Converting a Product to a Sum

Question

How can I convert $$\prod_{i=1}^k\left[\sum_{j=0}^{i-1}x^j\right]$$ to a sum? I have been trying to solve this product by inductive reasoning but I figured it was too complex... Is there an agebraic/deductive way of converting this expression to a sum?

Answer 1

The sums are just geometric series which you can sum, getting $$\prod_{i=1}^k\left[\sum_{j=0}^{i-1}x^j\right]=\prod_{i=1}^k\frac{1-x^i}{1-x}$$ The denominator is just $(1-x)^k$. You can expand the numerator as a sum getting $$1-\sum_{i=1}^kx^i+\sum_{i=1}^{k-1}\sum_{j=i+1}^kx^{i+j}-\sum_{i=1}^{k-2}\sum_{j=i+1}^{k-1}\sum_{m=j+1}^kx^{i+j+m}+\ldots$$ but I don't think that is helping.

Answer 2

$$\prod_{i=1}^k\left[\sum_{j=0}^{i-1}x^j\right]=\prod_{i=1}^k\frac{x^i-1}{x-1}=\frac{\displaystyle\prod_{i=1}^k(x^i-1)}{(x-1)^k}.$$

Now the expression

$$\prod_{i=1}^k(x^i-1)$$

yields irregular polynomials.

$$x-1$$

$$x^3-x^2-x+1$$

$$x^6-x^5-x^4+x^2+x-1$$

$$x^{10}-x^9-x^8+2x^5-x^2-x+1$$

$$x^{15}-x^{14}-x^{13}+x^{10}+x^9+x^8-x^7-x^6-x^5+x^2+x-1$$

$$\cdots$$