MATH-59

CHANGELOG.md

@@ -9,6 +9,12 @@ and this project adheres to [Semantic Versioning v2.0.0](https://semver.org/spec
 
 ### Changed
 
+- Formulae fix
+
+## [1.0.2] - 2024-09-23
+
+### Changed
+
 - Readability improvements
 - Update keywords
 - Add source files link

src/sections/03-introduction-and-main-results.tex

@@ -60,10 +60,6 @@
 \end{align*}
 From the other side, the theorem~\eqref{thm_odd_power_by_macaulays_convolution_strict} provides an odd-power
 polynomial identity as follows
-\begin{equation*}
-    x^{2m+1} = \sum_{r=0}^{m} \coeffA{m}{r} \sum_{k=1}^{x} k^r (x-k)^r
-\end{equation*}
-so that
 \begin{equation*}
     -1 + x^{2m+1} = \sum_{r=0}^{m} \coeffA{m}{r} \convPower{n}{r}{x}
     = \sum_{r=0}^{m} \coeffA{m}{r} \sum_{k=1}^{x-1} k^r (x-k)^r
@@ -83,17 +79,17 @@
 \begin{equation*}
     \begin{split}
     (x-2a)
-        ^{2m+1} + 1 = \sum_{r=0}^{m} \coeffA{m}{r} ((t-k)^r \ast (t-k)^r)[x] \\
-        = \sum_{r=0}^{m} \coeffA{m}{r} \sum_{k=a}^{x-a-1} (k-a)^r (x-k-a)^r
+        ^{2m+1} + 1 &= \sum_{r=0}^{m} \coeffA{m}{r} ((t-k)^r \ast (t-k)^r)[x] \\
+                    &= \sum_{r=0}^{m} \coeffA{m}{r} \sum_{k=a}^{x-a} (k-a)^r (x-k-a)^r
     \end{split}
 \end{equation*}
 Similarly, the following binomial holds.
 For every $n > 0$
 \begin{equation*}
     \begin{split}
     (x-2a)
-        ^{2m+1} - 1 = \sum_{r=0}^{m} \coeffA{m}{r} ((t-k)^r \ast (t-k)^r)[x] \\
-        = \sum_{r=0}^{m} \coeffA{m}{r} \sum_{k=a+1}^{x-a-1} (k-a)^r (x-k-a)^r
+        ^{2m+1} - 1 &= \sum_{r=0}^{m} \coeffA{m}{r} ((t-k)^r \ast (t-k)^r)[x] \\
+                    &= \sum_{r=0}^{m} \coeffA{m}{r} \sum_{k=a+1}^{x-a-1} (k-a)^r (x-k-a)^r
     \end{split}
 \end{equation*}
 This manuscript does not contain any historical context about the polynomial $\polynomialP{m}{b}{x}$,

src/sections/07-relation-between-binomial-theorem-and-discrete-convolution.tex

@@ -61,17 +61,17 @@
 \begin{equation}
     \label{eq:parametric-identity}
     \begin{split}
-        (x-2a)^{2m+1} + 1 = \sum_{r=0}^{m} \coeffA{m}{r} ((t-k)^r \ast (t-k)^r)[x] \\
-                          = \sum_{r=0}^{m} \coeffA{m}{r} \sum_{k=a}^{x-a-1} (k-a)^r (x-k-a)^r
+        (x-2a)^{2m+1} + 1 &= \sum_{r=0}^{m} \coeffA{m}{r} ((t-k)^r \ast (t-k)^r)[x] \\
+                          &= \sum_{r=0}^{m} \coeffA{m}{r} \sum_{k=a}^{x-a} (k-a)^r (x-k-a)^r
     \end{split}
 \end{equation}
 Similarly, the following binomial holds.
 For every $n > 0$
 \begin{equation}
     \label{eq:parametric-identity-strict}
     \begin{split}
-        (x-2a)^{2m+1} - 1 = \sum_{r=0}^{m} \coeffA{m}{r} ((t-k)^r \ast (t-k)^r)[x] \\
-                          = \sum_{r=0}^{m} \coeffA{m}{r} \sum_{k=a+1}^{x-a-1} (k-a)^r (x-k-a)^r
+        (x-2a)^{2m+1} - 1 &= \sum_{r=0}^{m} \coeffA{m}{r} ((t-k)^r \ast (t-k)^r)[x] \\
+                          &= \sum_{r=0}^{m} \coeffA{m}{r} \sum_{k=a+1}^{x-a-1} (k-a)^r (x-k-a)^r
     \end{split}
 \end{equation}
 To validate equations~\eqref{eq:parametric-identity} and~\eqref{eq:parametric-identity-strict}