Gamma average of an algebraic factor

For $a,b>0$ , define the Gamma average profile

F_{a,b}(z)=\frac1{\Gamma(a)}\int_0^\infty e^{-v}v^{a-1}(1+zv)^{-b}\,dv,\qquad z\ge0.

It is positive, at most one, and satisfies $F_{a,b}(0)=1$ . The normalization uses the Gamma integral.

Relation to Tricomi U

For $z>0$ , substituting $t=zv$ gives

F_{a,b}(z)=z^{-a}U(a,1+a-b,z^{-1}).

The original integral, unlike this transformed expression, directly defines the endpoint value at zero. No values at negative $z$ are supplied by the displayed real integral: the factor $1+zv$ then reaches zero within the integration range.