A note on Leibniz rule for difference quotient

TL;DR

The paper addresses the problem of establishing a Leibniz rule for difference quotients. It defines the forward difference operator $δ_{λ}$, proves the two-term product rule, and then derives a general Leibniz rule for $δ_{λ}^{r}$ via induction; it extends the result to $n$ functions using the multiplicative operator $L_{λ}=I+λ δ_{λ}$. The key result is the explicit formula $δ_{λ}^{r}(f_{1} \cdots f_{n}) = (1/λ^{r}) \sum_{k=0}^{r} (-1)^{r-k} C(r,k) L_{λ}^{k}(f_{1}) \cdots L_{λ}^{k}(f_{n})$, with the limit $λ \to 0$ recovering the classical Leibniz rule and connections to binomial inversion and Euler-Seidel relations via generating functions. The work provides an operator-based, combinatorial framework for higher-order difference quotients relevant to discrete calculus and related polynomial theories.

Abstract

In this note, we derive a Leibniz rule for difference quotient.

Theorems & Definitions (5)

• Theorem 1.1
• proof
• Remark 1.2
• Theorem 1.3
• Remark 1.4