Knowlpedia

Theorem

Differential equation for the Gamma average

An integration-by-parts identity gives a second-order equation for the normalized Gamma integral profile.

Statement

The F=Fa,bF=F_{a,b}, a,b>0a,b>0, satisfies

z2F′′+[1+(a+b+1)z]F′+abF=0(z≥0),z^2F''+[1+(a+b+1)z]F'+abF=0 \qquad(z\ge0),

where derivatives at zero are one-sided.

Integral proof

Insert the differentiated integral formulas into the left side. The result is

bΓ(a)∫0∞ddv[e−vva(1+zv)−b−1]dv.\frac b{\Gamma(a)}\int_0^\infty \frac{d}{dv}\left[e^{-v}v^a(1+zv)^{-b-1}\right]dv.

The integrand's primitive vanishes at both endpoints, so the expression is zero. This argument proves the differential equation directly for the function; checking a formal Taylor recurrence alone would not prove it, since the endpoint Taylor series diverges.