Alternating Ordinary Generating Functions

HYPERSPECIES

Alternate Geometric Series

$\displaystyle \dfrac{x^k}{1+x^{j+1}} = \sum_{n \geq k+1} (-1)^{n-1-k} \, x^{(n-1)(j+1)} = \sum_{n \geq 1} (-1)^{n-1} \, x^{(n-1)(j+1)+k} \\ \\ = x^k - x^{j+1+k} + x^{2(j+1)+k} - x^{3(j+1)+k} \pm \cdots \$

$\displaystyle \dfrac{x^k}{1+x} = \sum_{n \geq k+1} (-1)^{n-1-k} \, x^{n-1} = \sum_{n \geq 1} (-1)^{n-1} \, x^{n-1+k} \\ \\ = x^k - x^{k+1} + x^{k+2} - x^{k+3} \pm \cdots \$

$\displaystyle \dfrac{1}{1+x^{j+1}} = \sum_{n \geq 1} (-1)^{n-1} \, x^{(n-1)(j+1)} \\ \\ = 1 - x^{j+1} + x^{2(j+1)} - x^{3(j+1)} \pm \cdots \$

$\displaystyle \dfrac{x^k}{(1+x^{j+1})^2} = \sum_{n \geq k+1} (-1)^{n-1-k} \, (n-k) \, x^{(n-1)(j+1)} = \sum_{n \geq 1} (-1)^{n-1-k} \, n \, x^{(n-1)(j+1)+k} \\ \\ = x^k - 2x^{j+1+k} + 3x^{2(j+1)+k} - 4x^{3(j+1)+k} \pm \cdots \$

$\displaystyle \dfrac{x^k}{(1+x)^2} = \sum_{n \geq k+1} (-1)^{n-1-k} \, (n-k) \, x^{n-1} = \sum_{n \geq 1} (-1)^{n-1} \, n \, x^{n-1+k} \\ \\ = x^k - 2x^{k+1} + 3x^{k+2} - 4x^{k+3} \pm \cdots \$

$\displaystyle \dfrac{1}{(1+x^{j+1})^2} = \sum_{n \geq 1} (-1)^{n-1} \, n \, x^{(n-1)(j+1)} \\ \\ = 1 - 2x^{j+1} + 3x^{2(j+1)} - 4x^{3(j+1)} \pm \cdots \$

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