# Free Riordan Array transform software

Does anybody know of free/open software that can directly deal with Riodran Array transformations? I am trying to use them to analyze some combinatorial identities. I tried both reduce-algebra and Maxima and they can't even do the simple operations in a combinatorial sense. For instance $\frac{\partial^{n}\left(\left(1+x\right)^{m}\right)}{\partial x^{n}}$ is (more or less obviously) $\frac{m!}{\left(m-n\right)!}\left(1+x\right)^{m-n}$ or ${\color{black}{\frac{\Gamma(m+1)}{\Gamma(-(n-1-m))}}}$ ; yet both packages fail this simple test.
I need this to evaluate/generate things like
$\left[t^{n}\right]\left(1+x\right)^{m}=\frac{m!}{\left(m-n\right)!n!}\left[t^{0}\right]\left(1+x\right)^{\left(m-n\right)}$
I don't mind a little programing but I want to handle automatically generated expressions like:
$\left[t^{k}\right]\frac{\left(t^{m}\right)}{\left(1-t\right)n}\left(1+z\cdot t\right)^{p}$
with some confidence; and without delving deep into the pattern recognition algorithms.
I did find that powerseries(exp,x,o) in maxima provides a combinatorial answer for the test case I mentioned; after some extraction and interpretation. It doesn't handle more than one (1-x)^m term; but that's start. For those who try it: the answer is a series in the Beta function; but that is just a different representation of the binomial function analyticaly extended.

• Not sure how close this is to what you're asking for, but there is this: riordancalculator.com Nov 29, 2021 at 6:43
• @AlexanderBurstein Thanks for the link, :) It's not what I wanted, but I have bookmarked it. It's not irrelevant but not symbolic. I did find a breakdown; I might write/program it up. It's interesting; in the end it's amenable to an extensive rewrite process. My phrase above "pattern recognition algorithms" was a mistake/miss-guess, but forgivable. If somebody gripes, I will remove it. If I do program it, I wonder if anybody here would be interested. Nov 29, 2021 at 17:16