Certain new integral formulas involving the generalized Bessel functions (Q2877802)

Let \(w_{\nu } (z)\) be the generalized Bessel function of the first kind NEWLINE\[NEWLINEw_{\nu } (z)=\sum _{n=0}^{\infty }\frac{(-c)^{n} }{n!\Gamma (n+\nu +(1+b)/2))} \left(\frac{z}{2} \right)^{2n+\nu } .NEWLINE\]NEWLINE The authors evaluate in terms of \({}_{2} \Psi _{3} \) two integrals involving this function. In Theorem 1 they evaluate the integral NEWLINE\[NEWLINE\int _{0}^{\infty }x^{\mu -1} \left(x+a+\sqrt{\, x^{2} +2ax} \right) ^{-\lambda } w_{\nu } \left(y/\left(x+a+\sqrt{\, x^{2} +2ax} \right)\right)\, \, d{\kern 1pt} x .NEWLINE\]NEWLINE The second one is similar, with \(xy\) in the place of \(y\). Some particular cases are considered.