Difference between revisions of "Jackson q-Bessel (1)"
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The Jackson $q$-Bessel function $J_{\nu}^{(1)}$ is defined by | The Jackson $q$-Bessel function $J_{\nu}^{(1)}$ is defined by | ||
$$J_{\nu}^{(1)}(x;q)=\dfrac{(q^{\nu+1};q)_{\infty}}{(q;q)_{\infty}} \left( \dfrac{x}{2} \right)^{\nu} {}_0\phi_1 \left(-;q^{\nu+1};q,-\dfrac{x^2}{4} \right),$$ | $$J_{\nu}^{(1)}(x;q)=\dfrac{(q^{\nu+1};q)_{\infty}}{(q;q)_{\infty}} \left( \dfrac{x}{2} \right)^{\nu} {}_0\phi_1 \left(-;q^{\nu+1};q,-\dfrac{x^2}{4} \right),$$ | ||
| − | where $(\xi,q)_{\infty}$ denotes the [[Q-Pochhammer|$q$-Pochhammer symbol]] and $\ | + | where $(\xi,q)_{\infty}$ denotes the [[Q-Pochhammer|$q$-Pochhammer symbol]] and ${}_0\phi_1$ denotes the [[Basic hypergeometric phi|basic hypergeometric $\phi$]]. |
=Properties= | =Properties= | ||
Latest revision as of 21:38, 17 June 2017
The Jackson $q$-Bessel function $J_{\nu}^{(1)}$ is defined by $$J_{\nu}^{(1)}(x;q)=\dfrac{(q^{\nu+1};q)_{\infty}}{(q;q)_{\infty}} \left( \dfrac{x}{2} \right)^{\nu} {}_0\phi_1 \left(-;q^{\nu+1};q,-\dfrac{x^2}{4} \right),$$ where $(\xi,q)_{\infty}$ denotes the $q$-Pochhammer symbol and ${}_0\phi_1$ denotes the basic hypergeometric $\phi$.
