Strong Frobenius structures associated with q-difference operators

Abstract 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 $$L_q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:math> has a strong Frobenius structure for a prime p with period h and if L is the p -adic differential operator obtained from $$L_q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:math> 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 $$f(q,z)\in {\mathbb {Z}}[q][[z]]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> <mml:mo>∈</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>[</mml:mo> <mml:mi>q</mml:mi> <mml:mo>]</mml:mo> <mml:mo>[</mml:mo> <mml:mo>[</mml:mo> <mml:mi>z</mml:mi> <mml:mo>]</mml:mo> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> 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.