Hi, fellow physicists (to be). This is my first post on the forum, so I hope I get it right. If not so, please let me know :)(adsbygoogle = window.adsbygoogle || []).push({});

introduction to the problem

At the moment I am working on my physics bachelor's thesis at the theoretical department of my university (Amsterdam). My thesis focusses on a certain supersymmetric 1D lattice model, on which spinless fermions can be placed. While working in Mathematica, I found an exact expression for the occupation of sub-levels on the chain of length l (l being the number of sites on the lattice). The expression I found (f(l)), however, is a product over indices, which is not very insightful. My supervisor has asked me to rewrite this (f(l)) into a power of l. To do this, he said that I should use Stirling's approximation.

The problem

The function I found is

\begin{equation}

f(l) = \frac{2}{5} {(-1)}^l \prod_{i=0}^{l-3} \frac{3(l-i)-2}{3(l-i)-1}.

\end{equation}

While being not skilled in rewriting this, Mathematica rewrote f(l) in terms of Gamma functions. leaving out the pre-factors, I am left with

\begin{equation}

f(l)={(-1)}^l \frac{3l-2}{3l-1} \frac{\Gamma(\frac{4}{3}-l)}{\Gamma(\frac{5}{3}-l)}.

\end{equation}

On this expression I wanted to use Sterling's approximation for Gamma functions:

\begin{equation}

\Gamma(z)=\sqrt{\frac{2π}{z}}(\frac{z}{e})^{z}.

\end{equation}

The trouble that arises is that this approximation is only valid for positive arguments z of Gamma.

Attempt to Solve

To solve this, I tried to use the recursion relation

\begin{equation}

\Gamma(z)=\frac{\Gamma(z+1)}{z}

\end{equation}

If I use this recursion relation, however, I'm back at the indexed product I started with.

Does anyone on this forum know of a different approach, some sparks of creativity or otherwise good tips? All help is very welcome.

greetings,

Ludo

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Stirling's approximation for Gamma functions with a negative argument

**Physics Forums | Science Articles, Homework Help, Discussion**