A Unified Framework for $p$-adic Congruences and $p$-adic Interpolations of Sequences

In this paper, we treat the problem of determining when a sequence $\{c_n\}_{n=0}^{\infty}$ with the exponential generating function $F(t)$ satisfies the congruence $c_{n+(p^h-1)p^k}\equiv c_n \pmod{p^{k+r}}$ for large $n$ and when its canonical interpolating functions are locally analytic. This typ