Summation of Products of n Numbers taken m at a time with Repetitions/Lemma 1

Theorem
Let $a, b \in \Z$ be integers such that $b \ge a$.

Let $U$ be a set of $n = b - a + 1$ numbers $\left\{ {x_a, x_{a + 1}, \ldots, x_b}\right\}$.

Let $m \in \Z_{>0}$ be a (strictly) positive integer.

Let:

That is, $h_m$ is the product of all $m$-tuples of elements of $U$ taken $m$ at a time.

Let $G \left({z}\right)$ be the generating function for the sequence $\left\langle{h_m}\right\rangle$.

Then:

Proof
Let us set up the generating function for the sequence $\left\langle{h_m}\right\rangle$:

For each $r \in \left\{ {a, a + 1, \ldots, b}\right\}$, the product of $x_r$ taken $m$ at a time is simply ${x_r}^m$.

Thus for $n = 1$ we have:
 * $h_m = {x_r}^m$

Let the generating function for such a $\left\langle{h_m}\right\rangle$ be $G_r \left({z}\right)$.

From Generating Function for Sequence of Powers of Constant:
 * $G_r \left({z}\right) = \dfrac 1 {1 - x_r z}$

By Product of Summations, we have:


 * $\displaystyle \sum_{a \mathop \le j_1 \mathop \le \cdots \mathop \le j_m \mathop \le b} x_{j_1} \cdots x_{j_m} = \prod_{k \mathop = a}^b \sum_{r \mathop = 1}^m x_{k_r}$

Hence: