Total positivity, Grassmannian and modified Bessel functions

Abstract

A rectangular matrix is called totally positive, if all its minors are positive. A point of a real Grassmanian manifold $G_{l,m}$ of $l$-dimensional subspaces in $\mathbb R^m$ is called strictly totally positive, if one can normalize its Pl\"ucker coordinates to make all of them positive. The totally positive matrices and the subsets of strictly totally positive points in Grassmanian manifolds arise in many domains of mathematics, mechanics and physics. F.R.Gantmacher and M.G.Krein considered totally positive matrices in the context of classical mechanics. Total positivity was used for construction of solutions of the Kadomtsev-Petviashvili (KP) partial differential equation by T.M.Malanyuk, M.Boiti, F.Pemperini, A.Pogrebkov, Y.Kodama, L.Williams. Different problems of mathematics, mechanics and physics led to constructions of totally positive matrices due to many mathematicians, including F.R. Gantmacher, M.G.Krein, I.J.Schoenberg, S.Karlin, A.E.Postnikov and ourselves. In our case totally positive matrices were constructed for solution of problems on model of the overdamped Josephson effect in superconductivity and double confluent Heun equations. In our previous paper we have proved that certain determinants formed by modified Bessel functions of the first kind are positive on the positive semi-axis. In the present paper we give a new result: a construction of multidimensional families of totally positive matrices formed by values of modified Bessel functions with non-negative integer indices. Their columns are numerated by the indices of the modified Bessel functions, and their rows are numerated by their arguments.