Hahn–Exton q-Bessel function explained

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 ^{( 3 )} _{\nu} (x;q) = (x ^{\nu(q} ^{\nu+1} ;q) _{infty})/((q;q) _{infty}) \sum _{k\ge} (( -1 ) ^{kq} ^{k(k+1)/2} x ^{2k})/((q ^{\nu+1} ;q) _{k(q;q)} _{k}) = ((q ^{\nu+1} ;q) _{infty})/((q;q) _{infty}) x ^{\nu} {} _{1\phi} ^{\nu +1} ^{1 ( 0 ;q} ;q,qx ^{2).} \phi is the basic hypergeometric function.

Properties

Zeros

Koelink and Swarttouw proved that

J ^{( 3 )} _{\nu} (x;q) has infinite number of real zeros.They also proved that for \nu> -1 all non-zero roots of J ^{( 3 )} _{\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 ^{( 3 )} _{\nu} (x;q) , see . The symmetric q-derivative of J ^{( 3 )} _{\nu} (x;q) is described on .

Recurrence Relation

The Hahn–Exton q-Bessel function has the following recurrence relation (see):

J ^{( 3 )} _{\nu +1} (x;q) = \left( (1- q ^{\nu})/(x) + x\right)J ^{( 3 )} _{\nu} (x;q) - J ^{( 3 )} _{\nu -1} (x;q).

Alternative Representations

Integral Representation

The Hahn–Exton q-Bessel function has the following integral representation (see):

J ^{( 3 )} _{\nu} (z;q) = (z ^{\nu})/(\sqrt{\pi log q ^{-2}) }\int_^\frac\,dx.(a _{1,a} _{2, … ,a} _{n;q)} _{infty} : = (a _{1;q)} _{infty} (a _{2;q)} _{infty} … (a _{n;q)} _{infty} .

Hypergeometric Representation

The Hahn–Exton q-Bessel function has the following hypergeometric representation (see):

J ^{( 3 )} _{\nu} (x;q) = x ^{\nu} ((x ^{2} q;q) _{infty})/((q;q) _{infty}) _{1\phi} ^{2} ^{1 ( 0 ;x} q;q,q ^{\nu+1} ). This converges fast at x\to inf ty . It is also an asymptotic expansion for \nu\to inf ty