Abstract:
In this paper, we discuss the problem
of the number of representations of
positive integers as sums of triangular
numbers.
The method we use is similar to Rankin's
way in studying the sum of squares
representation of positive integers.
We decompose the theta function
$q^{k}\psi ^{4k}(q)\psi ^{2k}({q^2})$
into an Eisenstein series and a cusp form
to give an asymptotic formula for
$t_{4k,2k}(n)$.
Moreover, we obtain concrete
formulas for
$k = 2,4$,
respectively, by using a linear combination
of the divisor function and the coefficient
of an
$\eta$-product.
