Combinatorial congruences and ψ-operators

The ψ-operator for ( ,Γ)-modules plays an important role in the study of Iwasawa theory via Fontaineʹs big rings. In this note, we prove several sharp estimates for the ψ-operator in the cyclotomic case. These estimates immediately imply a number of sharp p-adic combinatorial congruences, one of which extends the classical congruences of Fleck (1913) and Weisman [Some congruences for binomial coefficients, Michigan Math. J. 24 (1977) 141–151].