\[{{}_{0}{\mathbf{F}}_{1}}\left({-\atop\alpha+1};\frac{(x-1)z}{2}\right)\*{{}_{0%
}{\mathbf{F}}_{1}}\left({-\atop\beta+1};\frac{(x+1)z}{2}\right)=\left(\tfrac{1%
}{2}(1-x)z\right)^{-\frac{1}{2}\alpha}J_{\alpha}\left(\sqrt{2(1-x)z}\right)\*%
\left(\tfrac{1}{2}(1+x)z\right)^{-\frac{1}{2}\beta}I_{\beta}\left(\sqrt{2(1+x)%
z}\right)=\sum_{n=0}^{\infty}\frac{P^{(\alpha,\beta)}_{n}\left(x\right)}{%
\Gamma\left(n+\alpha+1\right)\Gamma\left(n+\beta+1\right)}z^{n},\]