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If $$z\in {\mathbb {C}}$$ and $$1\le n$$ is a natural number then $$\begin{aligned} \sum _{d_1 d_2 =n} (1-z^{p_1})\cdots (1-z^{p_m}) z^{q_1 e_{1}+\cdots +q_i e_{i} }=1, \end{aligned}$$ where $$d_1=p_1^{r_1}\dots p_m^{r_m }$$ , $$d_2=q_1^{e_1}\dots q_i^{e_i }$$ are the prime decompositions of $$d_1, d_2$$ . This is one of the identities involving arithmetic functions that we prove using ideas from the paper of Dressler and van de Lune [3].

If fn is a free parameter, we give a combinatorial closed form solution of the recursion (n + 1)2un+1 − fnun − n 2un−1 = 0, n ≥ 1, and a related generating function. This is used to give a solution to the Apéry type sequence rnn3 + rn−1nαn3 −3α2n+α+ 2θon − θo+ rn−2(n − 1)3 = 0, n ≥ 2, for certain parameters α, θ. We show from another viewpoint two independent solutions of the last recursion related to certain modular forms associated with a problem of conformal mapping: Let f(τ) be a conformal map of a zero-angle hyperbolic quadrangle to an open half plane with values 0, ρ, 1, ∞ (0 < ρ < 1) at the cusps and define t = t(τ) := 1ρf(τ)f(τ)−ρ(τ)−1. Then the function E(τ) = 12πif0(τ)f(τ)11 −f(τ)ρ is a solution, as a generating function in the variable t, of the above recurrence. In other words, E(τ) = r0 +r1t+r2t 2 +. . . , where r0 = 1, r1 = −θ, α = 2− 4ρ. Fil: Panzone, Pablo Andres. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentina

We prove that the plane can be tiled with equilateral triangles and regular hexagons of integer sides using exactly one of each family. Fil: Panzone, Pablo Andres. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentina