Difference between revisions of "Relationship between Bessel J and hypergeometric 0F1"
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Revision as of 18:36, 20 May 2015
Theorem: The following formula holds: $$J_{\nu}(z) = \left( \dfrac{z}{2} \right)^{\nu} \dfrac{1}{\Gamma(\nu+1)} {}_0F_1 \left(-;\nu+1;-\dfrac{z^2}{4} \right),$$ where $J_{\nu}$ denotes the Bessel function of the first kind, $\Gamma$ denotes the Gamma function and ${}_0F_1$ denotes the hypergeometric pFq.
Proof: █