6 problems

Spectral-polynomial identity. The two polynomials are equal:

Let $Q(E)$ be the polynomial whose zeros characterize those values of $E$ for which the equation in the preceding theorem has a solution in the space ${\mathcal F}$, and let…

Let $q$ be the nome in the series expansions … where the functions $\mathcal{P}^{(\ell)}_n(z)$ and coefficients $\mathcal{E}^{(\ell)}_n$ arise in the constructed solutions of the n…

The non-stationary Lamé equation is a special-function model arising in integrable systems and mathematical physics, and the non-stationary Heun equation is its broader elliptic co…

Consider a generalized Heun equation … where $\deg Q_{k+1}(z)=k+1$ and $\deg Q_i(z)\leq i$ for $i=2,3,\ldots,k$. For each sufficiently large positive integer $n$, let $\mu_n$ be th…

Let $Q(z)$ have roots $a_1,a_2,a_3$, and let $\mu$ be the limiting root-counting measure from the Shapiro–Tater conjecture. For $i\in\{1,2,3\}$, let $\gamma_i$ be the curve consist…