$$ _{1}F_{2}\left(n-\frac{1}{2};n+1,2n+1;-x^{2}\right)=\frac{2^{2n+1}\Gamma^{2}(n+1)}{x^{2n}(2n+1)}\left(\left(xJ^{'}_{n}(x)+\frac{J_{n}(x)}{2}\right)^{2}+\left(x^{2}-n^{2}+\frac{1}{4}\right)J_{n}^{2}(x)\right)$$I just came to confront this $ _{1}F_{2}$ hypergeometric series while i was integrating a square of a Bessel Function.My question is there a way to prove the above $ _{1}F_{2}$ hypergeometric series in terms of bessel function. I tried methods like taking the square of a bessel function which inturn can be presented in terms of $ _{1}F_{2}$ hypergeometric series but nothing worked out,I hope if anyone could come up with a technique or a reference mathematical paper regarding to this question.
$\endgroup$ 2 Reset to default1F2 Hypergeometric and Bessel function
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