# Expressing upper incomplete gamma function of half-integer order in terms of gamma function?

N. M. Temme, "Special Functions" (Wiley 1996) gives the following expression that expresses the upper incomplete gamma function in terms of the ordinary gamma function, for integer orders:

$$\Gamma(n,z) = \Gamma(n) e^{-z} \sum_{m=0}^{n-1} \frac{z^m}{m!} \\ n=1,2,...$$

My experiments indicate that this is a convenient way to compute the upper incomplete gamma function for small integer orders as the computation appears to be numerically stable. I tried orders up to n=50 and a wide range of real z.

Is there a similar expression that allows the straightforward computation of the upper incomplete gamma function in terms of the ordinary gamma function, for half-integer orders, that is, $\Gamma(n+\frac{1}{2},z)$? I am aware that $\Gamma(\frac{1}{2},z) = \sqrt{\pi} erfc(\sqrt{z})$.

• Apart from your question. How it csn be said that the above relation is numerically stable? As, the R.H.S also involves $\Gamma(n)$.
– kaka
Mar 24 '14 at 1:50
• C++ provides a function tgamma() that computes $\Gamma(x)$ accurately, if a good math library is provided. The remaining computation does not involve subtractive cancellation and excessive round-off error, so a straight rendering of the above into C++ code (for n <= 50) seems to be accurate to about 15 decimal digits if the entire computation is performed in double precision. I am not a numerical analyst, just a computer science guy, so I determined the stability empirically (which is why I stated that it "appears" to be stable). I took care to avoid intermediate underflow in exp(-z). Mar 24 '14 at 3:15

According to Maple, for nonnegative integers $n$ $$\Gamma(1/2+n, t) = \text{pochhammer}(1/2,n) \sqrt{\pi}\; \text{erfc}(\sqrt{t}) + t^{n-1/2} e^{-t} \sum_{k=0}^{n-1} \text{pochhammer}(1/2-n,k) (-t)^{-k}$$
Note that $\text{pochhammer}(1/2,k) = 2^{1-2k} \dfrac{(2k-1)!}{(k-1)!}$.
• Could you please clarify the definition of $pochhammer (\frac{1}{2}-n, k)$ ? I am not familiar with this and can't determine it from the above (there is n on the left-hand side but k on the right-hand side, and I have been unable to track it down in literature so far. Thanks. Mar 24 '14 at 1:40
• For any nonnegative integer $k$, $\text{pochhammer}(z,k) = \prod_{j=0}^{k-1} (z + j) = z (z+1) \ldots (z + k-1)$ (with $\text{pochhammer}(z,0) = 1$). This is $\Gamma(z+k)/\Gamma(z)$ if $z$ is not a nonpositive integer. Mar 24 '14 at 5:53
• Thanks for the clarification. With $pochhammer(z,k) = \Gamma(z+k) / \Gamma(z)$ the straightforward translation into C++ works fine. Mar 24 '14 at 7:16
• In a now deleted answer, a poster pointed out this reference: M.A. Chaudhry, N.M. Temme, E.J.M. Veling, "Asymptotics and closed form of a generalized incomplete gamma function", Journal of Computational and Applied Mathematics 67 (1996) 371-379. On page 379, the following formula is given, which seems to be a more concise variant of the above: $$\Gamma(\alpha,x) = \Gamma(\alpha)\left\{erfc(\sqrt{x})+e^{-x} \sum_{j=0}^{\alpha-\frac{3}{2}}\frac{x^{j+\frac{1}{2}}}{(j+\frac{1}{2})\Gamma({j+\frac{1}{2}})}\right\}, \space \space \space \alpha = \frac{3}{2},\frac{5}{2}, ...,$$ Aug 14 '15 at 16:29