Spread Polynomials in Rational Trigonometry

• Spread polynomials are defined as polynomial sequences in rational trigonometry that generalize angle multiplication via sine square identities.
• They combine Chebyshev, Fibonacci, and Lucas formulations to exhibit cyclotomic-style factorizations and notable divisibility properties.
• A bivariate extension reveals deep combinatorial patterns and matrix representations connecting to classical number sequences like the Catalan numbers.

Spread polynomials are polynomial sequences arising in Wildberger’s rational trigonometry, where they encode the multiplication law for spreads in the same way that Chebyshev polynomials encode angle multiplication. The classical family $S_n(x)$ is characterized by

$S_n(\sin^2\theta)=\sin^2(n\theta),$

and a normalized variant

$Z_n(x)=4S_n\!\left(\frac{x}{4}\right)$

satisfies

$Z_n(4\sin^2\theta)=4\sin^2(n\theta).$

Recent work places these polynomials in a unified algebraic framework involving Chebyshev, Fibonacci, and Lucas polynomials, proves a cyclotomic-style factorization theory, and introduces a bivariate deformation $Z_n(x,s)$ that recovers the classical normalized family at $s=-1$ (Herbig et al., 2023, Cigler, 13 Jul 2025, Cigler et al., 2024, Cigler, 6 Aug 2025).

1. Classical definition and rational-trigonometric meaning

The classical spread polynomials $S_n(x)$ are defined recursively by

$$S_{0}(x)=0,\qquad S_{1}(x)=x,\qquad S_{n}(x)=2(1-2x)S_{n-1}(x)-S_{n-2}(x)+2x,$$

and satisfy the identity

$S_n(x)=\frac{1-T_n(1-2x)}{2},$

where $T_n$ is the Chebyshev polynomial of the first kind (Herbig et al., 2023). This Chebyshev representation makes the trigonometric evaluation immediate: $S_n(\sin^2\theta)=\sin^2(n\theta),$0 Accordingly, spread polynomials are the rational-trigonometric analogue of the Chebyshev polynomials (Herbig et al., 2023).

In Wildberger’s rational trigonometry, lengths are replaced by quadrances and angles by spreads. Within that framework, $S_n(\sin^2\theta)=\sin^2(n\theta),$1 encodes repeated equal spreads in a configuration of $S_n(\sin^2\theta)=\sin^2(n\theta),$2 concurrent lines: if neighboring spreads are equal to $S_n(\sin^2\theta)=\sin^2(n\theta),$3, then the spread between the extreme lines is

$S_n(\sin^2\theta)=\sin^2(n\theta),$4

(Cigler et al., 2024). The defining identity therefore has both a trigonometric and a geometric interpretation.

The Chebyshev relation also clarifies why spread polynomials exhibit composition behavior and why their roots and factor structure are closely tied to trigonometric algebraic numbers. This perspective is central in later treatments that convert spread-polynomial questions into Lucas- and cyclotomic-polynomial questions (Cigler et al., 2024).

2. Normalization, zpread polynomials, and base-change structure

A central normalization is

$S_n(\sin^2\theta)=\sin^2(n\theta),$5

called the “zpread polynomials” in one exposition (Herbig et al., 2023). With

$S_n(\sin^2\theta)=\sin^2(n\theta),$6

the normalization becomes

$S_n(\sin^2\theta)=\sin^2(n\theta),$7

so $S_n(\sin^2\theta)=\sin^2(n\theta),$8 acts as the base-change operator between the basis

$S_n(\sin^2\theta)=\sin^2(n\theta),$9

and the power basis

$Z_n(x)=4S_n\!\left(\frac{x}{4}\right)$0

(Herbig et al., 2023).

The normalization is not merely cosmetic. It is chosen so that the transition matrix has integer entries and smaller coefficients than the unscaled spread matrix (Herbig et al., 2023). The corresponding generating function follows from the Chebyshev generating function and is

$Z_n(x)=4S_n\!\left(\frac{x}{4}\right)$1

This generating series is one route to the explicit matrix description of the normalized family (Herbig et al., 2023).

The matrix $Z_n(x)=4S_n\!\left(\frac{x}{4}\right)$2, defined by

$Z_n(x)=4S_n\!\left(\frac{x}{4}\right)$3

has entries

$Z_n(x)=4S_n\!\left(\frac{x}{4}\right)$4

where $Z_n(x)=4S_n\!\left(\frac{x}{4}\right)$5 are even-dimensional pyramidal numbers (Herbig et al., 2023). The same paper states that $Z_n(x)=4S_n\!\left(\frac{x}{4}\right)$6 is a Riordan array, with inverse expressed in terms of the Catalan generating function $Z_n(x)=4S_n\!\left(\frac{x}{4}\right)$7 and the central binomial generating function $Z_n(x)=4S_n\!\left(\frac{x}{4}\right)$8. This embeds spread polynomials into the same matrix-and-base-change framework used there for Chebyshev polynomials and Catalan triangles (Herbig et al., 2023).

3. Fibonacci and Lucas polynomial formulations

A decisive simplification is the identification of the normalized spread polynomials with Lucas-type polynomials. In one notation,

$Z_n(x)=4S_n\!\left(\frac{x}{4}\right)$9

where $Z_n(4\sin^2\theta)=4\sin^2(n\theta).$0 is the “minus” Lucas polynomial satisfying

$Z_n(4\sin^2\theta)=4\sin^2(n\theta).$1

and related to Chebyshev by

$Z_n(4\sin^2\theta)=4\sin^2(n\theta).$2

(Cigler, 13 Jul 2025). In a parallel notation used in the factorization paper,

$Z_n(4\sin^2\theta)=4\sin^2(n\theta).$3

with $Z_n(4\sin^2\theta)=4\sin^2(n\theta).$4 the Lucas polynomial for the recurrence $Z_n(4\sin^2\theta)=4\sin^2(n\theta).$5 (Cigler et al., 2024).

The first values are

$Z_n(4\sin^2\theta)=4\sin^2(n\theta).$6

so $Z_n(4\sin^2\theta)=4\sin^2(n\theta).$7 is monic of degree $Z_n(4\sin^2\theta)=4\sin^2(n\theta).$8 (Cigler, 13 Jul 2025). The same framework yields the composition law

$Z_n(4\sin^2\theta)=4\sin^2(n\theta).$9

which is the polynomial reformulation of repeated angle multiplication under the substitution $Z_n(x,s)$0 (Cigler et al., 2024).

The Lucas viewpoint also gives a linear recurrence of order three: $Z_n(x,s)$1 and a Cassini-type identity

$Z_n(x,s)$2

(Cigler, 13 Jul 2025). These relations are not ad hoc: they arise because $Z_n(x,s)$3 is expressed through a Lucas sequence with a characteristic structure that produces a cubic recurrence rather than the second-order recurrence visible at the Chebyshev level (Cigler, 13 Jul 2025).

A further identity with arithmetic consequences is

$Z_n(x,s)$4

and, in particular,

$Z_n(x,s)$5

where $Z_n(x,s)$6 is the Fibonacci number (Cigler et al., 2024). This links spread-polynomial values at specific arguments to classical divisibility sequences.

4. Cyclotomic factorization and arithmetic structure

A major development is the proof of a conjecture of Goh and Wildberger on the factorization of spread polynomials. Defining the monic version

$Z_n(x,s)$7

the factorization paper introduces polynomials $Z_n(x,s)$8 by

$Z_n(x,s)$9

where $s=-1$0 is the minimal polynomial of $s=-1$1. Then

$s=-1$2

so $s=-1$3 is the minimal polynomial of $s=-1$4 (Cigler et al., 2024).

The conjectured divisor-product factorization becomes

$s=-1$5

with

$s=-1$6

This is the theorem proved in the paper (Cigler et al., 2024). It shows that spread polynomials admit a genuine analogue of cyclotomic factorization, indexed by divisors and governed by minimal polynomials of trigonometric algebraic numbers.

The proof proceeds by transferring the factorization of $s=-1$7 to $s=-1$8. A key ingredient is Grubb’s factorization

$s=-1$9

which reflects the multiplicity pattern of the roots of $S_n(x)$0 (Cigler et al., 2024).

The factors can also be computed effectively. For odd $S_n(x)$1,

$S_n(x)$2

so odd-index factors are read off directly from Lucas polynomials. For powers of $S_n(x)$3,

$S_n(x)$4

and for odd $S_n(x)$5 and $S_n(x)$6,

$S_n(x)$7

(Cigler et al., 2024). The factorization theory thus has both structural and computational content.

This arithmetic structure propagates to Fibonacci numbers. Since

$S_n(x)$8

the paper writes

$S_n(x)$9

so the primitive-part factorization of Fibonacci numbers is controlled by the same factors $$S_{0}(x)=0,\qquad S_{1}(x)=x,\qquad S_{n}(x)=2(1-2x)S_{n-1}(x)-S_{n-2}(x)+2x,$$0 that govern spread-polynomial factorization (Cigler et al., 2024).

5. The bivariate extension $$S_{0}(x)=0,\qquad S_{1}(x)=x,\qquad S_{n}(x)=2(1-2x)S_{n-1}(x)-S_{n-2}(x)+2x,$$1

The bivariate generalization introduced in 2025 defines

$$S_{0}(x)=0,\qquad S_{1}(x)=x,\qquad S_{n}(x)=2(1-2x)S_{n-1}(x)-S_{n-2}(x)+2x,$$2

where $$S_{0}(x)=0,\qquad S_{1}(x)=x,\qquad S_{n}(x)=2(1-2x)S_{n-1}(x)-S_{n-2}(x)+2x,$$3 is the bivariate Lucas polynomial associated with the recurrence

$$S_{0}(x)=0,\qquad S_{1}(x)=x,\qquad S_{n}(x)=2(1-2x)S_{n-1}(x)-S_{n-2}(x)+2x,$$4

and initial conditions $$S_{0}(x)=0,\qquad S_{1}(x)=x,\qquad S_{n}(x)=2(1-2x)S_{n-1}(x)-S_{n-2}(x)+2x,$$5, $$S_{0}(x)=0,\qquad S_{1}(x)=x,\qquad S_{n}(x)=2(1-2x)S_{n-1}(x)-S_{n-2}(x)+2x,$$6 (Cigler, 6 Aug 2025). The same paper gives the equivalent and especially useful representation

$$S_{0}(x)=0,\qquad S_{1}(x)=x,\qquad S_{n}(x)=2(1-2x)S_{n-1}(x)-S_{n-2}(x)+2x,$$7

with $$S_{0}(x)=0,\qquad S_{1}(x)=x,\qquad S_{n}(x)=2(1-2x)S_{n-1}(x)-S_{n-2}(x)+2x,$$8 the corresponding bivariate Fibonacci polynomial (Cigler, 6 Aug 2025).

The first values are

$$S_{0}(x)=0,\qquad S_{1}(x)=x,\qquad S_{n}(x)=2(1-2x)S_{n-1}(x)-S_{n-2}(x)+2x,$$9

This exhibits the coefficient pattern that later reappears in the closed expansion

$S_n(x)=\frac{1-T_n(1-2x)}{2},$0

(Cigler, 6 Aug 2025).

The classical normalized family is recovered by specialization: $S_n(x)=\frac{1-T_n(1-2x)}{2},$1 and hence

$S_n(x)=\frac{1-T_n(1-2x)}{2},$2

The bivariate family is therefore a strict extension of the classical spread polynomials, with the second parameter $S_n(x)=\frac{1-T_n(1-2x)}{2},$3 deforming the characteristic data of the underlying Fibonacci/Lucas system (Cigler, 6 Aug 2025).

A striking feature is that $S_n(x)=\frac{1-T_n(1-2x)}{2},$4 satisfies an order-three recurrence,

$S_n(x)=\frac{1-T_n(1-2x)}{2},$5

and has ordinary generating function

$S_n(x)=\frac{1-T_n(1-2x)}{2},$6

(Cigler, 6 Aug 2025). The paper also derives parity-type identities

$S_n(x)=\frac{1-T_n(1-2x)}{2},$7

and a Cassini-like relation

$S_n(x)=\frac{1-T_n(1-2x)}{2},$8

(Cigler, 6 Aug 2025).

The bivariate construction preserves the trigonometric meaning only through the classical specialization $S_n(x)=\frac{1-T_n(1-2x)}{2},$9. The paper explicitly states that it does not introduce a new geometric quantity beyond spread; rather, it packages the classical trigonometric behavior into a more flexible algebraic family controlled by Fibonacci and Lucas polynomials (Cigler, 6 Aug 2025).

6. Combinatorial context, related structures, and terminological boundaries

Spread polynomials sit at the intersection of several classical polynomial technologies. Their Chebyshev realization explains the trigonometric identities; their Lucas realization explains composition, cubic recurrences, and factorization; and their matrix formulation organizes coefficient arrays in terms of pyramidal numbers and Riordan arrays (Herbig et al., 2023, Cigler, 13 Jul 2025, Cigler et al., 2024). In the bivariate case, the coefficient array

$T_n$0

is noted to be present in the OEIS and in work of Burstein–Shapiro, indicating that the sequence belongs to a broader ecosystem of combinatorial polynomial arrays (Cigler, 6 Aug 2025).

One recurrent theme is divisibility. The normalized polynomials satisfy a composition law, admit divisor-indexed factorization, and control primitive factors of Fibonacci numbers at the specialization $T_n$1 (Cigler et al., 2024). This supports the description of spread polynomials as a divisibility-type polynomial sequence with explicit primitive factors and recursive computability (Cigler et al., 2024).

A common terminological confusion concerns the word “spread.” In matrix analysis, the spread of a matrix means

$T_n$2

and for real-rooted polynomials the corresponding notion is the span between largest and smallest roots. A 2019 paper on these inequalities explicitly states that it does not define a special new class of polynomials by the name “spread polynomials” (Sharma et al., 2019). Likewise, “spread complexity” in recent quantum-information work refers to average Krylov position in a Lanczos chain and is formulated through orthogonal polynomials and spectral measures; it is unrelated to Wildberger’s spread polynomials despite the shared word “spread” (Caputa et al., 16 Sep 2025).

Within its own domain, however, the subject has become increasingly coherent. The modern picture is that the classical polynomials $T_n$3, the normalized polynomials $T_n$4, and the bivariate family $T_n$5 are different layers of a single algebraic structure. The classical layer is trigonometric, the normalized layer is arithmetic and combinatorial, and the bivariate layer makes the governing Fibonacci/Lucas mechanism explicit (Cigler, 6 Aug 2025).