On determinants of modified Bessel functions and entire solutions of double confluent Heun equations

We investigate the question on existence of entire solutions of well-known linear differential equations that are linearizations of nonlinear equations modeling the Josephson effect in superconductivity. We consider the modified Bessel functions I_j(x) of the first kind, which are Laurent series coefficients of the analytic function family ${{\text{e}}^{\frac{x}{2}\left(z+\frac{1}{z}\right)}}$. For every $l\geqslant 1$ we study the family parametrized by $k,n\in {{\mathbb{Z}}^{l}}$, ${{k}_{1}}>\cdots >{{k}_{l}}$, ${{n}_{1}}>\cdots >{{n}_{l}}$ of $(l\times l)$-matrix functions formed by the modified Bessel functions of the first kind ${{a}_{ij}}(x)={{I}_{{{k}_{j}}-{{n}_{i}}}}(x)$, $i,j=1,\ldots,l$. We show that their determinants f_{k, n}(x) are positive for every $l\geqslant 1$, $k,n\in {{\mathbb{Z}}^{l}}$ as above and x  >  0. The above determinants are closely related to a sequence (indexed by l) of families of double confluent Heun equations, which are linear second order differential equations with two irregular singularities, at zero and at infinity. Buchstaber and Tertychnyi have constructed their holomorphic solutions on $\mathbb{C}$ for an explicit class of parameter values and conjectured that they do not exist for other parameter values. They have reduced their conjecture to the second conjecture saying that if an appropriate second similar equation has a polynomial solution, then the first one has no entire solution. They have proved the latter statement under the additional assumption (third conjecture) that ${{f}_{k,n}}(x)\ne 0$ for $k=(l,\ldots,1)$, $n=(l-1,\ldots,0)$ and every x  >  0. Our more general result implies all the above conjectures, together with their corollary for the overdamped model of the Josephson junction in superconductivity: the description of adjacency points of phase-lock areas as solutions of explicit analytic equations.