Sums of squares and sequences of modular forms

Let hₙ (v) be the sequence of rational functions with hₙ (v) v-nhₙ (v) + (n-1) h₍-₁ (v) -vh₍-₁' (v) +v (v (vh₍-₁ (ₕ) ) ') '4=0 for n>0 and h₀ (v) =1. We prove that hₙ (v) has a pole at v=1n if and only if n is a sum of two squares of integers. Moreover, if r₂ (n) =\#\ (a, b) Z²: a²+b²=n\, then we derive the formula v=1/nReshₙ (v) = (-1) ^n-1r₂ (n) n16ⁿ. The results are then generalized to arbitrary modular forms with respect to (2) and as a consequence we obtain a new criterion for Lehmer's conjecture for Ramanujan's -function.