The Incomplete Gamma and Confluent Hypergeometric Functions (Part 1)

Calculus 1 Minute

Yes, the title of this post is a mouthful.

While working on a research project, a trail of citations led me to this curious equality in the Digital Library of Mathematical Functions:

\gamma(a,z) = \displaystyle \frac{z^a e^{-z}}{a} M(1, 1+a, z),

where the incomplete gamma function \gamma(a,z) is

\gamma(a,z) = \displaystyle \int_0^z t^{a-1} e^{-t} \, dt

and the confluent hypergeometric function M(a,b,z) is

M(a,b,z) = \displaystyle 1+\sum_{s=1}^\infty \frac{a(a+1)\dots(a+s-1)}{b(b+1)\dots (b+s-1)} \frac{z^s}{s!}.

While I didn’t doubt that this was true — I don’t doubt this has been long established — I had an annoying problem: I didn’t really believe it. The gamma function

\Gamma(a) = \displaystyle \int_0^\infty t^{a-1} e^{-t} \, dt

is a well-known function with the famous property that

\Gamma(n+1) = n!

for non-negative integers n; this is often seen in calculus textbooks as an advanced challenge using integration by parts. The incomplete gamma function \gamma(a,z) has the same look as \Gamma(a), except that the range of integration is from 0 to z (and not \infty). The gamma function appears all over the place in mathematics courses.

The confluent hypergeometric function, on the other hand, typically arises in mathematical physics as the solution of the differential equation

z f''(z) + (b-z) f'(z) - af(z) = 0.

As I’m not a mathematical physicist, I won’t presume to state why this particular differential equation is important — except that it appears to be a niche equation that arises in very specialized applications.

So I had a hard time psychologically accepting that these two functions were in any way related.

While ultimately unimportant for advancing mathematics, this series will be about the journey I took to directly confirm the above equality.

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Published by John Quintanilla

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.

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