Difference between revisions of "Jackson q-Bessel (1)"

From specialfunctionswiki
Jump to: navigation, search
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 series phi|basic hypergeometric series $\phi$]].
 
where $(\xi,q)_{\infty}$ denotes the [[Q-Pochhammer|$q$-Pochhammer symbol]] and $\phi$ denotes the [[Basic hypergeometric series phi|basic hypergeometric series $\phi$]].
 +
 +
[[Category:SpecialFunction]]

Revision as of 18:54, 24 May 2016

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 series $\phi$.