# Series expansion of Beta function in Mathematica

How one should do the series expansion of Beta function $$B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$$ around an arbitrary negative integer let's say $$x=-k$$ and $$y=-l$$? I want a symbolic expression of the coefficients of the expansion in terms of $$k$$ and $$l$$.

• may be relevant: math.stackexchange.com/questions/3195847/…
– ydd
Commented Aug 8, 2023 at 22:20
• @ydd That is relevant. But I want to avoid knowing all those details. I thought one should be able to directly expand the function around arbitrary points in mathematica.
– Hkw
Commented Aug 9, 2023 at 4:18

You can try directly expanding the function. For example, the first three terms:

Series[Beta[w, v], {w, -k, 3}, {v, -l, 3}]


That produces a large expression about -k and -l in terms of Gamma and PolyGamma functions.

• Plugging in integer values produces Indeterminate coefficient values though: Series[Beta[w, v], {w, -k, 1}, {v, -l, 1}] /. {l -> 1, k -> 1} $\left(\text{Indeterminate}+\text{Indeterminate} (v+1)+O\left((v+1)^2\right)\right)+(w+1) \left(\text{Indeterminate}+\text{Indeterminate} (v+1)+O\left((v+1)^2\right)\right)+O\left((w+1)^2\right)$
– ydd
Commented Aug 8, 2023 at 22:16
• @Lexington1776 Doing the series expansion doesn't really work. Even the leading term is not being reproduced by the way you are suggesting.
– Hkw
Commented Aug 9, 2023 at 4:12
• @ydd The beta function is singular at those negative integers. Try doing the expansion directly around -1. Not using the l->1. You will get some coefficients* 1/x+k..
– Hkw
Commented Aug 9, 2023 at 4:15