Word Count : 326

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To solve this, we are asked to: 1. Express $5^x$ in terms of falling factorials. 2. Evaluate the Stirling transform $mS_5$ for $m = 0,1,2,3,4,5$, i.e., compute: $$ mS_5 = \sum_{k=0}^5 S(5,k) \cdot m^{\underline{k}} $$ where $S(n, k)$ is the Stirling number of the second kind, and $m^{\underline{k}}$ is the falling factorial defined as: $$ m^{\underline{k}} = m(m-1)(m-2)\dots(m-k+1) $$ --- ### ✅ Step 1: Express $5^x$ in terms of falling factorials We use the identity: $$ a^x = \sum_{k=0}^{x} S(x, k) ___ ________ _____ ________ _________ _____ ______.
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