How to prove this.Thank you in advance Let $\delta,\beta>0$ How to prove this \begin{align} & \int^1_0 \frac{w^{1-\beta}}{(1-w)^{1+\delta}} (-t.s w)^{\frac{-\delta}{2}} e^{-\frac{w}{1-w}(s+t)}J_\delta \left(\frac{(-t.s w)^{\frac{1}{2}}}{1-w} \right) \, dw \\[8pt] & = \frac{\Gamma(\beta+1)}{\Gamma(\delta+1)}\Phi(\beta,\delta+1,s)\Psi(\beta,\delta+1,s) \end{align} where $J_\delta$ is the Bessel function and \begin{align} \Psi(\alpha, \gamma ; z) = {} & \frac{1}{\Gamma(\alpha)} \int_0^\infty e^{-z t} t^{\alpha-1}(1+t)^{\gamma-\alpha-1} d t \\[6pt] & [\operatorname{Re} \alpha>0, \quad \operatorname{Re} z>0] \end{align} with $$ \Phi(\alpha, \gamma ; z)=\frac{2^{1-\gamma} e^{\frac{1}{2} z}}{\operatorname{B}(\alpha, \gamma-\alpha)} \int_{-1}^1(1-t)^{\gamma-\alpha-1}(1+t)^{\alpha-1} e^{\frac{1}{2} z t} \, dt $$
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