Spread polynomials

In the conventional language of trigonometry, the nth-degree spread polynomial Sn, 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 trigonometric functions of angles. The angle at which two lines meet is determined by a rational function of their slopes, known as the spread between the two lines. The spread is 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.

Contents

Identities

Explicit formulas

S_n(s) = s\sum_{k=0}^{n-1} {n \over n - k} {2n-1-k \choose k} (-4s)^{n-1-k}. (S. Goh)
S_n(s) = \frac{1}{2} - \frac{1}{4} \left ( 1-2s%2B2 \sqrt {s^2 -s} \right )^{n} - \frac{1}{4} \left ( 1-2s%2B2 \sqrt {s^2 -s} \right )^{-n}. (M. Hovdan)

From the definition it immediately follows that

S_n(s) = \sin^2\left(n\arcsin\left(\sqrt{s}\right)\right).

Recursion formula

S_{n%2B1}(s) = 2(1-2s) S_n(s) - S_{n-1}(s) %2B 2s.\,

Relation to Chebyshev polynomials

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

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

This implies

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

The second equality above follows from the identity

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

on Chebyshev polynomials.

Composition

The spread polynomials satisfy the composition identity

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

Coefficients in finite fields

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

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/8 unless n = m = 0, in which case it is π/4.

Generating functions

The ordinary generating function is

\sum_{n=1}^\infty S_n(s)x^n = {sx(1%2Bx) \over (1-x)^3 %2B 4sx(1-x)}.

The exponential generating function is

\sum_{n=1}^\infty {S_n(s)\over n!} x^n = {1 \over 2} e^x \left [ 1-e^{-2sx} \cos\left (2x \sqrt{s(1-s)}\right )\right ] .

Differential equation

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

s(1-s)y'' %2B (1/2-s)y' %2B n^2(y-1/2) = 0.\,

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.

Table of spread polynomials, with factorizations

The first several spread polynomials are as follows:

Failed to parse (PNG conversion failed;

check for correct installation of latex, dvips, gs, and convert): \begin{align} S_0(s) & = 0 \\[10pt] S_1(s) & = s \\[10pt] S_2(s) & = 4s-4s^2 \\ & = 4s(1-s) \\[10pt] S_3(s) & = 9s-24s^2+16s^3 \\ & = s(3-4s)^2 \\[10pt] S_4(s) & = 16s-80s^2+128s^3-64s^4 \\ & = 16s(1-s)(1-2s)^2 \\[10pt] S_5(s) & = 25s-200s^2+560s^3-640s^4+256s^5 \\ & = s(5-20s+16s^2)^2 \\[10pt] S_6(s) & = 36s-420s^2+1792s^3-3456s^4+3072s^5-1024s^6 \\ & = 4s(1-s)(1-4s)^2(3-4s)^2 \\[10pt] S_7(s) & = 49s-784s^2+4704s^3-13440s^4+19712s^5-14336s^6+4096s^7 \\ & = s(7-56s+112s^2-64s^3)^2 \\[10pt] S_8(s) & = 64s-1344s^2+10752s^3-42240s^4+90112s^5-106496s^6 \\ & {} \qquad + 65536s^7-16384s^8 \\ & = 64s(s-1)(1-2s)^2(1-8s+8s^2)^2 \\[10pt] S_9(s) & = 81s - 2160s^2 + 22176s^3 - 114048s^4 + 329472s^5 - 559104s^6 \\ & {} \qquad + 552960s^7 - 294912s^8 + 65536s^9 \\ & = s(-3+4s)^2(-3+36s-96s^2+64s^3)^2 \\[10pt] S_{10}(s) & = 100s - 3300s^2 + 42240s^3 - 274560s^4 + 1025024s^5 \\ {} & \qquad - 2329600s^6 + 3276800s^7 - 2785280s^8 + 1310720s^9 - 262144s^{10} \\ & = 4s(1-s)(5 - 20s+16s^2)^2(1-12s+16s^2)^2\\[10pt] S_{11}(s) & = 121s - 4840s^2 + 75504s^3 - 604032s^4 + 2818816s^5 \\ {} & \qquad -8200192s^6 + 15319040s^7 - 18382848s^8 + 13697024s^9 -5767168s^{10} + 1048576s^{11}\\ & = s(11 -220s + 1232s^2 -2816s^3 +2816s^4 -1024s^5)^2 \end{align}

References