New modular equations in the spirit of Ramanujan

We give proofs for new Ramanujan-type modular equations discovered by M. Somos and establish applications of some of them.     

Iterates of increasing sequences of positive integers

It is proved that for fixed integers ${K \ge 2, D \ge 2, {\rm if} (D - 1) \mid K}$ , then there exists a unique increasing sequence (a(n)) _{n ≥ K} of positive integers such that $$\underset{K {\rm times}}{\underbrace{a ( a ( \dots a(a}}(n)) \dots)) = Dn$$ otherwise, there are uncountably many increasing sequences of positive integers (a(n)) satisfying this iterated functional equation. This generalizes recent results of Propp and Allouche–Rampersad–Shallit.