Polynomial identities involving Pascal's triangle rows
Résumé
In this short report we consider the famous binomial identity
\[
2^n = \sum_{k=0}^{n}\binom{n}{k}
\]
Based on it, the following binomial identities are derived
\begin{gather*}
m^n = \sum_{k=0}^{n} \sum_{j=0}^{k} \binom{n}{k} \binom{k}{j} (-1)^{k-j} m^j,\\
m^n = \sum_{k=0}^{n} \sum_{j=0}^{k} \binom{n}{j} \binom{n-j}{k-j} (-1)^{k-j} m^j,\\
\end{gather*}
where $\binom{n}{k}$ are binomial coefficients and $(m, \ n)$ are non-negative integers.
Fichier principal
PolynomialIdentitiesInvolvingPascalsTriangleRows.pdf (276.22 Ko)
Télécharger le fichier
Origine | Fichiers produits par l'(les) auteur(s) |
---|