How to Expand the Lower Incomplete Gamma Function for s < 0

For positive $x$, the lower incomplete gamma function has a useful power series even when $s<0$. The main caveat is that its usual integral definition no longer converges at zero. The series instead gives the function by analytic continuation in the parameter $s$.

For negative noninteger $s$, the expansion is

$$\gamma(s,x)=x^s\sum_{n=0}^{\infty}\frac{(-x)^n}{n!(s+n)}.$$

It converges for every finite positive $x$. The excluded values $s=0,-1,-2,\ldots$ are poles of $\gamma(s,x)$ as a function of $s$.

Why the integral needs continuation

When $\operatorname{Re}(s)>0$, the lower incomplete gamma function is defined by

$$\gamma(s,x)=\int_0^x t^{s-1}e^{-t}\,dt.$$

Near $t=0$, the exponential approaches $1$, so the integrand behaves like $t^{s-1}$. The integral converges at zero only when $\operatorname{Re}(s)>0$. For real $s<0$, inserting that value directly into the integral gives a divergent improper integral; it does not define $\gamma(s,x)$ there.

The gamma function itself, and the incomplete gamma functions, can be extended meromorphically in $s$. The series below is one way to express that extension. For real positive $x$, there is no branch choice involved in $x^s$. For complex $x$, a branch of $x^s$ must be specified, commonly the principal branch.

Deriving the series

Start in the region where the integral converges and expand the exponential:

$$e^{-t}=\sum_{n=0}^{\infty}\frac{(-t)^n}{n!}.$$

Termwise integration gives

$$\begin{aligned} \gamma(s,x) &=\int_0^x t^{s-1}\sum_{n=0}^{\infty}\frac{(-t)^n}{n!}\,dt\\ &=\sum_{n=0}^{\infty}\frac{(-1)^n}{n!}\frac{x^{s+n}}{s+n}\\ &=x^s\sum_{n=0}^{\infty}\frac{(-x)^n}{n!(s+n)}. \end{aligned}$$

This derivation is initially valid for $\operatorname{Re}(s)>0$. The right-hand side is a meromorphic function of $s$, with possible poles where a denominator $s+n$ vanishes. It therefore supplies the analytic continuation to negative noninteger values of $s$. For fixed such $s$, the factorial in the denominator ensures convergence for every finite $x$.

Worked example: $s=-\tfrac12$

Set $s=-\tfrac12$. The series becomes

$$\gamma\!\left(-\frac12,x\right) =x^{-1/2}\left( -2-2x+\frac{x^2}{3}-\frac{x^3}{15}+\frac{x^4}{84}-\cdots \right).$$

For example, at $x=\tfrac14$ this gives the convergent representation

$$\gamma\!\left(-\frac12,\frac14\right) =2\left( -2-\frac12+\frac{1}{48}-\frac{1}{960}+\frac{1}{21504}-\cdots \right).$$

Each coefficient comes directly from $1/[n!(n-\tfrac12)]$, with the alternating factor $(-x)^n$. The leading behavior as $x\to0^+$ is $-2x^{-1/2}$; the function is not finite at zero for this negative parameter.

Recurrence as another way to evaluate it

The recurrence

$$\gamma(s+1,x)=s\gamma(s,x)-x^s e^{-x}$$

can relate a negative noninteger parameter to one with a larger real part. Solving for the desired value gives

$$\gamma(s,x)=\frac{\gamma(s+1,x)+x^s e^{-x}}{s},\qquad s\ne0.$$

For instance, this expresses $\gamma(-\tfrac12,x)$ in terms of $\gamma(\tfrac12,x)$ and an elementary term. The recurrence is useful for computation, but it does not remove the poles at nonpositive integer $s$.

For large positive $x$ and noninteger $s$, another representation is $\gamma(s,x)=\Gamma(s)-\Gamma(s,x)$. The upper incomplete gamma function $\Gamma(s,x)$ decays exponentially as $x$ grows, so $\gamma(s,x)$ approaches $\Gamma(s)$. This large-$x$ behavior is distinct from the power series, which is often most convenient for moderate or small $x$.

Common mistakes

• Using the integral for $s<0$. Its divergence at zero means the integral cannot be used as an ordinary improper integral in that range.
• Using the series at a pole. If $s=-m$ for a nonnegative integer $m$, the term with $n=m$ has denominator zero. The meromorphic function has a pole there; the displayed formula does not give a finite value.
• Calling the series an asymptotic expansion. For fixed admissible $s$ and finite $x$, it is a convergent power series (with the prefactor $x^s$), not merely a large- or small-$x$ approximation.
• Ignoring the parameter convention. Some contexts discuss a finite part or a regularized value at a pole. That requires an additional convention and is not the ordinary meromorphic value of $\gamma(s,x)$.