finite difference

Calculus of Finite Differences Part-2

Achyut Bharadwaj, Saee Patil, Valentio Iverson, Danny Li
August 2021
calculus, finite difference, sequences, patterns, polynomials, chain rule

Acknowledgements # This work was carried out as part of a research project at PROMYS-2021. I am a grateful recipient of the Mehta Fellowship to the PROMYS programs in 2021 and 2022. Definition and Preliminaries Notation # Let us define a polynomial $f(x)$ to be $$f(x) = \sum_{i = 0}^n a_i x^i$$ for some $a_i \in \mathbb{C}$, for all $0 \le i \le n$ and $n \in \mathbb{N}_0$. We define the degree of a polynomial to be the largest non-negative integer $n$ such that $a_n \not= 0$ and let degree of the polynomial $f(x) = 0$ be $0$. ...

Calculus of Finite Differences Part-1

Achyut Bharadwaj, Saee Patil, Valentio Iverson, Danny Li
August 2021
calculus, finite difference, sequences, patterns, polynomials, chain rule

Acknowledgements # This work was carried out as part of a research project at PROMYS-2021. I am a grateful recipient of the Mehta Fellowship to the PROMYS programs in 2021 and 2022. Preliminary Definitions # Finding Patterns # (1) $$1,4,7,10,\cdots\;\;?$$ Here’s a harder one: (2) $$3,4,6,9,13,\cdots\;\;?$$ Notations # (3) $$\Delta\left(f\right)(x)=f(x+1)-f(x)$$ (4) $$\Delta^{n+1}(f)=\Delta\left(\Delta^n(f)\right)$$ What is $\Delta\left(f\right)$ when $f$ is $1,4,7,10,\cdots$? What is $\Delta\left(f\right)$ when $f$ is $3,4,6,9,13\cdots$? What is $\Delta^2(f)$ when $f$ is $3,4,6,9,13\cdots$? ...