reorder sections in chebyshev.1

posts/chebyshev/1/index.qmd

@@ -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
+The relationship between $p_n$ and the intermediate $f_d$, where *d* is a divisor of *n*,
-  either irreducible or the product of another polynomial and its reflection (potentially negated).
+  can be made explicit by a [Möbius inversion](https://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula).
-For example,
 
 $$
 \begin{align*}
-  p_9(z) &= f_3(z) \cdot g_9(z) \cdot -g_9(-z)
+  p_n(z) &= \prod_{d|n} f_n(z)
   \\
-  & \text{where } g_9(z) = z^3 -\ 3z -\ 1
+  \log( p_n(z) )
+    &= \log \left( \prod_{d|n} f_d(z) \right)
+    = \sum_{d|n} \log( f_d(z) )
+  \\
+  \log( f_n(z) ) &= \sum_{d|n} { \mu \left({n \over d} \right)}
+    \log( p_d(z) )
+  \\
+  f_n(z) &= \prod_{d|n} p_d(z)^{ \mu (n / d) }
+  \\[10pt]
+  f_6(z) = g_6(z)
+    &= p_6(z)^{\mu(1)}
+    p_3(z)^{\mu(2)}
+    p_2(z)^{\mu(3)}
+  \\
+  &= {p_6(z) \over p_3(z) p_2(z)}
 \end{align*}
 $$
 
-These reflections can be observed in the Chebyshev polynomials for $n = 3, 5, 7, 9$,
+Unfortunately, it's difficult to apply this technique across our whole series.
-  strongly implying that it occurs on the odd terms.
+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
 -------