A partition formula for the integer coefficients of the theta function nome

Abstract:

In elliptic function theory, the nome q can be given as a power series in $\varepsilon$ with integer coefficients, $q = {\Sigma _{n \geqslant 0}}{\delta _n}{\varepsilon ^{4n + 1}}$. Heretofore, the first 14 coefficients were calculated with considerable difficulty. In this paper, an explicit and general formula involving partitions is given for all the ${\delta _n}$. A table of the first 59 of these integers is given. The table is of number-theoretical interest as well as useful for calculating complete and incomplete elliptic integrals.