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Is there a generalized formula for the product of falling factorial polynomials $(x)_n$? From Wikipedia the product of two falling factorials can be expressed as:

\begin{equation} (x)_m (x)_n = \sum\limits_{k=0}^{m}\binom{m}{k}\binom{n}{k}k! \cdot (x)_{m+n-k} \end{equation}

I am wondering if there is a more general formula for $T$ products. I.e. for some integers $\alpha_1, \alpha_2, ... \alpha_T$:

\begin{equation} \prod\limits_{t=1}^{T}(\alpha_t + x - 1)_{\alpha_t} \end{equation}

Is there a concise expression for the coefficients $c_k$ of the polynomial that results:

\begin{equation} \prod\limits_{t=1}^{T}(\alpha_t + x - 1)_{\alpha_t} = \sum\limits_{k=T}^{\sum \alpha_i}c_k x^k \end{equation}

The Wikipedia page referenced "Specializations of MacMahon symmetric functions and the polynomial algebra" but aside from this I haven't found many other references for these types of formulas. Is there a recommended reference book for these types of combinatorial expressions? Thank you!

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I'm not sure if you can get a simple expression, but the product in question can be expressed as $\alpha_1!\cdots\alpha_T!$ times the coefficient of $z_1^{\alpha_1}\cdots z_T^{\alpha_T}$ in $$\left( (1-z_1)\cdots (1-z_T)\right)^{-x}.$$ In terms of elementary symmetric polynomials of $z_i$, $$c_k = \frac{(-1)^k}{k!} \alpha_1!\cdots\alpha_T!\ [z_1^{\alpha_1}\cdots z_T^{\alpha_T}]\ \log(1-e_1+e_2-\dots)^k.$$

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