Difference between revisions of "Jackson q-Bessel (1)"
From specialfunctionswiki
| Line 2: | Line 2: | ||
$$J_{\nu}^{(1)}(x;q)=\dfrac{(q^{\nu+1};q)_{\infty}}{(q;q)_{\infty}} \left( \dfrac{x}{2} \right)^{\nu} {}_0\phi_1 \left(-;q^{\nu+1};q,-\dfrac{x^2}{4} \right),$$ | $$J_{\nu}^{(1)}(x;q)=\dfrac{(q^{\nu+1};q)_{\infty}}{(q;q)_{\infty}} \left( \dfrac{x}{2} \right)^{\nu} {}_0\phi_1 \left(-;q^{\nu+1};q,-\dfrac{x^2}{4} \right),$$ | ||
where $(\xi,q)_{\infty}$ denotes the [[Q-Pochhammer|$q$-Pochhammer symbol]] and $\phi$ denotes the [[Basic hypergeometric phi|basic hypergeometric $\phi$]]. | where $(\xi,q)_{\infty}$ denotes the [[Q-Pochhammer|$q$-Pochhammer symbol]] and $\phi$ denotes the [[Basic hypergeometric phi|basic hypergeometric $\phi$]]. | ||
| + | |||
| + | =Properties= | ||
| + | |||
| + | =References= | ||
[[Category:SpecialFunction]] | [[Category:SpecialFunction]] | ||
Revision as of 21:37, 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 $\phi$ denotes the basic hypergeometric $\phi$.
