Endomorphisms of the binomial coalgebra

The algebra of formal Hurwitz series has been recently investigated in view to potential applications to differential algebra. In the case where the ground ring is a reduced ring of prime characteristic, this paper describes continuous endomorphisms of the algebra of Hurwitz series—or equivalently endomorphisms of the binomial coalgebra. This is done by investigating the coradical filtration of this coalgebra and components of linear maps between some graded modules that naturally arise in the problem. Our problem is tantamount to that of determining all polynomial sequences of binomial type. As such, it contains the study of sequences built by Carlitz in the context of his discovery of the Carlitz module.