Hirzebruch functional equation: classification of solutions

The Hirzebruch functional equation is $\sum _{i=1}^n\prod _{j\ne i} (1/f(z_j-z_i))=c$ with constant $c$ and initial conditions $f(0)=0$ and $f'(0)=1$. In this paper we find all solutions of the Hirzebruch functional equation for $n\leq 6$ in the class of meromorphic functions and in the class of series. Previously, such results have been known only for $n\leq 4$. The Todd function is the function determining the two-parameter Todd genus (i.e., the $\chi _{a,b}$-genus). It gives a solution to the Hirzebruch functional equation for any $n$. The elliptic function of level $N$ is the function determining the elliptic genus of level $N$. It gives a solution to the Hirzebruch functional equation for $n$ divisible by $N$. A series corresponding to a meromorphic function $f$ with parameters in $U\subset \mathbb C^k$ is a series with parameters in the Zariski closure of $U$ in $\mathbb C^k$, such that for the parameters in $U$ it coincides with the series expansion at zero of $f$. The main results are as follows: (1) Any series solution of the Hirzebruch functional equation for $n=5$ corresponds either to the Todd function or to the elliptic function of level $5$. (2) Any series solution of the Hirzebruch functional equation for $n=6$ corresponds either to the Todd function or to the elliptic function of level $2$, $3$, or $6$. This gives a complete classification of complex genera that are fiber multiplicative with respect to $\mathbb C\mathrm P^{n-1}$ for $n\leq 6$. A topological application of this study is an effective calculation of the coefficients of elliptic genera of level $N$ for $N=2,\dots ,6$ in terms of solutions of a differential equation with parameters in an irreducible algebraic variety in $\mathbb C^4$.