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 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 André 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 for infinitely many prime numbers.
