From 4ad94252939c991a5f0286d57d52066de6889687 Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Fri, 23 Jun 2023 21:16:14 +0200 Subject: [PATCH] coefficients examples --- mathematica/CoefficientsANotebook.nb | 67 +++++++++++++++++++++--- mathematica/CoefficientsAPackage.m | 4 ++ src/sections/coefficients_derivation.tex | 40 +++++++++++++- 3 files changed, 103 insertions(+), 8 deletions(-) diff --git a/mathematica/CoefficientsANotebook.nb b/mathematica/CoefficientsANotebook.nb index 02bda8f..153d803 100644 --- a/mathematica/CoefficientsANotebook.nb +++ b/mathematica/CoefficientsANotebook.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 94077, 2579] -NotebookOptionsPosition[ 76388, 2266] -NotebookOutlinePosition[ 76788, 2282] -CellTagsIndexPosition[ 76745, 2279] +NotebookDataLength[ 96190, 2634] +NotebookOptionsPosition[ 77829, 2309] +NotebookOutlinePosition[ 78229, 2325] +CellTagsIndexPosition[ 78186, 2322] WindowFrame->Normal*) (* Beginning of Notebook Content *) @@ -24,7 +24,7 @@ Cell[BoxData[ CellChangeTimes->{{3.796417857132905*^9, 3.7964178661028905`*^9}, { 3.806069920517039*^9, 3.8060699230180326`*^9}, 3.851522410135803*^9, 3.851618152361442*^9}, - CellLabel->"In[30]:=",ExpressionUUID->"c66212dc-a740-4d86-be65-4290e8e2102a"], + CellLabel->"In[36]:=",ExpressionUUID->"c66212dc-a740-4d86-be65-4290e8e2102a"], Cell[BoxData[ RowBox[{"(*", " ", @@ -2262,6 +2262,49 @@ Cell[BoxData[ Cell[BoxData["252"], "Output", CellChangeTimes->{3.896511273343273*^9}, CellLabel->"Out[47]=",ExpressionUUID->"c2480795-b6dc-47d0-8a1d-71da4fe8692e"] +}, Open ]], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"auxiliaryCoefficient", "[", + RowBox[{"0", ",", " ", "1"}], "]"}]], "Input", + CellChangeTimes->{{3.896534118720274*^9, 3.8965341212810135`*^9}}, + CellLabel->"In[37]:=",ExpressionUUID->"44e661a0-87e6-4994-89db-ec37c6d81559"], + +Cell[BoxData[ + FractionBox["1", "6"]], "Output", + CellChangeTimes->{3.8965341217505565`*^9}, + CellLabel->"Out[37]=",ExpressionUUID->"b081297b-e7d1-4b54-8713-af915651002c"] +}, Open ]], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"auxiliaryCoefficient", "[", + RowBox[{"0", ",", " ", "2"}], "]"}]], "Input", + CellChangeTimes->{{3.8965343464813366`*^9, 3.8965343486022835`*^9}, { + 3.896534393412722*^9, 3.8965343936352687`*^9}}, + CellLabel->"In[39]:=",ExpressionUUID->"db31c748-1d62-482b-b3cf-1d9fb1c6d2b6"], + +Cell[BoxData[ + FractionBox["1", "30"]], "Output", + CellChangeTimes->{3.896534349049232*^9, 3.896534393980844*^9}, + CellLabel->"Out[39]=",ExpressionUUID->"ecdff976-1ddb-4778-810d-4265e588e9ec"] +}, Open ]], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"auxiliaryCoefficient", "[", + RowBox[{"1", ",", " ", "3"}], "]"}]], "Input", + CellChangeTimes->{{3.8965345338093405`*^9, 3.89653455794691*^9}}, + CellLabel->"",ExpressionUUID->"75683dad-a61e-473b-9c4a-83816cb3e469"], + +Cell[BoxData[ + FractionBox["1", "42"]], "Output", + CellChangeTimes->{3.896534534646197*^9}, + CellLabel->"Out[40]=",ExpressionUUID->"4a748b9d-d16e-429b-a809-53ba37c4b150"] }, Open ]] }, WindowSize->{1269, 727}, @@ -2575,11 +2618,23 @@ Cell[75416, 2237, 151, 2, 32, "Output",ExpressionUUID->"362d3ff1-937f-4967-b965- }, Open ]], Cell[CellGroupData[{ Cell[75604, 2244, 202, 3, 28, "Input",ExpressionUUID->"2732d679-2000-41a9-be8d-ad84c0af4bca"], -Cell[75809, 2249, 175, 2, 54, "Output",ExpressionUUID->"6991b24f-bc73-4d7c-9c40-ec04f959e387"] +Cell[75809, 2249, 175, 2, 32, "Output",ExpressionUUID->"6991b24f-bc73-4d7c-9c40-ec04f959e387"] }, Open ]], Cell[CellGroupData[{ Cell[76021, 2256, 197, 3, 28, "Input",ExpressionUUID->"c43e12ab-d523-4793-b90b-e908e7637a6e"], Cell[76221, 2261, 151, 2, 32, "Output",ExpressionUUID->"c2480795-b6dc-47d0-8a1d-71da4fe8692e"] +}, Open ]], +Cell[CellGroupData[{ +Cell[76409, 2268, 248, 4, 28, "Input",ExpressionUUID->"44e661a0-87e6-4994-89db-ec37c6d81559"], +Cell[76660, 2274, 171, 3, 48, "Output",ExpressionUUID->"b081297b-e7d1-4b54-8713-af915651002c"] +}, Open ]], +Cell[CellGroupData[{ +Cell[76868, 2282, 301, 5, 28, "Input",ExpressionUUID->"db31c748-1d62-482b-b3cf-1d9fb1c6d2b6"], +Cell[77172, 2289, 192, 3, 48, "Output",ExpressionUUID->"ecdff976-1ddb-4778-810d-4265e588e9ec"] +}, Open ]], +Cell[CellGroupData[{ +Cell[77401, 2297, 239, 4, 28, "Input",ExpressionUUID->"75683dad-a61e-473b-9c4a-83816cb3e469"], +Cell[77643, 2303, 170, 3, 48, "Output",ExpressionUUID->"4a748b9d-d16e-429b-a809-53ba37c4b150"] }, Open ]] } ] diff --git a/mathematica/CoefficientsAPackage.m b/mathematica/CoefficientsAPackage.m index eb7542f..4e444f2 100644 --- a/mathematica/CoefficientsAPackage.m +++ b/mathematica/CoefficientsAPackage.m @@ -27,6 +27,8 @@ 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]; @@ -58,6 +60,8 @@ 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[ ] diff --git a/src/sections/coefficients_derivation.tex b/src/sections/coefficients_derivation.tex index 4e4c069..f0dfac9 100644 --- a/src/sections/coefficients_derivation.tex +++ b/src/sections/coefficients_derivation.tex @@ -146,8 +146,8 @@ \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*} @@ -168,4 +168,40 @@ & = \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} \ No newline at end of file