Solving the hypergeometric system of Okubo type in terms of a certain generalized hypergeometric function

Abstract

We introduce one scalar function f of a complex variable and finitely many parameters, which allows to represent all solutions of the so-called hypergeometric system of Okubo type under the assumption that one of the two coefficient matrices has all distinct eigenvalues. In the simplest non-trivial situation, f is equal to the hypergeometric function, while in other more complicated cases it is related, but not equal, to the generalized hypergeometric functions. In general, however, this function appears to be a new higher transcendental one. The coefficients of the power series of f about the origin can be explicitly given in terms of a generalized version of the classical Pochhammer symbol, involving two square matrices that in general do not commute. The function can also be characterized by a Volterra integral equation, whose kernel is expressed in terms of the solutions of another hypergeometric system of lower dimension.

Promoter-mediated Transcriptional Dynamics

“…10 at steady state is also solvable. In some cases, the steady-state distributions can be expressed by confluent hypergeometric functions (67)(68)(69)(70). Refer to the Supporting Material.…”

On the Solutions of Okubo-Type Systems

“…have been studied extensively in the literature [1][2][3][4][5][6][7][8][9][10][11][12]. In [1], the authors study system [13] under the assumption that A 0 has all distinct eigenvalues and introduce one scalar function that allows representing all solutions of the system. is function is believed to be a new higher transcendental function and it satisfies a Volterra integral equation.…”

“…Let A 1 (t) � (a ij (t)) n i,j�1 and let us assume that a 11 (t) ≡ 0. For the sake of simplicity, we write the matrices involved in [1] in the form,…”

“…It is shown in [13] that every linear (scalar) differential equation with a finite number of distinct regular singularities and one irregular singularity can be reduced to system of the form [1] with λ i ≠ λ j . For instance, let us consider system [1] with…”

“…. , y n (t)) T of [1] which is holomorphic at the origin and we are interested in y 1 (t). We first study the reduced Okubo-type system and find its fundamental solution (see eorem 1).…”