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## Total positivity, Grassmannian and modified Bessel functions

A rectangular matrix is called totally positive, (according to F. R.

Gantmacher and M. G. Krein) if all its minors are positive. A point of a

real Grassmannian manifold G(l,m) of l-dimensional subspaces in Rm is called

strictly totally positive (according to A. E. Postnikov) if one can normalize

its Pl¨ucker coordinates to make all of them positive. The totally positive

matrices and the strictly totally positive Grassmannians, that is, the subsets

of strictly totally positive points in Grassmannian manifolds arise in many

areas: in classical mechanics (see the book of F. R. Gantmacher and M. G.

Krein); in a wide context of analysis, differential equations and probability

theory (see the book of S. Karlin); in physics, for example, in construction of

solutions of the Kadomtsev-Petviashvili (KP) partial differential equation (see

a paper by T. M. Malanyuk, a paper by M. Boiti, F. Pemperini, A. Pogrebkov,

a paper of Y. Kodama, L. Williams). Different problems of mathematics,

mechanics and physics led to constructions of totally positive matrices by many

mathematicians, including F. R. Gantmacher, M. G. Krein, I. J. Schoenberg,

S. Karlin, A. E. Postnikov and ourselves. One-dimensional families of totally

positive matrices whose entries are modified Bessel functions of the first kind

have arisen in our study (in collaboration with S. I. Tertychnyi) of model of

the overdamped Josephson effect in superconductivity and double confluent

Heun equations related to it.

In the present paper we give a new construction of multidimensional families

of totally positive matrices different from the above-mentioned families.

Their entries are again formed by values of modified Bessel functions of the

first kind, but now 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. This yields new multidimensional families of

strictly totally positive points in all the Grassmannian manifolds. These families

represent images of explicit injective mappings of the convex open simplex

{x = (x1, . . . , xl) ∈ Rl | 0 < x1 < · · · < xl} ⊂ Rl to the Grassmannian

manifolds G(l,m), l <m.