# Do any specializations of variables give valid equalities of series and products involving Witt symmetric functions?

Formally, Witt symmetric functions $w_n(x_1,x_2,...)$ ($n\geqslant1$) can be defined by $$\prod_n(1-w_nt^n)=1+\sum_k(-1)^ke_kt^k=\prod_j(1-x_jt),$$ where $e_k(x_1,x_2,...)$ are the elementary symmetric functions.

Taken literally, if the series in the middle or the product on the right converges in some area giving an analytic function $f(t)$ with zeros $r_j=1/x_j$ (or maybe even both converge and give the same function), the above equality makes one expect that to this function in some sense corresponds another one, with zeros in vertices of regular $n$-gons inscribed in circles of radii $|w_n(x_1,x_2,...)|^{-\frac1n}$ centered at the origin. Maybe even in some cases these functions may coincide.

Can this actually happen? Does the above formal equality correspond to some analytic fact when the zeros just cannot coincide?

Let us consider a concrete example, $x_j=\frac1{j^2}$, $f(t)=\frac{\sin(\pi z)}{\pi z}$ (where $z^2=t$). What does one obtain on the left?

Found later

In "On symmetric functions related to Witt vectors and the free Lie algebra" Reutenauer states Schur positivity conjecture for Witt symmetric functions (later solved by several people); as a consequence, the numbers $d_n$ determined by $$e^{-t}=\prod_{n\geqslant1}(1+\frac{d_n}{n!}t^n)$$ come out integers, positive for $n>1$, being degrees of representations of symmetric groups corresponding to $-w_n$. These numbers begin $-1$, $1$, $2$, $9$, $24$, $130$, $720$, $8505$, $35840$, $412776$, $3628800$,... (sequence A006973 on OEIS); for $n$ prime, $d_n=(n-1)!$, but I could not find any explicit expression for general $n$. So another (not quite particular case but still obviously closely related) question - where does the product converge, and where does it actually give $e^{-t}$?