\frac\int_^\frac\,dx.$
:$(a_1,a_2,\cdots,a_n;q)_:=(a_1;q)_(a_2;q)_\cdots(a_n;q)_.$

Hypergeometric Representation

The Hahn–Exton ''q''-Bessel function has the following hypergeometric representation (see ):
:$J_^(x;q)=x^\frac\ _1\phi_1(0;x^2 q;q,q^).$
This converges fast at $x\to\infty$. It is also an asymptotic expansion for $\nu\to\infty$.

References

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{{DEFAULTSORT:Hahn-Exton q-Bessel function
Special functions
Q-analogs