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

Examples of Summation of Products of n Numbers taken m at a time with Repetitions

Order 2

$\ds \sum_{i \mathop = a}^b \sum_{j \mathop = a}^i x_i x_j = \dfrac 1 2 \paren {\paren {\sum_{i \mathop = a}^b x_i}^2 + \paren {\sum_{i \mathop = a}^b {x_i}^2} }$

Order 3

$\ds \sum_{i \mathop = a}^b \sum_{j \mathop = a}^i \sum_{k \mathop = a}^j x_i x_j x_k = \dfrac { {S_1}^3} 6 + \dfrac {S_1 S_2} 2 + \dfrac {S_3} 3$

where:

$\ds S_r := \sum_{k \mathop = a}^b {x_k}^r$

Order 4

$\ds \sum_{a \mathop \le j_1 \mathop \le j_2 \mathop \le j_3 \mathop \le j_4 \mathop \le b} x_{j_1} x_{j_2} x_{j_3} x_{j_4} = \dfrac { {S_1}^4} {24} + \dfrac { {S_1}^2 S_2} 4 + \dfrac { {S_2}^2} 8 + \dfrac {S_1 S_3} 3 + \dfrac {S_4} 4$

where:

$\ds S_r := \sum_{k \mathop = a}^b {x_k}^r$.