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An Algebraic-Operator Construction of the Half-Derivative on a Graded Monomial Space. Part I: Recurrence, Double Factorials, the Wallis Product, and Normalization

This paper constructs a half-order differentiation operator on a graded monomial space using an algebraic recurrence relation derived from the composition property $D^{1/2} \circ D^{1/2} = D$ , determining the necessary normalization constant by matching the result to the standard Riemann-Liouville formula without initially invoking integrals or the Gamma function.

Original authors: Davit Kapanadze

Published 2026-07-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Davit Kapanadze

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

The Mystery of the "Half-Step" in Math

Imagine you are walking down a path, and every time you take a step, you move forward exactly one meter. In the world of mathematics, this is like taking a "derivative," a tool that measures how fast something is changing. If you have a simple curve, taking a derivative tells you its slope. If you take the derivative again, you get the "second derivative," which tells you how that slope is changing. It's a very orderly system: you take one step, you get a new number; you take two steps, you get another.

But what if you wanted to take a "half-step"? What if you could stop exactly halfway between your starting point and your next full step? This question has puzzled mathematicians for centuries. It belongs to a field called "fractional calculus," which asks: Can we split a mathematical operation into smaller, non-whole pieces? Think of it like a music player that usually skips whole songs, but someone wants to invent a button that plays exactly half a song. The problem is tricky because, unlike a song, mathematical operations don't always split neatly. If you take a "half-step" twice, you must land exactly where you would have if you took one full step. If you don't, the math breaks. This paper dives into that exact puzzle: how to build a machine that takes a perfect "half-step" in the world of numbers.

The Algebraic Puzzle of the Half-Derivative

The paper by Davit Kapanadze tackles this puzzle by building a "half-derivative" machine from scratch, but with a twist: he refuses to use the usual heavy machinery of calculus, like complex integrals or special functions that most mathematicians rely on. Instead, he treats the problem like a giant algebraic puzzle, focusing on a specific family of numbers called "monomials." You can think of monomials as the building blocks of curves, like $x$ , $x^2$ , $x^3$ , and even weird ones like $x^{1/2}$ or $x^{3/2}$ .

The author starts with a simple rule: if you apply this mysterious "half-derivative" operator twice, it must act exactly like a normal, full derivative. If you start with $x^2$ , take a half-step, and then take another half-step, you must end up with the same result as taking one full derivative of $x^2$ . By forcing the math to obey this rule, the author discovers a hidden pattern. The numbers that multiply the terms (the coefficients) aren't random; they follow a strict rhythm. When you work out the numbers for the first step and the second step, they link together in a chain.

This chain leads to something called "double factorials." If a normal factorial is multiplying all numbers down to 1 (like $5 \times 4 \times 3 \times 2 \times 1$ ), a double factorial skips every other number (like $5 \times 3 \times 1$ ). The paper shows that the "half-step" coefficients are built entirely out of these skipping-number patterns. It's like finding that the secret code to a half-step is just a specific way of arranging odds and evens.

However, there is a catch. The math tells the author exactly how the numbers relate to each other, but it leaves one number floating in the air: a "normalization constant." Think of this like a volume knob. The author knows exactly how the left speaker and right speaker must balance to create the right sound, but the overall volume of the whole system is still up to a choice. The math says, "If you turn the volume up on the first half-step, you must turn it down on the second half-step by the exact same amount, so they cancel out." This means the final result (the full derivative) is always correct, no matter what volume you pick, but the "half-step" itself looks different depending on the setting.

To solve this, the author looks at a famous mathematical pattern called the "Wallis product," which is an infinite multiplication of fractions that slowly approaches the number $\pi$ (pi). The paper shows that the ratios of the numbers in the half-step formula naturally form this Wallis pattern. While the Wallis product doesn't tell the author exactly what the volume knob should be, it provides a clue. By comparing his result to a well-known classical formula (the Riemann–Liouville derivative), the author decides to set the volume knob to a specific value: $1/\sqrt{\pi}$ .

Once this knob is set, the paper reveals the final, clean formula. For any power of $x$ , the half-derivative is now a precise, calculable number. For example, the half-derivative of $x$ is $\frac{2}{\sqrt{\pi}}x^{1/2}$ , and the half-derivative of $x^2$ is $\frac{8}{3\sqrt{\pi}}x^{3/2}$ . If you apply this formula twice, it perfectly reproduces the standard derivative.

The paper is very careful to state what it has not done. It has not created a universal tool that works for every possible function in the universe. It has not found an "integral kernel" (a way to calculate the half-step using an area under a curve) or a way to handle complex boundary conditions. It has not proven that this operator works for real-world physics problems like how heat spreads or how a rubber band stretches over time. The author explicitly states that these are open problems. The result is strictly an algebraic construction for a specific set of mathematical building blocks.

In the end, the paper proves that you can build a "half-derivative" using only algebra and the logic of composition, without needing the usual heavy calculus tools. It shows that the numbers follow a beautiful, rhythmic pattern involving double factorials and $\pi$ . However, it also warns that while this works perfectly for these specific mathematical toys, we don't yet know if this specific "half-step" machine is the right one to use for the messy, complicated problems of the real world. The door is open for future explorers to see if this algebraic key fits the locks of physics and engineering.

Technical Summary: An Algebraic–Operator Construction of the Half-Derivative on a Graded Monomial Space

Problem Statement
The paper addresses the construction of a half-order differentiation operator ( $D^{1/2}$ ) acting on a specific graded monomial space spanned by non-negative integer and half-integer powers of $x$ ( $x > 0$ ). The central problem is to define an operator such that its composition with itself yields the ordinary first derivative ( $D^{1/2} \circ D^{1/2} = D$ ). Unlike classical approaches in fractional calculus, this construction seeks to derive the operator's action purely from algebraic and compositional requirements, avoiding initial assumptions regarding integral kernels, limiting processes (such as finite differences), or the Gamma function.

Methodology
The author employs a strictly algebraic–operator approach restricted to linear operators that lower the exponent of a monomial by exactly one-half. The methodology proceeds through the following steps:

Operator Formulation: The action of the half-derivative on a monomial $x^\beta$ is postulated as $D^{1/2} x^\beta = c(\beta)x^{\beta-1/2}$ , where $c(\beta)$ is an unknown coefficient.
Derivation of Recurrence: By applying the operator twice and equating the result to the ordinary derivative ( $D x^\beta = \beta x^{\beta-1}$ ), the fundamental recurrence relation is derived:
$c(\beta)c(\beta-1/2) = \beta$
Coefficient Families: Expanding this recurrence generates two linked families of coefficients: one for integer powers ( $n$ ) and one for half-integer powers ( $n-1/2$ ). These families are expressed in terms of double factorials and a free normalization constant $c_0 = c(0)$ .
Structural Analysis: The ratio of even to odd double factorials within the coefficients is shown to generate a partial Wallis product. The paper analyzes the limit of this product as $n \to \infty$ , noting its convergence to $\pi/2$ .
Normalization Selection: While the recurrence and Wallis product determine the relative structure of the coefficients, they do not fix the global scale ( $c_0$ ). The author selects a specific positive normalization by imposing compatibility with the standard monomial formula for the left Riemann–Liouville half-derivative with a lower limit of 0. This yields $c_0 = 1/\sqrt{\pi}$ .
Verification: The constructed operator is verified on monomials and their finite linear combinations to ensure that $(D^{1/2})^2$ reproduces the ordinary derivative exactly.

Key Contributions and Results

Algebraic Derivation: The paper successfully derives the coefficient structure of the half-derivative without invoking integral definitions or the Gamma function during the derivation phase. The Gamma function appears only in the final section as a notational comparison to classical results.
Recurrence and Double Factorials: The work establishes that the coefficients for integer powers $n$ are given by:
$c(n) = c_0 \frac{(2n)!!}{(2n-1)!!}$
and for half-integer powers by:
$c(n-1/2) = \frac{(2n-1)!!}{2c_0(2n-2)!!}$
Role of the Wallis Product: The analysis demonstrates that the structural connection between the coefficients and $\pi$ arises naturally through the Wallis product limit. However, the paper clarifies that the Wallis product alone is insufficient to determine the normalization constant $c_0$ ; an external compatibility condition is required.
Normalization and Equivalence: With the choice $c_0 = 1/\sqrt{\pi}$ , the resulting monomial formula coincides with the Riemann–Liouville definition for all considered powers. For positive integer powers ( $n \ge 1$ ), it also coincides with the Caputo definition. However, for the constant function ( $n=0$ ), the construction agrees with the Riemann–Liouville rule (yielding a non-zero result proportional to $x^{-1/2}$ ) and differs from the Caputo rule (which yields zero).
Compositional Consistency: The paper proves that for any polynomial $P(x)$ in the graded space, the composition of two half-derivative steps exactly recovers the ordinary derivative: $(D^{1/2})^2 P(x) = P'(x)$ .

Significance and Scope
The paper claims significance primarily as a constructive proof that a half-order operator satisfying the composition law can be derived purely from algebraic recurrence relations on a graded monomial space. It isolates the "compositional structure" from the "normalization choice," showing that the latter is a free parameter in the algebraic framework but must be fixed by external criteria (such as compatibility with classical integral definitions) to match established theories.

The author explicitly states the boundaries of this work:

Scope: The results are valid strictly on the graded monomial space and its finite linear combinations.
Limitations: The construction does not define the operator on general function spaces, nor does it provide an integral or convolution kernel, a Laplace/Fourier transform representation, or a theory of non-locality.
Physical Interpretation: The paper does not claim physical applications. It notes that while fractional calculus is used in viscoelasticity, anomalous diffusion, and biological modeling, the present algebraic construction cannot yet be applied to these fields until an integral representation, causality properties, and initial condition structures are established.
Normalization Freedom: The normalization constant $c_0$ is shown to act as a similarity transformation parameter that rescales the integer and half-integer sectors reciprocally. Different choices of $c_0$ yield different intermediate representations but the same final first derivative upon composition.

In conclusion, the paper provides a rigorous algebraic foundation for the half-derivative on monomials, confirming that the standard Riemann–Liouville coefficients emerge naturally from the composition requirement when a specific normalization is applied, while leaving the extension to broader functional analysis and physical modeling as open problems.

The Mystery of the "Half-Step" in Math

The Algebraic Puzzle of the Half-Derivative

Technical Summary: An Algebraic–Operator Construction of the Half-Derivative on a Graded Monomial Space

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