Polynomials of binomial type and Lucas' theorem (Q2790907)

The author presents various constructions of sequences of polynomials satisfying the binomial theorem (BT sequences for short) in finite characteristic based on the theory of additive functions. Recall that a sequence \(\{P_i(X)\}\) of polynomials over a field \(F\) is a BT sequence if and only if for all \(n\geq 0\), \(P_n(X+Y)\equiv \sum_0^n\binom{n}{i}P_i(X)P_{n-i}(Y)\). The interest in the subject comes from the measure theory in finite characteristic, where the Carlitz polynomials, which are created out of additive functions via digit expansions, satisfy the binomial theorem as a consequence of Lucas' famous congruence: if \(F\) has characteristic \(p,q\) is a power of \(p\) and \(m=\sum_im_iq^i\), \(n=\sum_jn_jq^j\) are two integers written \(q\)-adically, then \(\binom{m}{n}\equiv\Pi_i\binom{m_i}{n_i}\mod p\).NEWLINENEWLINEIn this note the author shows how the Carlitz approach gives rise to a large subspace of BT sequences which is closed under multiplication in the algebra of divided power series. Moreover, still using additive polynomials, he obtains many other examples of BT sequences which do not arise from the Carlitz method. He discusses various actions on the space of divided elements and how they relate to his constructions. Along the way, he derives a decomposition of the divided power series associated to Dirac measures. It is an open question whether these constructions produce enough BT sequences out of additive functions to generate all of them.