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A note on Leibniz rule for difference quotient

This paper presents a derivation of the Leibniz rule specifically applied to difference quotients.

Original authors: Taekyun Kim, Dae san Kim

Published 2026-02-03

📖 4 min read🧠 Deep dive

Original authors: Taekyun Kim, Dae san Kim

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to understand how a smooth, flowing river changes as it moves. In the real world, water flows continuously. But in the world of computers and numerical math, we can't always handle "smooth" flow. Instead, we chop the river into a series of stepping stones (a grid) and look at the water level at each stone.

This paper is about a specific mathematical tool called the Difference Quotient. Think of this as a way to measure the "slope" or "change" between two stepping stones instead of a smooth curve. If you stand on stone $x$ and look at stone $x+\lambda$ (where $\lambda$ is the distance between them), the difference quotient tells you how much the value changed divided by the distance.

The authors, Taekyun Kim and Dae San Kim, are tackling a classic problem: What happens when you multiply two things together and then measure their change?

The "Product Rule" for Stepping Stones

In standard calculus (smooth math), there is a famous rule called the Leibniz Rule (or Product Rule). It tells you how to find the slope of a product, like $f(x) \times g(x)$ . It says the slope is roughly:
(Slope of f) $\times$ (Value of g) + (Value of f) $\times$ (Slope of g).

However, when you are working with "stepping stones" (discrete math) instead of a smooth river, the math gets messier. The authors discovered that the simple rule doesn't quite work because the stones are separated by a gap.

The Discovery:
They found a new, more complex formula (Theorem 1.1) that acts like a "super-product rule" for these stepping stones.

The Analogy: Imagine you are trying to calculate the change in the total weight of two people walking across a bridge. In smooth math, you just add their individual changes. But on a bumpy bridge (the difference quotient), the way they step affects each other. The authors' formula adds a "correction term" (involving the distance $\lambda$ ) to account for the gap between the stones.
The Result: They proved that if you take the change of a product ( $f \times g$ $f \times g$ ), it equals the sum of three parts:
1. The change in $f$ times the change in $g$ (scaled by the gap size).
2. The change in $f$ times the current value of $g$ .
3. The current value of $f$ times the change in $g$ .

Scaling Up: The "Multi-Tool"

The paper doesn't stop at two people walking across the bridge. The authors asked: What if we have a whole group of people ( $n$ functions) walking together?

They introduced a clever "magic box" (an operator they call $L_\lambda$ ).

The Metaphor: Think of $L_\lambda$ as a machine that takes a function and adds its "step" to it. The authors proved that this machine is multiplicative. If you put a group of people into the machine, the machine's output for the whole group is exactly the same as the product of the machine's outputs for each individual person.
The Payoff: Because this machine works so neatly, they could use it to reverse-engineer the answer for the "difference quotient" of a huge group of functions. They derived a formula (Theorem 1.3) that tells you exactly how to calculate the change for a product of any number of functions, not just two.

The "Smooth" Connection

The paper ends with a nice "sanity check." The authors show that if you make the distance between the stepping stones ( $\lambda$ ) smaller and smaller until it vanishes (approaching zero), their complicated new formulas magically shrink down to become the classical Leibniz rule you learn in high school calculus.

Summary

In simple terms, this paper is a recipe book for calculating how things change when they are multiplied together, specifically when you are forced to look at them in "chunks" rather than as a smooth flow.

They fixed the "Product Rule" for chunky math.
They generalized it to handle any number of items multiplied together.
They showed that their "chunky" math perfectly connects back to the "smooth" math we already know when the chunks get tiny.

The authors mention this is useful for working with "degenerate versions of special polynomials and numbers," which is a fancy way of saying these formulas help solve specific, tricky problems in advanced number theory and algebra, but the core contribution is the new rule for handling multiplication in discrete steps.

Technical Summary: A Note on Leibniz Rule for Difference Quotient

Problem Statement
The Finite Difference Method (FDM) is a standard numerical technique for solving differential equations by approximating continuous derivatives with discrete difference quotients. While the classical Leibniz rule for derivatives is well-established, this note addresses the derivation of a corresponding Leibniz rule specifically for the difference quotient operator. The authors aim to establish a rigorous product rule for the $r$ -th order difference quotient of a product of functions, extending the known first-order product rule to arbitrary orders and to products of $n$ functions. The work is motivated by potential applications to degenerate versions of special polynomials and numbers, referencing recent literature in this domain.

Methodology
The authors define the difference quotient operator $\delta_\lambda$ for a complex-valued function $f$ and a nonzero real number $\lambda$ as:
$(\delta_\lambda f)(x) = \frac{f(x+\lambda) - f(x)}{\lambda}$
The methodology proceeds in two primary stages:

Inductive Derivation for Two Functions:
Starting with the first-order product rule derived from the definition of $\delta_\lambda$ :
$\delta_\lambda(fg) = \lambda(\delta_\lambda f)(\delta_\lambda g) + (\delta_\lambda f)g + f(\delta_\lambda g)$
The authors employ mathematical induction to generalize this to the $r$ -th order difference quotient, $\delta_\lambda^r(fg)$ . The proof involves applying the first-order rule to the terms generated in the inductive step and utilizing combinatorial identities involving binomial coefficients to rearrange the resulting sums.
Extension to $n$ Functions via Operator Algebra:
To extend the rule to a product of $n$ functions ( $f_1 f_2 \cdots f_n$ ), the authors introduce a multiplicative operator $L_\lambda = I + \lambda\delta_\lambda$ , where $I$ is the identity operator. They demonstrate that $L_\lambda$ is multiplicative on the algebra of complex-valued functions, satisfying $L_\lambda(fg) = L_\lambda(f)L_\lambda(g)$ . By expressing the difference quotient in terms of this operator ( $\delta_\lambda = \frac{L_\lambda - I}{\lambda}$ ) and applying the binomial theorem, they derive a general formula for the $r$ -th order difference quotient of a product of $n$ functions.

Key Contributions and Results

Theorem 1.1 (Leibniz Rule for Two Functions):
The paper establishes a closed-form expression for the $r$ -th order difference quotient of a product of two functions $f$ and $g$ :
$\delta_\lambda^r(fg) = \sum_{l=0}^r \lambda^{r-l} \binom{r}{l} \sum_{k=0}^l \binom{l}{k} (\delta_\lambda^{r-k} f)(\delta_\lambda^{k+r-l} g)$
The authors verify that taking the limit $\lambda \to 0$ recovers the classical Leibniz rule for derivatives.
Theorem 1.3 (Leibniz Rule for $n$ Functions):
The authors generalize the result to a product of $n$ functions. The formula is expressed using the operator $L_\lambda$ :
$\delta_\lambda^r(f_1 f_2 \cdots f_n) = \frac{1}{\lambda^r} \sum_{k=0}^r \binom{r}{k} (-1)^{r-k} L_\lambda^k(f_1) L_\lambda^k(f_2) \cdots L_\lambda^k(f_n)$
where $L_\lambda^k(f_i) = (I + \lambda\delta_\lambda)^k(f_i) = \sum_{j_i=0}^k \binom{k}{j_i} \lambda^{j_i} \delta_\lambda^{j_i} f_i$ .
Operator Relations and Binomial Inversion:
The paper highlights the relationship between the difference quotient operator and the $L_\lambda$ operator. By defining generating functions $A_n(t)$ and $\bar{A}_n(t)$ , the authors show that the binomial relations between the two operators can be concisely expressed as $\bar{A}_n(t) = e^t A_n(t)$ , linking the discrete difference calculus to Euler-Seidel matrix properties.

Significance and Claims
The paper claims to provide a complete derivation of the Leibniz rule for difference quotients, filling a gap in the explicit formulation for higher-order differences and multiple functions. The significance of the work is framed within the context of its utility for "degenerate versions of some special polynomials and numbers." The authors do not present new numerical experiments or specific applications to differential equations in this note; rather, they provide the theoretical algebraic framework intended to support such applications in future research. The results are presented as a direct extension of classical calculus concepts to the discrete domain, with explicit formulas ready for application in the cited areas of special polynomials.

The "Product Rule" for Stepping Stones

Scaling Up: The "Multi-Tool"

The "Smooth" Connection

Summary

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