Formula:KLS:01.14:13

<< Formula:KLS:01.14:12

formula in Some q-analogues of special functions

Formula:KLS:01.14:14 >>

${\displaystyle{\displaystyle{\displaystyle J^{(2)}_{\nu}\!\left(z;q\right):=\frac{\left(q^{\nu+1};q\right)_{\infty}}{\left(q;q\right)_{\infty}}\left(\frac{z}{2}\right)^{\nu}\,\qHyperrphis{0}{1}@@{-}{q^{\nu+1}}{q}{-\frac{q^{\nu+1}z^{2}}{4}}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle J^{(2)}_{q}}}}$ : Jackson ${\displaystyle{\displaystyle{\displaystyle q}}}$-Bessel function 2 : http://drmf.wmflabs.org/wiki/Definition:JacksonqBesselII
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}$ : basic hypergeometric (or ${\displaystyle{\displaystyle{\displaystyle q}}}$-hypergeometric) function : http://dlmf.nist.gov/17.4#E1

Bibliography

Equation in Section 1.14 of KLS.

URL links

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<< Formula:KLS:01.14:12

formula in Some q-analogues of special functions

Formula:KLS:01.14:14 >>