  In this paper we study certain real functions defined in a very simple way by
Zagier as sums of infinite powers of quadratic polynomials with integer
coefficients. These functions give the even parts of the period polynomials of
the modular forms which are the coefficients in Fourier expansion of the kernel
function for Shimura-Shintani correspondence. We prove two conjectures of
Zagier showing that the sums converge exponentially. We also prove unexpected
results on the representation of these functions as sums over simple or reduced
quadratic forms and the positive or negative continued fraction of the
variable. These arise from more general results on polynomials of even degree.
Especially we give the even part of the Eichler integral on a real number x of
any cusp form for PSL(2,Z) in terms of the even part of its period polynomial
and the continued fraction of x.

Number Theory