ag.algebraic geometry – Todd polynomials

Let $T_k(x_1,ldots,x_n)$ be the Todd polynomials, $e_k(x_1,ldots,x_n)$ the elementary symmetric polynomials and $p_k(x_1,ldots, x_n)$ the power sums of degree $k$.

We have the following generating formulas
begin{align*}
sum_{kgeq 0}T_k(x_1,ldots,x_n)t^k = prod_{i=1}^nfrac{tx_i}{1-e^{-tx^i}},,\
sum_{kgeq 0}e_k(x_1,ldots,x_n)t^k = prod_{i=1}^n(1+tx_i),,\
sum_{kgeq 0}frac{1}{k!}p_k(x_1,ldots,x_n)t^k = sum_{i=1}^ne^{tx_i},.
end{align*}

There is an explicit relation between $(1/k!)p_k$ and $e_k$ in terms of Newton’s identities which can be expressed using generating series (see e.g. wikipedia). Is there some similar expression for $T_k$ in terms of $e_k$ or $p_k$?

For example if $X$ is a hyperkähler complex manifold. Replacing $x_i$ with $alpha_i$ the roots of $c(TX)$, one can show that (see (3.13)):
$$
td(X) = text{exp}Big(-2sum_{ngeq0} b_{2n}text{ch}_{2n}(TX)Big),,
$$

where $b_{2n}$ are the modified Bernoulli numbers. Is there possibly even a less explicit formula without any assumptions on the geometry?