From c6d6d5a639d2e5a906e49bf93496b8a68868fa60 Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Sat, 21 Sep 2024 17:56:43 +0200 Subject: [PATCH 01/13] update ci | update citation | add changelog | cleanups --- .editorconfig | 12 +++++++++ .github/workflows/build-and-deploy-pdf.yml | 30 +++++++++------------ .github/workflows/build-pdf.yml | 27 ++++++++++--------- .gitignore | 14 +++++----- BIBTEX-CITATION.bib | 9 ++++--- CHANGELOG.md | 10 +++++++ CITATION.cff | 19 ------------- README.md | 18 +++++++++++++ img/template_example.PNG | Bin 115908 -> 0 bytes img/template_example2.PNG | Bin 180929 -> 0 bytes 10 files changed, 79 insertions(+), 60 deletions(-) create mode 100644 .editorconfig create mode 100644 CHANGELOG.md delete mode 100644 CITATION.cff delete mode 100644 img/template_example.PNG delete mode 100644 img/template_example2.PNG diff --git a/.editorconfig b/.editorconfig new file mode 100644 index 0000000..8389293 --- /dev/null +++ b/.editorconfig @@ -0,0 +1,12 @@ +root = true + +[*] +charset = utf-8 +indent_style = space +indent_size = 4 +insert_final_newline = true +trim_trailing_whitespace = true +end_of_line = lf + +[*.{xml,yml}] +indent_size = 2 diff --git a/.github/workflows/build-and-deploy-pdf.yml b/.github/workflows/build-and-deploy-pdf.yml index f81c54e..2241875 100644 --- a/.github/workflows/build-and-deploy-pdf.yml +++ b/.github/workflows/build-and-deploy-pdf.yml @@ -4,15 +4,11 @@ on: push: tags: - v* - branches: - - develop - - master - - main - - release/* workflow_dispatch: env: FILE_NAME: "PolynomialIdentityInvolvingBTandFaulhaber" + ACTIONS_RUNNER_DEBUG: false jobs: build-pdf: @@ -21,34 +17,34 @@ jobs: steps: - name: Checkout - uses: actions/checkout@v3 + uses: actions/checkout@v4 with: fetch-depth: 0 - name: Install GitVersion - uses: gittools/actions/gitversion/setup@v0.9.15 + uses: gittools/actions/gitversion/setup@v1.1.1 with: versionSpec: '5.x' - name: Determine Version - uses: gittools/actions/gitversion/execute@v0.9.15 + uses: gittools/actions/gitversion/execute@v1.1.1 - name: Print SemVer run: | - echo ${{ env.GITVERSION_SEMVER }} - echo ${{ env.GitVersion_InformationalVersion }} - echo ${{ env.GitVersion_EscapedBranchName }} - newVersion="${{ env.GITVERSION_SEMVER }}+${{ env.GITVERSION_BRANCHNAME }}.${{ env.GITVERSION_SHORTSHA }}" + echo "SemVer: ${{ env.GitVersion_SemVer }}" + echo "BranchName: ${{ env.GitVersion_BranchName }}" + echo "ShortSha: ${{ env.GitVersion_ShortSha }}" + newVersion="${{ env.GitVersion_SemVer }}+${{ env.GitVersion_BranchName }}.${{ env.GitVersion_ShortSha }}" echo "Next version: $newVersion" - name: Update version.tex shell: bash run: | - newVersion="${{ env.GITVERSION_SEMVER }}+${{ env.GITVERSION_BRANCHNAME }}.${{ env.GITVERSION_SHORTSHA }}" + newVersion="${{ env.GitVersion_SemVer }}+${{ env.GitVersion_BranchName }}.${{ env.GitVersion_ShortSha }}" sed -i "s|Local-0.1.0|$newVersion|" "src/sections/version.tex" - name: Build PDF - uses: xu-cheng/latex-action@v2 + uses: xu-cheng/latex-action@v3 with: root_file: "${{ env.FILE_NAME }}.tex" working_directory: src @@ -58,7 +54,7 @@ jobs: ls -lsa src - name: Upload artifacts - uses: actions/upload-artifact@v3 + uses: actions/upload-artifact@v4 with: name: "${{ env.FILE_NAME }}-${{ env.GitVersion_InformationalVersion }}" path: | @@ -68,7 +64,7 @@ jobs: src/sections - name: Upload artifacts PDF - uses: actions/upload-artifact@v3 + uses: actions/upload-artifact@v4 with: name: "${{ env.FILE_NAME }}-PDF-${{ env.GitVersion_InformationalVersion }}" path: | @@ -92,4 +88,4 @@ jobs: env: USERNAME: "kolosovpetro" EMAIL: "kolosovp94@gmail.com" - MESSAGE: "CICD deploy of PDF document" \ No newline at end of file + MESSAGE: "CICD deploy of ${{ env.FILE_NAME }} PDF document" diff --git a/.github/workflows/build-pdf.yml b/.github/workflows/build-pdf.yml index ca9d81a..6d968b1 100644 --- a/.github/workflows/build-pdf.yml +++ b/.github/workflows/build-pdf.yml @@ -15,6 +15,7 @@ on: env: FILE_NAME: "PolynomialIdentityInvolvingBTandFaulhaber" + ACTIONS_RUNNER_DEBUG: false jobs: build-pdf: @@ -23,34 +24,34 @@ jobs: steps: - name: Checkout - uses: actions/checkout@v3 + uses: actions/checkout@v4 with: fetch-depth: 0 - name: Install GitVersion - uses: gittools/actions/gitversion/setup@v0.9.15 + uses: gittools/actions/gitversion/setup@v1.1.1 with: versionSpec: '5.x' - name: Determine Version - uses: gittools/actions/gitversion/execute@v0.9.15 + uses: gittools/actions/gitversion/execute@v1.1.1 - name: Print SemVer run: | - echo ${{ env.GITVERSION_SEMVER }} - echo ${{ env.GitVersion_InformationalVersion }} - echo ${{ env.GitVersion_EscapedBranchName }} - newVersion="${{ env.GITVERSION_SEMVER }}+${{ env.GITVERSION_BRANCHNAME }}.${{ env.GITVERSION_SHORTSHA }}" + echo "SemVer: ${{ env.GitVersion_SemVer }}" + echo "BranchName: ${{ env.GitVersion_BranchName }}" + echo "ShortSha: ${{ env.GitVersion_ShortSha }}" + newVersion="${{ env.GitVersion_SemVer }}+${{ env.GitVersion_BranchName }}.${{ env.GitVersion_ShortSha }}" echo "Next version: $newVersion" - name: Update version.tex shell: bash run: | - newVersion="${{ env.GITVERSION_SEMVER }}+${{ env.GITVERSION_BRANCHNAME }}.${{ env.GITVERSION_SHORTSHA }}" + newVersion="${{ env.GitVersion_SemVer }}+${{ env.GitVersion_BranchName }}.${{ env.GitVersion_ShortSha }}" sed -i "s|Local-0.1.0|$newVersion|" "src/sections/version.tex" - name: Build PDF - uses: xu-cheng/latex-action@v2 + uses: xu-cheng/latex-action@v3 with: root_file: "${{ env.FILE_NAME }}.tex" working_directory: src @@ -60,18 +61,18 @@ jobs: ls -lsa src - name: Upload artifacts - uses: actions/upload-artifact@v3 + uses: actions/upload-artifact@v4 with: name: "${{ env.FILE_NAME }}-${{ env.GitVersion_InformationalVersion }}" - path: | + path: | src/${{ env.FILE_NAME }}.tex src/${{ env.FILE_NAME }}.bbl src/${{ env.FILE_NAME }}.bib src/sections - name: Upload artifacts PDF - uses: actions/upload-artifact@v3 + uses: actions/upload-artifact@v4 with: name: "${{ env.FILE_NAME }}-PDF-${{ env.GitVersion_InformationalVersion }}" path: | - src/${{ env.FILE_NAME }}.pdf \ No newline at end of file + src/${{ env.FILE_NAME }}.pdf diff --git a/.gitignore b/.gitignore index 9cc9137..c8d630a 100644 --- a/.gitignore +++ b/.gitignore @@ -28,12 +28,12 @@ ## Generated if empty string is given at "Please type another file name for output:" .idea/ ## Bibliography auxiliary files (bibtex/biblatex/biber): -*.bbl -*.bcf -*.blg -*-blx.aux -*-blx.bib -*.run.xml +#*.bbl +#*.bcf +#*.blg +#*-blx.aux +#*-blx.bib +#*.run.xml ## Build tool auxiliary files: *.fdb_latexmk @@ -292,4 +292,4 @@ TSWLatexianTemp* #*Notes.bib /publish -*.zip \ No newline at end of file +*.zip diff --git a/BIBTEX-CITATION.bib b/BIBTEX-CITATION.bib index df2f2e2..ba7d494 100644 --- a/BIBTEX-CITATION.bib +++ b/BIBTEX-CITATION.bib @@ -1,7 +1,8 @@ @misc{kolosov2023polynomial, - title = {Polynomial identity involving Binomial Theorem and Faulhaber's formula Source files}, - author = {Kolosov, Petro}, - journal = {GitHub}, - howpublished = "\url{https://github.com/kolosovpetro/PolynomialIdentityInvolvingBTandFaulhaber}", + title = {{Polynomial identity involving Binomial Theorem and Faulhaber's formula}}, + author = {{Kolosov, Petro}}, + journal = {Figshare}, + number = {https://doi.org/10.6084/m9.figshare.2386}, + note = {\url{https://kolosovpetro.github.io/pdf/PolynomialIdentityInvolvingBTandFaulhaber.pdf}}, year = {2023} } \ No newline at end of file diff --git a/CHANGELOG.md b/CHANGELOG.md new file mode 100644 index 0000000..b4ebec6 --- /dev/null +++ b/CHANGELOG.md @@ -0,0 +1,10 @@ +# Changelog + +All notable changes to this project will be documented in this file. + +The format is based on [Keep a Changelog](https://keepachangelog.com/en/1.0.0/), +and this project adheres to [Semantic Versioning v2.0.0](https://semver.org/spec/v2.0.0.html). + +## [1.0.0] - In progress + +- Update CI/CD actions diff --git a/CITATION.cff b/CITATION.cff deleted file mode 100644 index d0ca70b..0000000 --- a/CITATION.cff +++ /dev/null @@ -1,19 +0,0 @@ -# This CITATION.cff file was generated with cffinit. -# Visit https://bit.ly/cffinit to generate yours today! - -cff-version: 1.2.0 -title: Polynomial identity involving Binomial Theorem and Faulhaber's formula Source files -message: >- - If you use this software, please cite it using the - metadata from this file. -type: software -authors: - - given-names: Petro - family-names: Kolosov - email: kolosovp94@gmail.com - orcid: 'https://orcid.org/0000-0002-6544-8880' -identifiers: - - type: url - value: >- - https://github.com/kolosovpetro/github-latex-template - description: Polynomial identity involving Binomial Theorem and Faulhaber's formula diff --git a/README.md b/README.md index 31ae467..f255c3a 100644 --- a/README.md +++ b/README.md @@ -2,3 +2,21 @@ ![abs](./img/abstract.PNG) +## Build and run in Intellij IDEA under Windows + +- Install `MikTeX`: https://miktex.org/download +- Update `MikTeX` +- Install `SumatraPDF` viewer: https://www.sumatrapdfreader.org/download-free-pdf-viewer +- Path to SumatraPDF: `C:\Program Files\SumatraPDF` +- Install `Intellij IDEA Ultimate`: https://www.jetbrains.com/idea/download/#section=windows +- Activate `Intellij IDEA Ultimate` +- Install `TeXiFy IDEA` plugin: https://plugins.jetbrains.com/plugin/9473-texify-idea +- Clone this repository locally: `https://github.com/kolosovpetro/github-latex-template.git` +- Open `github-latex-template` folder in `Intellij IDEA Ultimate` and configure as follows + - LaTeX Configuration + ![LaTeX Configuration](./img/latex_configuration.PNG "LaTeX Configuration") + - BibTeX Configuration + ![BibTeX Configuration](./img/bibtex_configuration.PNG "BibTeX Configuration") +- Configure Inverse Search in `Intellij IDEA` for SumatraPDF: `Tools -> LaTeX -> Configure Inverse Search` +- Compile document using `Shift + F10` + diff --git a/img/template_example.PNG b/img/template_example.PNG deleted file mode 100644 index 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a/src/sections/05_conclusions/conclusions.tex b/src/sections/05_conclusions/conclusions.tex index 9f64639..1a64432 100644 --- a/src/sections/05_conclusions/conclusions.tex +++ b/src/sections/05_conclusions/conclusions.tex @@ -5,9 +5,9 @@ n^{2m+1} = \sum_{k=1}^{n} \mathbf{A}_{m,0} k^0 (n-k)^0 + \mathbf{A}_{m,1}(n-k)^1 + \cdots + \mathbf{A}_{m,m} k^m (n-k)^m \] -In particular, the coefficients $\coeffA{m}{r}$ may be evaluated both ways, -by constructing and solving a system of linear equations or applying recurrence relations; -all these approaches are explained with examples -in the sections~\ref{sec:approach-via-system-of-linear-equations} and~\ref{sec:approach-via-recursion}, -respectively. -Moreover, to validate the results, there are supplementary Mathematica programs provided at~\cite{kolosov2023github}. \ No newline at end of file +In particular, the coefficients $\coeffA{m}{r}$ can be evaluated both ways, +by constructing and solving certain system of linear equations or by deriving recurrence relation; +all these approaches are examined providing examples +in the sections~\ref{sec:approach-via-system-of-linear-equations} and~\ref{sec:approach-via-recursion}. +Moreover, to validate the results, +there are supplementary Mathematica programs provided at~\cite{kolosov2023github}. diff --git a/src/sections/abstract.tex b/src/sections/abstract.tex index 338a1c4..3d415c1 100644 --- a/src/sections/abstract.tex +++ b/src/sections/abstract.tex @@ -1,7 +1,11 @@ -In this manuscript we show that for every $n\geq 1, \; n,m\in\mathbb{N}$ +In this manuscript, we have shown that for every $n\geq 1, \; n,m\in\mathbb{N}$ there are coefficients $\mathbf{A}_{m,0}, \mathbf{A}_{m,1}, \ldots, \mathbf{A}_{m,m}$ such that the polynomial identity holds \[ n^{2m+1} = \sum_{k=1}^{n} \mathbf{A}_{m,0} k^0 (n-k)^0 + \mathbf{A}_{m,1}(n-k)^1 + \cdots + \mathbf{A}_{m,m} k^m (n-k)^m \] +In particular, the coefficients $\coeffA{m}{r}$ can be evaluated both ways, +by constructing and solving certain system of linear equations or by deriving recurrence relation; +all these approaches are examined providing examples. +To validate the results, there are supplementary Mathematica programs. From 704c89e0e7172144817d3d130c796a4bceb7cffa Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Sat, 21 Sep 2024 19:22:02 +0200 Subject: [PATCH 06/13] update footnote --- CHANGELOG.md | 1 + ...ynomialIdentityInvolvingBTandFaulhaber.pdf | Bin 362444 -> 362459 bytes src/sections/01_introduction/footnote.tex | 13 ++++++++----- 3 files changed, 9 insertions(+), 5 deletions(-) diff --git a/CHANGELOG.md b/CHANGELOG.md index ee4093e..7a41e8a 100644 --- a/CHANGELOG.md +++ b/CHANGELOG.md @@ -13,3 +13,4 @@ and this project adheres to [Semantic Versioning v2.0.0](https://semver.org/spec - Add footnote with GitHub repository link - Update abstract - Update conclusions +- Update footnote diff --git a/out/PolynomialIdentityInvolvingBTandFaulhaber.pdf b/out/PolynomialIdentityInvolvingBTandFaulhaber.pdf index 8b60feedc6064920af04910c8f220ebaf957e006..3b78052a9914dc962c20125ffb1a632570c7a699 100644 GIT binary patch delta 8189 zcmai22~>~i_a~aDM7u2_lqKLw6(ux9vNye1QWR0~Wl$5+NQDX6 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-Another approach to determine the coefficients $\coeffA{m}{r}$ was provided by Dr. Max Alekseyev +Another approach to determine the coefficients $\coeffA{m}{r}$ was proposed by Dr. Max Alekseyev in MathOverflow discussion~\cite{alekseyev2018mathoverflow}. Generally, the idea was to determine the coefficients $\coeffA{m}{r}$ recursively starting from the base case $\coeffA{m}{m}$ up to $\coeffA{m}{r-1}, \ldots, \coeffA{m}{0}$ via previously determined values. @@ -151,4 +151,4 @@ It is as well interesting to notice that row sums of the $\coeffA{m}{r}$ give powers of $2$ \begin{equation*} \sum_{r=0}^{m} \coeffA{m}{r} = 2^{2m+1} -\end{equation*} \ No newline at end of file +\end{equation*} diff --git a/src/sections/04_coefficients_via_recursion_examples/coefficients_recursion_derivation_example5.tex b/src/sections/04_coefficients_via_recursion_examples/coefficients_recursion_derivation_example5.tex index b953cad..5f8314b 100644 --- a/src/sections/04_coefficients_via_recursion_examples/coefficients_recursion_derivation_example5.tex +++ b/src/sections/04_coefficients_via_recursion_examples/coefficients_recursion_derivation_example5.tex @@ -30,4 +30,5 @@ &= -1386 \cdot \frac{1}{6} + 660 \cdot \frac{1}{30} + 2772 \cdot \frac{5}{66} = 1 \end{split} \end{equation*} -\end{examp} \ No newline at end of file +\end{examp} +As expected. 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z2Oh}4J`N+`K%e}*^#%pHZ1V946f@xY#t7*-gXeIRFgG_rI2?pBrA% Date: Sat, 21 Sep 2024 19:30:33 +0200 Subject: [PATCH 09/13] fix grammar: introduction --- CHANGELOG.md | 3 +++ ...ynomialIdentityInvolvingBTandFaulhaber.pdf | Bin 362491 -> 362514 bytes src/sections/01_introduction/introduction.tex | 12 ++++++------ 3 files changed, 9 insertions(+), 6 deletions(-) diff --git a/CHANGELOG.md b/CHANGELOG.md index 7a41e8a..debcc98 100644 --- a/CHANGELOG.md +++ b/CHANGELOG.md @@ -14,3 +14,6 @@ and this project adheres to [Semantic Versioning v2.0.0](https://semver.org/spec - Update abstract - Update conclusions - Update footnote +- Grammar fix abstract +- Grammar fix conclusions +- Grammar fix introduction diff --git a/out/PolynomialIdentityInvolvingBTandFaulhaber.pdf b/out/PolynomialIdentityInvolvingBTandFaulhaber.pdf index 2fe9be78af61b98e01f29fc0c640504b7838bea5..9cf3e65e8793e925ed59aa0774a79980009deb37 100644 GIT binary patch delta 10392 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z`y>B=5cvO(aJuXQ89^;bKt`?KU>yQpNvL}Ps_8O7W@zj7fTuy4kf2gYpoC!2EJnZ( zOx+7O8BH+?V^HkDK(1BRFu~G;bYZ>KC>SSK!-O{A^Ew!q%~=@eBsg=0tj7Pl#8P&} eirKD*cDfw$aIv+Uje;jgf;C$~L2v!)+5ZRUKw>EX diff --git a/src/sections/01_introduction/introduction.tex b/src/sections/01_introduction/introduction.tex index 469e4d3..57f3775 100644 --- a/src/sections/01_introduction/introduction.tex +++ b/src/sections/01_introduction/introduction.tex @@ -17,7 +17,7 @@ \end{center} \caption{Table of finite differences of the polynomial $n^3$.} \label{tab:table} \end{table} -We can observe easily that finite differences +We can easily observe that finite differences \footnote{\input{sections/01_introduction/footnote}} of the polynomial $n^3$ may be expressed according to the following relation, via rearrangement of the terms @@ -32,7 +32,7 @@ \Delta(n^3) &= 1+6\cdot0+6\cdot1+6\cdot2+6\cdot3+\cdots+6\cdot n \end{split} \end{align} -Furthermore, the polynomial $n^3$ is identical to +Furthermore, the polynomial $n^3$ is equivalent to \begin{align*} n^3 &= [1+6\cdot0]+[1+6\cdot0+6\cdot1]+[1+6\cdot0+6\cdot1+6\cdot2]+\cdots \\ &+[1+6\cdot0+6\cdot1+6\cdot2+\cdots+6\cdot(n-1)] @@ -46,15 +46,15 @@ \label{eq:cube_identity} n^3 = \sum_{k=1}^{n} 6k(n-k) + 1 \end{equation} -Assume that equation~\eqref{eq:cube_identity} has an implicit form as follows +Assume that equation~\eqref{eq:cube_identity} has the following implicit form \begin{equation} \label{eq:pattern} n^3 = \sum_{k=1}^{n} \coeffA{1}{1} k^1(n-k)^1 + \coeffA{1}{0} k^0(n-k)^0, \end{equation} where $\coeffA{1}{1} = 6$ and $\coeffA{1}{0} = 1$, respectively. Note that here the power of $3$ is actually defined by $2m+1$ where $m=1$. -So is there a generalization of the relation~\eqref{eq:pattern} for all positive odd powers $2m+1, \; m=0,1,2,\dots$? -Therefore, let be a conjecture +So, is there a generalization of the relation~\eqref{eq:pattern} for all positive odd powers $2m+1, \; m=0,1,2,\dots$? +Therefore, let us propose a conjecture \begin{conj} For every $n\geq 1, \; n,m\in\mathbb{N}$ there are coefficients $\coeffA{m}{0}, \coeffA{m}{1}, \ldots, \coeffA{m}{m}$ such that \begin{equation} @@ -62,4 +62,4 @@ n^{2m+1} = \sum_{k=1}^{n} \coeffA{m}{0} k^0 (n-k)^0 + \coeffA{m}{1} (n-k)^1 + \cdots + \coeffA{m}{m} k^m (n-k)^m \end{equation} -\end{conj} \ No newline at end of file +\end{conj} From 6324542161eb304edacefeb2104ca8e9c433b404 Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Sat, 21 Sep 2024 19:33:20 +0200 Subject: [PATCH 10/13] grammar fix: section 2 --- CHANGELOG.md | 1 + ...ynomialIdentityInvolvingBTandFaulhaber.pdf | Bin 362514 -> 362516 bytes .../coefficients_via_system_of_equations.tex | 10 +++++----- 3 files changed, 6 insertions(+), 5 deletions(-) diff --git a/CHANGELOG.md b/CHANGELOG.md index debcc98..f0f85b4 100644 --- a/CHANGELOG.md +++ b/CHANGELOG.md @@ -17,3 +17,4 @@ and this project adheres to [Semantic Versioning v2.0.0](https://semver.org/spec - Grammar fix abstract - Grammar fix conclusions - Grammar fix introduction +- Grammar fix section 2 diff --git a/out/PolynomialIdentityInvolvingBTandFaulhaber.pdf b/out/PolynomialIdentityInvolvingBTandFaulhaber.pdf index 9cf3e65e8793e925ed59aa0774a79980009deb37..228a5d91cf2f246d45a1facc17f2d676daed5dff 100644 GIT binary patch delta 9930 zcmaiY2{cvR`~R!V^AJf$q8CN?4ENk~iOfS86Vf1aRAv#8B0kxBb6w?{IEw{jGIZt!MA^+0XFV&+|DaDM2J9L8Scq2D;+}vqG|j6gzFE z?D_se*T}cL<>`jau0tIvF?WUvviy>^lLXy-6ccTB`;9KmsoqSGmj5Ku+VfN`k-0HE z+^1=V>3(=IljLo+K&o1x-=$aTm-p^e6FtOhAIE%-?F_WWM^)5>ZQ?R5yJ6jtb&COh zOVd+7pL*V}jWwnP9LHo{{PbPO_rT|aY5GI#w*>E99x!RM$@CMrv;E+y&6g(2%ARer 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diff --git a/src/sections/03_coefficients_via_recursion/coefficients_recursion_derivation.tex b/src/sections/03_coefficients_via_recursion/coefficients_recursion_derivation.tex index 4312e8d..fa9629c 100644 --- a/src/sections/03_coefficients_via_recursion/coefficients_recursion_derivation.tex +++ b/src/sections/03_coefficients_via_recursion/coefficients_recursion_derivation.tex @@ -6,8 +6,8 @@ \begin{equation*} \sum_{k=1}^{n} k^{p} = \frac{1}{p+1}\sum_{j=0}^{p} \binom{p+1}{j} \bernoulli{j} n^{p+1-j} \end{equation*} -it is very important to note that summation bound is $p$ while binomial coefficient upper bound is $p+1$. -It means that we cannot skip summation bounds unless we do some trick as +it is very important to note that the summation bound is $p$ while binomial coefficient upper index is $p+1$. +It means that we cannot skip summation bounds unless we use some trick as \begin{equation*} \begin{split} \sum_{k=1}^{n} k^{p} @@ -38,7 +38,7 @@ \frac{(-1)^r}{j} \binom{r}{2r-j+1}, & \text{if } j>0. \end{cases}\label{eq:combinatorial-identity} \end{equation} -An elegant proof of the above binomial identity is provided at~\cite{scheuer2023mathstackexchange}. +An elegant proof of the above binomial identity is provided in~\cite{scheuer2023mathstackexchange}. In particular, the equation~\eqref{eq:combinatorial-identity} is zero for $0< t \leq j$. So that taking $j=0$ we have \begin{equation*} @@ -48,7 +48,7 @@ &- \left[ \sum_{t=0}^{r} \binom{r}{t} \frac{(-1)^t}{t+r+1} \bernoulli{t+r+1} n^{r-t} \right] \end{split} \end{equation*} -Now let's simplify the double summation applying the identity~\eqref{eq:combinatorial-identity} +Now let's simplify the double summation by applying the identity~\eqref{eq:combinatorial-identity} \begin{equation*} \begin{split} \sum_{k=1}^{n} k^{r} (n-k)^{r} @@ -83,10 +83,6 @@ &\sum_{r} \coeffA{m}{r} \left[ \frac{1}{(2r+1) \binom{2r}r} n^{2r+1} \right] + 2 \sum_{r} \coeffA{m}{r} \left[ \sum_{d} \frac{(-1)^r}{2r-2d} \binom{r}{2d+1} \bernoulli{2r-2d} n^{2d+1} \right] - n^{2m+1} = 0 -% \\ -% &\sum_{r=0}^{m} \coeffA{m}{r} \left[ \frac{1}{(2r+1) \binom{2r}r} n^{2r+1} \right] -% + 2 \sum_{r=0}^{m} \coeffA{m}{r} \left[ \sum_{d=0}^{(r-1)/2} \frac{(-1)^r}{2r-2d} \binom{r}{2d+1} \bernoulli{2r-2d} n^{2d+1} \right] -% - n^{2m+1} = 0 \end{split}\label{eq:equation7} \end{equation} Taking the coefficient of $n^{2m+1}$ in~\eqref{eq:equation7}, we get From 4b2912c4ca9daed419c042bf970b48ed22c5f8e8 Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Sat, 21 Sep 2024 19:38:18 +0200 Subject: [PATCH 12/13] grammar fix: conclusions --- CHANGELOG.md | 1 + ...ynomialIdentityInvolvingBTandFaulhaber.pdf | Bin 362502 -> 362500 bytes src/sections/05_conclusions/conclusions.tex | 3 +-- 3 files changed, 2 insertions(+), 2 deletions(-) diff --git a/CHANGELOG.md b/CHANGELOG.md index 77e6c70..75753a3 100644 --- a/CHANGELOG.md +++ b/CHANGELOG.md @@ -19,3 +19,4 @@ and this project adheres to [Semantic Versioning v2.0.0](https://semver.org/spec - Grammar fix introduction - Grammar fix section 2 - Grammar fix section 3 +- Grammar fix conclusions diff --git a/out/PolynomialIdentityInvolvingBTandFaulhaber.pdf b/out/PolynomialIdentityInvolvingBTandFaulhaber.pdf index b5ca039265901460d027852f5e25fbe81bd0f444..62450f690d7fa211602803864e848d181534b23f 100644 GIT binary patch delta 5560 zcmai03piD2_jhm%QphEV98wLH_I>ZeAmfrssF9LONp9tmOXYHMT*f6KQFc-Y)x=kF zgf8+CLKmaZ#SFO>O65}0)HMIs{HJ-Qz2|wp^El6T*8A@D{(fuy)_PY$y6TVVs+T^L zWv+mvO{jM%jyB&x5($8Cnry5jcd_AuWYq#Ow8Ad;N}|)=D21nqC)p-M0KR=+|!3Fpwv-&HFEPW6ye917YYHJO3P| z_Rm$5dA7MHYo;HScFrpv7+Pr%Wn?-(P5P0}62WxDr%AEM*~LTq?YM`f%?>Jb48sK- z`ba^(w6^V$_QNCnUb(YZ+Z}Xq2xT!F{M0nGoT-M#Xx%k%j9*bjX@gOSs}$ez2)Av0-LcgBB=RWElas_>qA>8hS|R95sc+_m;uMgH3RZITN% zxa-&JyDPhoay7Pjhd2AB)QSm)lqaV8e|fcdD);;Eiu`t5_elxyQ>{o2+M`()eXUY> z!p^)aCeOzvBXU=j*ab;=Jv16y6bT1g4 zJ{{}Y)+AmLtMJH4(<(=heYh*6F6?Y-!H7VVdv2Lgg&=&ZRJ~rKf^O}k#QW{XO)g0k z{!;Jzs#aXsFNU4W=AYP8vsgS$cdkWTM6XET@-eX4U0d!4&F|II&ps4!WIG+wWIB(( z*5Rq8oO|XdRj9D3!`c=y8q>{5n0H<(qGxzW<(BScd+yt<*HeQYV=kna0yP|!{@CW_ z$cDBGg$AE)#rWK$kXcQ2!}=WA-|X*KtTT0L3bR?d%|hBnG0ijGI#kB)`l6U`hs>b9 z)9R8JFC^zDZ4@cyKPdFjmvcwpL!w6Pn{xwoc@pjFt9AQ(Ez~@YSVL>YvRihHP2ia; 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all these approaches are examined providing examples in the sections~\ref{sec:approach-via-system-of-linear-equations} and~\ref{sec:approach-via-recursion}. -Moreover, to validate the results, -there are supplementary Mathematica programs available at~\cite{kolosov2023github}. +Moreover, to validate the results, supplementary Mathematica programs are available at~\cite{kolosov2023github}. From 87b80bf3291f59dd86d19dc2d662950e8ce14e58 Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Sat, 21 Sep 2024 19:39:42 +0200 Subject: [PATCH 13/13] update changelog --- CHANGELOG.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/CHANGELOG.md b/CHANGELOG.md index 75753a3..c775046 100644 --- a/CHANGELOG.md +++ b/CHANGELOG.md @@ -5,7 +5,7 @@ All notable changes to this project will be documented in this file. The format is based on [Keep a Changelog](https://keepachangelog.com/en/1.0.0/), and this project adheres to [Semantic Versioning v2.0.0](https://semver.org/spec/v2.0.0.html). -## [1.0.0] - In progress +## [1.0.0] - 2024-09-21 - Update CI/CD actions - Update citation

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kolosovpetro Date: Sat, 21 Sep 2024 17:58:25 +0200 Subject: [PATCH 02/13] update changelog --- CHANGELOG.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/CHANGELOG.md b/CHANGELOG.md index b4ebec6..cd854a8 100644 --- a/CHANGELOG.md +++ b/CHANGELOG.md @@ -8,3 +8,5 @@ and this project adheres to [Semantic Versioning v2.0.0](https://semver.org/spec ## [1.0.0] - In progress - Update CI/CD actions +- Update citation +- Update README.md From e9b6c2d00af40a4531ee57959152c8fbf8bc8158 Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Sat, 21 Sep 2024 18:01:55 +0200 Subject: [PATCH 03/13] compile document --- ...ynomialIdentityInvolvingBTandFaulhaber.bbl | 86 ++++++++++++++++++ ...ynomialIdentityInvolvingBTandFaulhaber.blg | 48 ++++++++++ ...ynomialIdentityInvolvingBTandFaulhaber.pdf | Bin 361613 -> 361453 bytes 3 files changed, 134 insertions(+) create mode 100644 out/PolynomialIdentityInvolvingBTandFaulhaber.bbl create mode 100644 out/PolynomialIdentityInvolvingBTandFaulhaber.blg diff --git a/out/PolynomialIdentityInvolvingBTandFaulhaber.bbl b/out/PolynomialIdentityInvolvingBTandFaulhaber.bbl new file mode 100644 index 0000000..273790d --- /dev/null +++ b/out/PolynomialIdentityInvolvingBTandFaulhaber.bbl @@ -0,0 +1,86 @@ +\begin{thebibliography}{10} + +\bibitem{tkaczyk2018problem} +{Tkaczyk, Albert}. +\newblock {About the problem of a triangle developing the polynomial function}. +\newblock Published electronically at + \href{https://www.linkedin.com/pulse/problem-triangle-developing-polynomial-function-fn-nm-albert-tkaczyk/}{\texttt{LinkedIn}}, + 2018. + +\bibitem{tkaczyk2018continuation} +{Tkaczyk, Albert}. +\newblock {On the problem of a triangle developing the polynomial function - + continuation}. +\newblock Published electronically at + \href{https://www.linkedin.com/pulse/problem-triangle-developing-polynomial-function-f-n-m-albert-tkaczyk/}{\texttt{LinkedIn}}, + 2018. + +\bibitem{kolosov2022106} +{Kolosov, Petro}. +\newblock {106.37 An unusual identity for odd-powers}. +\newblock {\em {The Mathematical Gazette}}, 106(567):509--513, 2022. + +\bibitem{abramowitz1988handbook} +Milton Abramowitz, Irene~A Stegun, and Robert~H Romer. +\newblock Handbook of mathematical functions with formulas, graphs, and + mathematical tables, 1988. + +\bibitem{beardon1996sums} +Alan~F Beardon. +\newblock {Sums of powers of integers}. +\newblock {\em {The American mathematical monthly}}, 103(3):201--213, 1996. + +\bibitem{kolosov2017third} +Petro Kolosov. +\newblock {Numerical triangle, row sums give third power, Entry A287326 in The + On-Line Encyclopedia of Integer Sequences}. +\newblock Published electronically at \url{https://oeis.org/A287326}, 2017. + +\bibitem{kolosov2018fifth} +Petro Kolosov. +\newblock {Numerical triangle, row sums give fifth power, Entry A300656 in The + On-Line Encyclopedia of Integer Sequences}. +\newblock Published electronically at \url{https://oeis.org/A300656}, 2018. + +\bibitem{kolosov2018seventh} +Petro Kolosov. +\newblock {Numerical triangle, row sums give seventh power, Entry A300785 in + The On-Line Encyclopedia of Integer Sequences}. +\newblock Published electronically at \url{https://oeis.org/A300785}, 2018. + +\bibitem{alekseyev2018mathoverflow} +{Alekseyev, Max}. +\newblock {MathOverflow answer 297916/113033}. +\newblock Published electronically at + \url{https://mathoverflow.net/a/297916/113033}, 2018. + +\bibitem{scheuer2023mathstackexchange} +{Scheuer, Markus}. +\newblock {MathStackExchange answer 4724343/463487}. +\newblock Published electronically at + \url{https://math.stackexchange.com/a/4724343/463487}, 2023. + +\bibitem{bateman1953higher} +Harry Bateman. +\newblock {\em {Higher transcendental functions [volumes i-iii]}}, volume~1. +\newblock McGRAW-HILL book company, 1953. + +\bibitem{kolosov2018numerator} +Petro Kolosov. +\newblock {Entry A302971 in The On-Line Encyclopedia of Integer Sequences}. +\newblock Published electronically at \url{https://oeis.org/A302971}, 2018. + +\bibitem{kolosov2018denominator} +Petro Kolosov. +\newblock {Entry A304042 in The On-Line Encyclopedia of Integer Sequences}. +\newblock Published electronically at \url{https://oeis.org/A304042}, 2018. + +\bibitem{kolosov2023github} +{Kolosov, Petro}. +\newblock {Polynomial identity involving Binomial Theorem and Faulhaber's + formula Source files}. +\newblock Published electronically at + \url{https://github.com/kolosovpetro/PolynomialIdentityInvolvingBTandFaulhaber}, + 2023. + +\end{thebibliography} diff --git a/out/PolynomialIdentityInvolvingBTandFaulhaber.blg b/out/PolynomialIdentityInvolvingBTandFaulhaber.blg new file mode 100644 index 0000000..55be42f --- /dev/null +++ b/out/PolynomialIdentityInvolvingBTandFaulhaber.blg @@ -0,0 +1,48 @@ +This is BibTeX, Version 0.99d +Capacity: max_strings=200000, hash_size=200000, hash_prime=170003 +The top-level auxiliary file: PolynomialIdentityInvolvingBTandFaulhaber.aux +Reallocating 'name_of_file' (item size: 1) to 6 items. +The 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