Section alpha_nat_coprime.
Let A_eq_3_12 n : exists u v, u*α (S n) + v*α n = 1.
End alpha_nat_coprime.
End Pell_inner.
(* ⊕  ⊗  ∸    ⊞ ⊠ ⊟ *)
Notation expoZ := (mscal Zmult 1).
Fact expoZ_power n x : expoZ n (Z.of_nat x) = Z.of_nat (power n x).
Fact mscal_Zplus n : mscal Zplus 0 n 1 = Z.of_nat n.
Notation "∑" := (msum MZ_plus MZ_zero).

Theorem MA_expo_A_binomial m k l :
(m = l * k)%nat
-> A m = ∑ (S l) (fun i => MZ_scal ( expoZ (l-i) (-1)
* Z.of_nat (binomial l i)
* expoZ i (α (S k))
* expoZ (l-i) (α k) )
(MZ_expo i B)).