Consider the polynomial algebra $\mathbb C[X]$. Then the set $\{1, X, X^2,\dots,\}$ forms a vector space basis for this algebra. In general, we know that the set $\{P_n(X) \in \mathbb C[X]: n \ge 0 \text{ and degree}(P_n) = n\}$ forms a basis. We consider the binomial basis defined as follows: $\binom{X}{n} = \frac{(X-n+1)(X-n+2)\cdots (X)}{1 \cdot 2 \cdot \, \cdots \, \cdot n }$. Then $\binom{X}{n}$ is a polynomial of degree $n$. Therefore $\{\binom{X}{n}: n \ge 0\}$ is yet another basis for $\mathbb C[X]$. I have the following questions.
- Is there any instance where the binomial basis is preferred over the usual basis?
- Is there any algorithms to switch between the usual and binomial basis?
- Is there any paper where they talk about the computation complexity of converting a polynomial expanded in binomial basis to the usual basis? Kindly share some references.
I have Theorem 1 and 2. Theorem 1 gives an expression for a polynomial in the usual basis and Theorem 2 gives an expression for the same polynomial in the binomial basis. I have to justify Theorem 1 is computationally preferred to calculate the polynomial over Theorem 2. But I am not sure how to approach this.
Thank you for your valuable time.
