Is there known generating function having this form: $\sum\frac{m-(i_1 + \ldots + i_k)}{i_1! \ldots i_k!} x_1^{i_1}\ldots x_k^{i_k}$?

Question

The sum is over n-dimensional simplex:

$$ \sum_{0\leq i_ 1+\ldots+i_n \leq m}\frac{m-(i_1 + \ldots + i_k)}{i_1! \ldots i_k!} x_1^{i_1}\ldots x_k^{i_k} $$

Does this function has a name?

Answer 1

Assuming $k=n$.

These functions are polynomials in $x_1, \dots x_n$, let's denote them $w_m(x_1,\dots x_n)$. I haven't heard about a name for them, but if we construct a generating function for them it turns out to have a pretty simple form:

\begin{align} & \sum_{m=0}^\infty w_m(x_1,\dots x_n) t^m = \\ &= \sum_{m=0}^\infty \sum_{0\le i_1+\dots i_n\le m} \frac{m-(i_1+\dots +i_n)}{i_1! \dots i_n!} x_1^{i_1}\dots x_n^{i_n} t^m = \\ &= \sum_{0\le i_1,\dots, i_n< \infty} \sum_{m=i_1+\dots i_n}^\infty \frac{m-(i_1+\dots +i_n)}{i_1! \dots i_n!} x_1^{i_1}\dots x_n^{i_n} t^m = \\ &= \sum_{0\le i_1,\dots, i_n<\infty} \sum_{m'=0}^\infty \frac{m'}{i_1! \dots i_n!} x_1^{i_1}\dots x_n^{i_n} t^{m'+(i_1+\dots+i_n)} = \\ &= \sum_{m'=0}^\infty m't^{m'} \sum_{0\le i_1,\dots, i_n<\infty} \frac{1}{i_1! \dots i_n!} (tx_1)^{i_1}\dots (tx_n)^{i_n} = \\ &= \frac{t}{(1-t)^2} e^{tx_1}\dots e^{tx_n} = \\ &= \frac{t\,e^{t(x_1+\dots+ x_n)}}{(1-t)^2}\end{align}

You have then $$ w_m(x_1,\dots, x_n) = v_m(x_1+\dots+x_n) $$ where $$v_m(z) = \frac{1}{m!} \left.\frac{d^m}{dt^m}\right|_{t=0} \frac{t e^{tz}}{(1-t)^2} $$