Spread polynomials

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In the conventional language of trigonometry, the nth-degree spread polynomial S_n, for n = 0, 1, 2, ..., may be characterized by the trigonometric identity

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

Although that is probably the simplest way to explain what spread polynomials are to those versed in well-known topics in mathematics, spread polynomials were introduced by Norman Wildberger for use in rational trigonometry, a subject in which one generally avoids the sine function and the other conventional trigonometric functions. The angle at which two lines meet is determined by a rational function of their slopes, known as the spread between the two lines, and equal to the square of the sine. The rational function can be identified without mentioning the sine function; see rational trigonometry for more on this.

1 Identities
2 Use in rational trigonometry
3 Table of spread polynomials
4 References

[edit] Identities

[edit] Explicit formula

An explicit formula, due to S. Goh, is

$S_n(s) = s\sum_{k=0}^{n-1} {n \over n - k} {2n-1-k \choose k} (-4s)^{n-1-k}.$

[edit] Recursion formula

$S_{n+1}(s) = 2(1-2s) S_n(s) - S_{n-1}(s) + 2s.\,$

[edit] Relation to Chebyshev polynomials

The spread polynomials are related to the Chebyshev polynomials of the first kind, T_n by the identity

$1 - 2S_n(s) = T_n(1 - 2s).\,$

This implies

$S_n(s) = {1 - T_n(1 - 2s) \over 2} = 1 - \left(T_n\left(\sqrt{1-s}\right)\right)^2.$

The second equality above follows from the identity

$2T_n(x)^2 - 1 = T_{2n}(x) \,$

on Chebyshev polynomials.

[edit] Composition

The spread polynomials satisfy the composition identity

$S_n(S_m(s)) = S_{nm}(s).\,$

[edit] Coefficients in finite fields

When the coefficients are taken to be members of the finite field F_p, then the sequence { S_n }_{n = 0, 1, 2, ...} of spread polynomials is periodic with period (p² − 1)/2. In other words, if k = (p² − 1)/2, then S_n + k = S_n, for all n.

[edit] Orthogonality

When the coefficients are taken to be real, then for n ≠ m, we have

$\int_0^1 \left(S_n(s) - {1 \over 2}\right) \left(S_m(s) - {1 \over 2}\right){ds \over \sqrt{s(1-s)}}=0.$

For n = m, the integral is π/4.

[edit] Generating function

$\sum_{n=1}^\infty S_n(s)x^n = {sx(1+x) \over (1-x)^3 + 4sx(1-x)}.$

[edit] Differential equation

S_n(s) satisfies the second order linear homogenous differential equation

s (1 - s) y'' + (1 / 2 - s) y' + n 2 (y - 1 / 2) = 0.

[edit] Use in rational trigonometry

Rational trigonometry is a recently introduced approach to trigonometry that eschews all transcendental functions (such as sine, cosine, etc.), all measurements of angles or compositions of rotations, and characterizes the separation between lines by a quantity called the "spread", which is a rational function of the slopes. Equality of angles between rays entails equality of spreads between lines. The spread between two lines is the square of the sine of the angle. The name "spread polynomials" comes from the use of these polynomials in rational trigonometry.