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reorder sections in chebyshev.1

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queue-miscreant 2025-06-19 02:53:41 -05:00
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161
posts/chebyshev/1/index.qmd
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@ -511,24 +511,105 @@ If this observation is legitimate, call the new term $f_n(z)$
and denote $p_n(z) = U_{n -\ 1}( z / 2 )$.
### Total Degrees
### Factorization Attempts
It can be observed that the new term is symmetric ($f(z) = f(-z)$), and is therefore
either irreducible or the product of another polynomial and its reflection (potentially negated).
For example,
The relationship between $p_n$ and the intermediate $f_d$, where *d* is a divisor of *n*,
can be made explicit by a [Möbius inversion](https://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula).
$$
\begin{align*}
p_9(z) &= f_3(z) \cdot g_9(z) \cdot -g_9(-z)
p_n(z) &= \prod_{d|n} f_n(z)
\\
& \text{where } g_9(z) = z^3 -\ 3z -\ 1
\log( p_n(z) )
&= \log \left( \prod_{d|n} f_d(z) \right)
= \sum_{d|n} \log( f_d(z) )
\\
\log( f_n(z) ) &= \sum_{d|n} { \mu \left({n \over d} \right)}
\log( p_d(z) )
\\
f_n(z) &= \prod_{d|n} p_d(z)^{ \mu (n / d) }
\\[10pt]
f_6(z) = g_6(z)
&= p_6(z)^{\mu(1)}
p_3(z)^{\mu(2)}
p_2(z)^{\mu(3)}
\\
&= {p_6(z) \over p_3(z) p_2(z)}
\end{align*}
$$
These reflections can be observed in the Chebyshev polynomials for $n = 3, 5, 7, 9$,
strongly implying that it occurs on the odd terms.
Unfortunately, it's difficult to apply this technique across our whole series.
Möbius inversion over series typically uses more advanced generating functions such as
[Dirichlet series](https://en.wikipedia.org/wiki/Dirichlet_series#Formal_Dirichlet_series)
or [Lambert series](https://en.wikipedia.org/wiki/Lambert_series).
However, naively reaching for these fails for two reasons:
In other words, if $f_n$ is the new polynomial introduced by $p_n$, then denote its conditional factorization $g_n$.
- We built our series of polynomials on a recurrence relation, and these series
are opaque to such manipulations.
- To do a proper Möbius inversion, we need these kinds of series over the *logarithm*
of each polynomial (*B* is a series over the polynomials themselves).
Ignoring these (and if you're in the mood for awful-looking math) you may note
the Lambert equivalence[^2]:
[^2]:
This equivalence applies to other polynomial series obeying the same factorization rule
such as the [cyclotomic polynomials](https://en.wikipedia.org/wiki/Cyclotomic_polynomial).
$$
\begin{align*}
\log( p_n(z) )
&= \sum_{d|n} \log( f_d(z) )
\\
\sum_{n = 1}^\infty \log( p_n ) x^n
&= \sum_{n = 1}^\infty \sum_{d|n} \log( f_d ) x^n
\\
&= \sum_{k = 1}^\infty \sum_{m = 1}^\infty \log( f_m ) x^{m k}
\\
&= \sum_{m = 1}^\infty \log( f_m ) \sum_{k = 1}^\infty (x^m)^k
\\
&= \sum_{m = 1}^\infty \log( f_m ) {x^m \over 1 - x^m}
\end{align*}
$$
Either way, the number-theoretic properties of this sequence are difficult to ascertain
without advanced techniques.
If research has been done, it is not easily available in the OEIS.
### Total Degrees
It can be also be observed that the new term is symmetric ($f(z) = f(-z)$), and is therefore
either irreducible or the product of polynomial and its reflection (potentially negated).
For example,
$$
p_9(z) = \left\{
\begin{matrix}
(z - 1)(z + 1)
& \cdot
& (z^3 - 3z - 1)(z^3 - 3z + 1)
\\
\shortparallel && \shortparallel
\\
f_3(z)
& \cdot
& f_9(z)
\\
\shortparallel && \shortparallel
\\
g_3(z) \cdot g_3(-z)
& \cdot
& g_9(z) \cdot -g_9(-z)
\end{matrix}
\right.
$$
These factor polynomials $g_n$ are the minimal polynomials of $2\cos( \pi / n )$.
Multiplying these minimal polynomials by their reflection can be observed in the Chebyshev polynomials
for $n = 3, 5, 7, 9$, strongly implying that it occurs on the odd terms.
Assuming this is true, we have
$$
f_n(z) = \begin{cases}
@ -647,68 +728,6 @@ Markdown(tabulate(
```
### Factorization Attempts
The relationship between $p_n$ and the intermediate $f_d$, where *d* is a divisor of *n*,
can be made explicit by a [Möbius inversion](https://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula).
$$
\begin{align*}
p_n(z) &= \prod_{d|n} f_n(z)
\\
\log( p_n(z) )
&= \log \left( \prod_{d|n} f_d(z) \right)
= \sum_{d|n} \log( f_d(z) )
\\
\log( f_n(z) ) &= \sum_{d|n} { \mu \left({n \over d} \right)}
\log( p_d(z) )
\\
f_n(z) &= \prod_{d|n} p_d(z)^{ \mu (n / d) }
\\[10pt]
f_6(z) = g_6(z)
&= p_6(z)^{\mu(1)}
p_3(z)^{\mu(2)}
p_2(z)^{\mu(3)}
\\
&= {p_6(z) \over p_3(z) p_2(z)}
\end{align*}
$$
Unfortunately, it's difficult to apply this technique across our whole series.
Möbius inversion over series typically uses more advanced generating functions such as
[Dirichlet series](https://en.wikipedia.org/wiki/Dirichlet_series#Formal_Dirichlet_series)
or [Lambert series](https://en.wikipedia.org/wiki/Lambert_series).
However, naively reaching for these fails for two reasons:
- We built our series of polynomials on a recurrence relation, and these series
are opaque to such manipulations.
- To do a proper Möbius inversion, we need these kinds of series over the *logarithm*
of each polynomial (*B* is a series over the polynomials themselves).
Ignoring these (and if you're in the mood for awful-looking math) you may note
the Lambert equivalence:
$$
\begin{align*}
\log( p_n(z) )
&= \sum_{d|n} \log( f_d(z) )
\\
\sum_{n = 1}^\infty \log( p_n ) x^n
&= \sum_{n = 1}^\infty \sum_{d|n} \log( f_d ) x^n
\\
&= \sum_{k = 1}^\infty \sum_{m = 1}^\infty \log( f_m ) x^{m k}
\\
&= \sum_{m = 1}^\infty \log( f_m ) \sum_{k = 1}^\infty (x^m)^k
\\
&= \sum_{m = 1}^\infty \log( f_m ) {x^m \over 1 - x^m}
\end{align*}
$$
Either way, the number-theoretic properties of this sequence are difficult to ascertain
without advanced techniques.
If research has been done, it is not easily available in the OEIS.
Closing
-------