Schur specialization by a generating function
= Schur specialization by a generating function
{c}
{title2=$s_\lambda^F=\det[f_{\lambda_i-i+j}]$}
Given $F(t)=\sum_{r\ge0}f_rt^r$ with $f_0=1$, put $f_r=0$ for $r<0$. The <Cauchy identity for symmetric functions> and <Jacobi–Trudi identity> give $\prod_iF(t_i)=\sum_\lambda s_\lambda^Fs_\lambda(t)$, degree by degree in the completion. For $F(t)=\prod_j(1-x_jt)^{-1}$ the coefficients are $s_\lambda(x)$; for $F(t)=\prod_j(1+x_jt)$ they are $s_{\lambda'}(x)$.