# 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?