Mu Function

$\displaystyle \mu(x,\beta)$	$\textstyle \equiv$	$\displaystyle \int_0^\infty {x^t t^\beta\,dt\over \Gamma(\beta+1)\Gamma(t+1)}$
$\displaystyle \mu(x,\beta,\alpha)$	$\textstyle \equiv$	$\displaystyle \int_0^\infty {x^{\alpha+t}t^\beta\,dt\over \Gamma(\beta+1)\Gamma(\alpha+t+1)},$

where $\Gamma(z)$ is the Gamma Function (Gradshteyn and Ryzhik 1980, p. 1079).

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, 1979.