Explicit Expansion for Certain Product of homogeneous Polynomials

Question

Let $\mu \in \mathbb{N}_0^n$ be a multi index and set $$P(X_1, \dots, X_n) = X_1^{\mu_1}(X_1 + X_2)^{\mu_2} \cdots (X_1 + \cdots +X_n)^{\mu_n} = \prod_{j=1}^n (X_1 + \cdots + X_j)^{\mu_j}.$$ Since $P$ is a homogeneous polynomial of degree $|\mu| = \mu_1 + \dots + \mu_n$, it is clear that $$ P(X_1, \dots, X_n) = \sum_{|\alpha| = |\mu|} C(\alpha, \mu) \,X_1^{\alpha_1} \cdots X_n^{\alpha_n}$$ for certain coefficients $C(\alpha, \mu)$.

While we expect that there should be a closed expression for $C(\alpha, \mu)$ in terms of binomial coefficients, we were unable to obtain such a formula. Does someone know the answer or a reference to the literature?

Answer 1

We can express the coefficient as a product of binomial coefficients by expanding by the binomial theorem, starting with the rightmost factor.

For example, with $n=3$ we have $$ \def\m#1{{\mu_{#1}}} \def\i#1{{i_{#1}}} \begin{aligned} X_1^\m1(X_1+X_2)^\m2(X_1+&X_2+X_3)^\m3 = X_1^\m1(X_1+X_2)^\m2\sum_{\i3}\binom{\m3}{\i3} (X_1+X_2)^{\m3-\i3}X_3^\i3\\ &= X_1^\m1\sum_{\i3}\binom{\m3}{\i3}(X_1+X_2)^{\m2+\m3-\i3}X_3^\i3\\ &=\sum_{\i2,\i3}X_1^{\m1+\m2+\m3-\i2-\i3}X_2^{\i2}X_3^{\i3} \binom{\m2+\m3-\i3}{\i2}\binom{\m3}{\i3}. \end{aligned} $$ The formula for general $n$ is similar.

Answer 2

Just for completion, we expand Gessel's suggestion to get: $$\prod_{j=1}^n\left(\sum_{i=1}^jX_i\right)^{\mu_j}= \sum_{i_2,\dots,i_n\geq0}X_1^{\vert\mu\vert-\vert I\vert}\prod_{k=2}^nX_k^{i_k}\prod_{j=2}^n\binom{\mu_n+\cdots+\mu_j-(i_n+\cdots+i_{j+1})}{i_j}$$ where $\vert\mu\vert=\mu_1+\cdots+\mu_n$ and $\vert I\vert=i_2+\dots+i_n$.

Convention: $i_n+\cdots+i_{n+1}=0$.