From 466a41668baa3ae3a02bdad6629efc956e1a38fc Mon Sep 17 00:00:00 2001 From: queue-miscreant Date: Thu, 19 Jun 2025 02:53:41 -0500 Subject: [PATCH] reorder sections in chebyshev.1 --- posts/chebyshev/1/index.qmd | 161 ++++++++++++++++++++---------------- 1 file changed, 90 insertions(+), 71 deletions(-) diff --git a/posts/chebyshev/1/index.qmd b/posts/chebyshev/1/index.qmd index c2a8476..9c9bd11 100644 --- a/posts/chebyshev/1/index.qmd +++ b/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 - 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 -------