While researching some different equations to use for smooth interpolation, I came across a generalized form of the smoothstep function that I've never seen before:
$$S_N(x) = x^{N+1}\sum_{n=0}^N \begin{pmatrix}N + n\\\ n\end{pmatrix} \begin{pmatrix}2N + 1\\N - n\end{pmatrix}(-x)^n$$
Further information on this equation is in the Wiki article here: Smoothstep - Wikipedia. I decided to work this out for N == 1 to get the cubic Hermite polynomial used frequently in smoothstep equations:
$$S_1(x) = 3x^2 - 2x^3$$
However, when I evaluate, I end up getting to this point:
$$S_1(x) = \begin{pmatrix}1\\\ 0\end{pmatrix}\begin{pmatrix}3\\\ 1\end{pmatrix}x^2 - \begin{pmatrix}2\\\ 1\end{pmatrix}\begin{pmatrix}3\\\ 0\end{pmatrix}x^3$$
I don't see how this would be equivalent. Wouldn't I need to dot product the column vectors? Doesn't this lead to the following result instead?
$$S_1(x) = 3x^2 - 6x^3$$
I just want this to make sense in my head, and right now I can't figure out how this is working. Thank you for helping with this!
I spoke with a mathematician recently and received an answer to this question. I assumed that column vectors were used in this formula, but these were actually representing a binomial coefficient Binomial Coefficient - Wiki.
A binomial coefficient is interpreted as:
$$\begin{pmatrix}n \\ k\end{pmatrix} = \frac{n!}{k!(n -k)!}$$
Given that 0! == 1, the previous evaluation I wrote above now makes sense and results in the correct cubic Hermite polynomial.