Integration problem related to Gamma function: $ \int_{0}^{\infty} u^{\alpha + b - 1} \exp\left(-ub + u^{\alpha}c\right)du $

Question

During my work on some statistics problem, I stumbled across the following integral:

$$ \int_{0}^{\infty} u^{\alpha + b - 1} \exp\left(-ub + u^{\alpha}c\right)du,\qquad \alpha, b, c>0 $$

I tried to get a closed form expression for this integral for general $\alpha$, but I have not been successful. That being said, for $\alpha=1$, this reduces to an integration problem closely related to the Gamma function and I was able to get a closed form expression for this case. Does anybody have some advice how to get a closed form expression for the case of $\alpha\neq 1$?

Answer 1

I assume there is also the condition $\alpha < 1 \lor (\alpha = 1 \land b > c)$. Then we can apply the general formula for an integral of a product of two Meijer G-functions: $$\int_0^\infty t^{\alpha + b - 1} e^{-b u + c u^\alpha} du = \int_0^\infty t^{\alpha + b - 1} G_{0, 1}^{1, 0} \left( b u \middle| { - \atop 0} \right) G_{0, 1}^{1, 0} \left( -c u^\alpha \middle| { - \atop 0} \right) du = \\ b^{-\alpha - b} H_{1, 1}^{1, 1} \left( -c b^{-\alpha} \middle| {(1 - \alpha - b, \alpha) \atop (0, 1)} \right),$$ where $H$ is the Fox H-function. If $\alpha = m/n$ is rational, the result can also be written as $G_{m, n}^{n, m}$, which can in this case be converted to a sum of hypergeometric functions by Slater's theorem.

Answer 2

Case $1$: $0<\alpha\leq1$

Then $\int_0^\infty u^{\alpha+b-1}e^{-ub+u^{\alpha}c}~du$

$=\int_0^\infty\sum\limits_{n=0}^\infty\dfrac{c^nu^{\alpha n+\alpha+b-1}e^{-bu}}{n!}~du$

$=\sum\limits_{n=0}^\infty\dfrac{c^n\Gamma(\alpha n+\alpha+b)}{b^{\alpha n+\alpha+b}n!}$ (can be obtained from List of integrals)

$=\dfrac{1}{b^{\alpha+b}}~_1\Psi_0\left[\begin{matrix}(\alpha+b,\alpha)\\-\end{matrix};\dfrac{c}{b^\alpha}\right]$ (according to For Wright function)