Strong Frobenius structures associated with q-difference operators

The notion of strong Frobenius structure is classically studied in the theory of $p$-adic differential operators. In the present work, we introduce a new definition of the notion of strong Frobenius structure for $q$-difference operators. The relevance of this definition is supported by two main results. The first one deals with \emph{confluence}. We show that if the $q$-difference operator $L_q$ has a strong Frobenius structure for a prime $p$ with period $h$ and if $L$ is the $p$-adic differential operator obtained from $L_q$ by letting $q$ tend to 1, then $L$ has a strong Frobenius structure for $p$ with period $h$. The second one deals with congruence modulo cyclotomic polynomials. We show that if $f(q,z)\in\mathbb{Z}[q][[z]]$ is a solution of a $q$-difference operator having strong Frobenius structure for $p$ then $f(q,z)$ satisfies some congruences modulo the $p$-th cyclotomic polynomial. Another definition of strong Frobenius structures associated with $q$-difference operators has been introduced by Andr\'e and Di Vizio and we also point out why their definition is not suitable for our applications: confluence and congruence modulo cyclotomic polynomials. Finally, we show that some $q$-hypergeometric operators of order 1 have a strong Frobenius strong for infinitely many primes numbers.