Let $\alpha\in \mathbb{Z}_p$ be an $p$-adic integer and define for $n\in \mathbb{Z}_{\geq 0}$ ${\alpha\choose n} := \frac{\alpha(\alpha-1)\cdot\ldots\cdot(\alpha-n+1)}{n!}.$
This is again a $p$-adic integer, and we can define $\alpha_1:=\sum_{n=0}^\infty \overline{{\alpha \choose n}} p^n$ and $\alpha_2:=\sum_{n=0}^\infty \overline{{\alpha \choose p^n}} p^n,$
where $\overline{(\cdot)}$ means reduction modulo $p$. How are $\alpha,\alpha_1,\alpha_2$ related? Are there formulas expressing this relation?
Thanks a lot!
Edit: Some thinking led me to the following conclusion: Using continuity of $x\mapsto {x\choose p^n}$ as a function $\mathbb{Z}_p\rightarrow \mathbb{Q}_p$ and Lucas' Theorem it follows that $\alpha=\alpha_2$. Does this seem correct?