Planar rooted trees and non-associative exponential series

Abstract

The grafting of finite planar reduced rooted trees is a system of m-ary operations, m⩾2, on the set PRT of those trees. We consider power series with rational coefficients in a single variable x where the set of monomials is {1} ∪ ̇ PRT , 1 is the empty tree and x denotes the tree with a single node. Grafting induces a string (· m ) m⩾2 of m-ary multiplications on this power series algebra Q{{x}} ∞ . We show that for any natural number k⩾2 there is a unique power series exp k ( x) ∈ Q{{x}} ∞ such that exp k(x) k= exp k(kx) and exp k (0)=exp′ k (0)=1, where exp′ k ( x) is the formal derivative of exp k ( x) with respect to x. We suggest to call it the k-ary non-associative exponential series. It follows that exp′ k(x)= exp k(x). Also it is shown that there is a unique generic exponential exp( q, x) where the coefficients are in the field Q(q) of rational functions in a single variable q over Q such that exp(k,x)= exp k(x) if k∈ N ⩾2. The limit lim q→∞ exp( q, x) exists and is equal to the classical-looking series ∑ m=0 ∞ x m m! ∈ Q{{x}} ∞. For any tree t∈ PRT the coefficient coeff t exp(q,x) of exp( q, x) relative to t is of the form [t] [n−1]! where n=deg( t), [ n−1]! is the q-factorial of n−1 and [ t] is a polynomial in q for which a closed formula is derived. As the topic of this paper is in the meeting point of combinatorics of trees and non-associative algebra, one can expect further interesting cross-relations.