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coefficients examples
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kolosovpetro committed Jun 23, 2023
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67 changes: 61 additions & 6 deletions mathematica/CoefficientsANotebook.nb
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4 changes: 4 additions & 0 deletions mathematica/CoefficientsAPackage.m
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binomialPowerSum::usage= "Binomial power sum \\sum_{t=0}^{r} (-1)^t \\binom{r}{t} n^{r-t} \\sum_{k=1}^{n} k^{t+r}"

auxiliaryCoefficient::usage = "Coefficient \\binom{d}{2r+1} \\frac{(-1)^{d-1}}{d-r}"

Begin["`Private`"]

Unprotect[Power];
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binomialPowerSum[r_, n_] := Sum[(-1)^t * Binomial[r, t] * n^(r-t) * faulhaberFormula[t+r, n], {t, 0, r}];

auxiliaryCoefficient[r_, d_]:= Binomial[d, 2r+1] * (-1)^(d-1) / (d-r) * BernoulliB[2d - 2r];

End[ ]

EndPackage[ ]
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40 changes: 38 additions & 2 deletions src/sections/coefficients_derivation.tex
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\begin{equation*}
\begin{split}
\coeffA{3}{1}
&= 3 \binom{2}{1} \sum_{d \geq 3} \coeffA{3}{d} \binom{d}{3} \frac{(-1)^{d-1}}{d} \bernoulli{2d-2} \\
&= 3 \binom{2}{1} \coeffA{3}{3} \binom{3}{3} \frac{(-1)^2}{2} \bernoulli{6}
&= 3 \binom{2}{1} \sum_{d \geq 3} \coeffA{3}{d} \binom{d}{3} \frac{(-1)^{d-1}}{d-1} \bernoulli{2d-2} \\
&= 3 \binom{2}{1} \coeffA{3}{3} \binom{3}{3} \frac{(-1)^2}{2} \bernoulli{4}
= 3 \cdot 140 \cdot (-\frac{1}{30}) = -14
\end{split}
\end{equation*}
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& = \frac{-14}{6} + \frac{140}{42} = 1
\end{split}
\end{equation*}
\end{examp}
\begin{examp}
Example for $\coeffA{m}{r}$ for $m=4$.
First we get $\coeffA{4}{4}$
\begin{equation*}
\coeffA{4}{4} = 9 \binom{8}{4}= 630
\end{equation*}
Then $\coeffA{4}{2} = 0, \; \coeffA{4}{3} = 0$
because $\coeffA{m}{d}$ is zero in the range $m/2 \leq d < m$ means that zero for $d$ in
$2 \leq d < 4$.
The $\coeffA{4}{1}$ coefficient is non-zero and calculated as
\begin{equation*}
\begin{split}
\coeffA{4}{1}
&= 3 \binom{2}{1} \sum_{d \geq 3} \coeffA{4}{d} \binom{d}{3} \frac{(-1)^{d-1}}{d-1} \bernoulli{2d-2} \\
&= 3 \binom{2}{1} \coeffA{4}{4} \binom{4}{3} \frac{(-1)^3}{3} \bernoulli{6}
= 3 \cdot 2 \cdot 630 \cdot 4 \cdot (-\frac{1}{3}) \cdot \frac{1}{42} = -120
\end{split}
\end{equation*}
Finally $\coeffA{4}{0}$ coefficient is
\begin{equation*}
\begin{split}
\coeffA{3}{0}
&= 1 \binom{0}{0} \sum_{d \geq 1} \coeffA{3}{d} \binom{d}{1} \frac{(-1)^{d-1}}{d} \bernoulli{2d}
= \sum_{d \geq 1} \coeffA{3}{d} \binom{d}{1} \frac{(-1)^{d-1}}{d} \bernoulli{2d} \\
& = \coeffA{3}{1} \binom{1}{1} \frac{(-1)^{1-1}}{1} \bernoulli{2}
+ \coeffA{3}{2} \binom{2}{1} \frac{(-1)^{2-1}}{2} \bernoulli{4}
+ \coeffA{3}{3} \binom{3}{1} \frac{(-1)^{3-1}}{3} \bernoulli{6} \\
& = \coeffA{3}{1} \bernoulli{2} - 2 \coeffA{3}{2} \frac{1}{2} \bernoulli{4}
+ 3 \coeffA{3}{3} \frac{1}{3} \bernoulli{6} \\
&= \frac{1}{6} \coeffA{3}{1}
+ \coeffA{3}{2} \frac{1}{30}
+ \coeffA{3}{3} \frac{1}{42} \\
& = \frac{-14}{6} + \frac{140}{42} = 1
\end{split}
\end{equation*}
\end{examp}

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