Cogrowth series of free products of finite and free groups

Abstract

Let A={a₁,…,a_n,a₁⁻¹,…,a_n⁻¹} and iteration of A denoted by A^{[starf ]} to be the set of words in A (including the empty word). Let S⊆A^{[starf ]}; then the growth function of the set S is the function Γ(l)=number of words in S of length l. For m≤n let $\vec {a}$=(a_i1,…,a_im), where i_k∈{1,…,n} are different; then the relative growth function with respect to $\vec {a}$ is the function Γ_{$\vec {a}$}(l,l₁,…,l_m)=number of words in S of length l+l₁+…+l_m having (for each k) l_k total occurrences of a_ik and a_ik⁻¹.