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.
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