 Positive polynomial recurrence for lattice Eisenstein sums

ID: positive-polynomial-recurrence-for-lattice-eisenstein-sums

℘ (z) = z^{- 2} + \sum_{n \geq 1} c_{n} z^{2 n}

, with

c_{1} = 3 G_{4}

c_{2} = 5 G_{6}

and

c_{n} = (2 n + 1) G_{2 n + 2}

℘^{''} = 6 ℘^{2} - g_{2} /2

. Comparing coefficients of

z^{2 n - 2}

gives the displayed recurrence for

n \geq 3

. Every denominator is positive, so induction expresses every higher lattice Eisenstein series as a polynomial in

G_{4}, G_{6}

with positive rational coefficients on its occurring monomials. In particular

G_{8} = 3 G_{4}^{2} /7

and

G_{10} = 5 G_{4} G_{6} /11

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