M-ary partition polynomials

Let M=(mᵢ)ᵢ₌₀∞ be a sequence of integers such that m₀=1 and mᵢ≥ 2 for i≥ 1. In this paper we study M-ary partition polynomials (pM(n,t))ₙ₌₀∞ defined as the coefficient in the following power series expansion: {align*} ∏ᵢ₌₀∞{1}{1-tq^{Mᵢ}} = ∑ₙ₌₀∞ pM(n,t)qⁿ, {align*} where Mᵢ=∏ⱼ₌₀ⁱmⱼ. In particular, we provide a detailed description of their rational roots and show, that all their complex roots have absolute values not greater than $2$. We also study arithmetic properties of M-ary partition polynomials. One of our main results says that if n=a₀+a₁M₁+⋯ +aₖMₖ is a (unique) representation such that aⱼ∈\0,… ,mⱼ₊₁-1\ for every j, then {align*} pM(n,t)≡ t^{a₀}∏ t^{aⱼ}f(aⱼ+1,t^{mⱼ-1}) {gₖ(t)}, {align*} where f(a,t):=tᵃ-1t-1 and gₖ(t):= (t^m₁+m₂-1f(m₂,t^m₁-1),… ,t^mₖ+mₖ₊₁-1f(mₖ₊₁,t^mₖ-1)). This is a polynomial generalisation of the well-known characterisation modulo m of the sequence of m-ary partition.