Difference between revisions of "Q-Bessel functions"
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(Created page with "The three Jackson $q$-Bessel functions are $$J_{\nu}^{(1)}(x;q) = \dfrac{(q^{\nu+1};q)_{\infty}}{(q;q)_{\infty}} \left( \dfrac{x}{2} \right)^{\nu} {}_2\Phi_1\left(0,0;q^{\nu+1...") |
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$$J_{\nu}^{(3)}(x;q) = \dfrac{(q^{\nu+1};q)_{\infty}}{(q;q)_{\infty}} \left( \dfrac{x}{2} \right)^{\nu} {}_1\Phi_1\left(0;q^{\nu+1};q,\dfrac{qx^2}{4} \right),$$ | $$J_{\nu}^{(3)}(x;q) = \dfrac{(q^{\nu+1};q)_{\infty}}{(q;q)_{\infty}} \left( \dfrac{x}{2} \right)^{\nu} {}_1\Phi_1\left(0;q^{\nu+1};q,\dfrac{qx^2}{4} \right),$$ | ||
| − | where $(q^{\nu+1};q)_{\infty}$ and $(q,q)_{\infty}$ are [[q-Pochhammer symbol | q-Pochhammer symbols]] and ${}_p\Phi_q$ denotes the [[ | + | where $(q^{\nu+1};q)_{\infty}$ and $(q,q)_{\infty}$ are [[q-Pochhammer symbol | q-Pochhammer symbols]] and ${}_p\Phi_q$ denotes the [[Hypergeometric q-series | hypergeometric $q$-series]]<br />. |
Revision as of 00:55, 29 August 2014
The three Jackson $q$-Bessel functions are
$$J_{\nu}^{(1)}(x;q) = \dfrac{(q^{\nu+1};q)_{\infty}}{(q;q)_{\infty}} \left( \dfrac{x}{2} \right)^{\nu} {}_2\Phi_1\left(0,0;q^{\nu+1};q,-\dfrac{x^2}{4} \right),$$
$$J_{\nu}^{(2)}(x;q) = \dfrac{(q^{\nu+1};q)_{\infty}}{(q;q)_{\infty}} \left( \dfrac{x}{2} \right)^{\nu} {}_0\Phi_1\left(-q^{\nu+1};q,-\dfrac{x^2q^{\nu+1}}{4} \right),$$
and
$$J_{\nu}^{(3)}(x;q) = \dfrac{(q^{\nu+1};q)_{\infty}}{(q;q)_{\infty}} \left( \dfrac{x}{2} \right)^{\nu} {}_1\Phi_1\left(0;q^{\nu+1};q,\dfrac{qx^2}{4} \right),$$
where $(q^{\nu+1};q)_{\infty}$ and $(q,q)_{\infty}$ are q-Pochhammer symbols and ${}_p\Phi_q$ denotes the hypergeometric $q$-series
.
