Fixpoint Zsum_digit f k :=
match k with
| O => Z0
| S k => (Zsum_digit f k + f (Z_of_nat k) * Zpower beta (Z_of_nat k))%Z
end.

Theorem Zsum_digit_digit :
forall n k,
Zsum_digit (Zdigit n) k = Z.rem n (Zpower beta (Z_of_nat k)).