Difference between revisions of "Jackson q-Bessel (1)"
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$$J_{\nu}^{(1)}(x;q)=\dfrac{(q^{\nu+1};q)_{\infty}}{(q;q)_{\infty}} \left( \dfrac{x}{2} \right)^{\nu} {}_0\phi_1 \left(-;q^{\nu+1};q,-\dfrac{x^2}{4} \right),$$ | $$J_{\nu}^{(1)}(x;q)=\dfrac{(q^{\nu+1};q)_{\infty}}{(q;q)_{\infty}} \left( \dfrac{x}{2} \right)^{\nu} {}_0\phi_1 \left(-;q^{\nu+1};q,-\dfrac{x^2}{4} \right),$$ | ||
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 $\phi$]]. | ||
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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 $\phi$.
