Let $F$ be any field (or even ring). The following formal power series identity (i.e., equality in $F[[x]]$) holds for any $j \ge 0$:
$(1-x)^{-j} = \sum_{i \ge 0} \binom{i +j -1}{i} x^i $
The other identity is the following: let $F_{p}$ be a finite field with $p$ elements. The following holds for each $1 \neq x \in F_{p}$:
$(1-x)^{-j} = \sum_{i \ge 0}^{p-1} \binom{i +j }{i} x^i$
Are the identities equivalent or related in some way? I feel that the first identity is more formal and concerns equality of coefficients (after multiplying both sides by $(1-x)^j$), yet the second identity is not only formal: $x$ can be plugged in and both sides give an element in the field.
Any insights are welcome - I'm not sure what I'm expecting.
Context: I've encountered the first identity - which is quite useful - while studying generating functions. I've learned of the second identity in a paper by Waterhouse about a matrix with elements $a_{i,j} = \binom{i+j}{i, j}$.