Monotonic Ordinary Generating Functions

HYPERSPECIES

Geometric Series

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

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

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

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

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

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

Pages: 1 2 3 4 5 6 7 8