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 $\phi$ denotes the [[Basic hypergeometric series phi|basic hypergeometric series $\phi$]].
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where $(\xi,q)_{\infty}$ denotes the [[Q-Pochhammer|$q$-Pochhammer symbol]] and ${}_0\phi_1$ denotes the [[Basic hypergeometric phi|basic hypergeometric $\phi$]].
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=Properties=
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=References=
  
 
[[Category:SpecialFunction]]
 
[[Category:SpecialFunction]]

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 $\phi$.

Properties

References