# Kruglov V. E.

### Solution of a Linear Second-Order Differential Equation with Coefficients Analytic in the Vicinity of a Fuchsian Zero Point

Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1381-1393

We obtain a solution of a second-order differential equation with coefficients analytic near a Fuchsian zero point. This solution is expressed via the hypergeometric functions and the fractional-order hypergeometric functions introduced in this paper.

### Construction of a fundamental system of solutions of a linear finite-order difference equation

Ukr. Mat. Zh. - 2009. - 61, № 6. - pp. 777-794

We present an efficient algorithm for the construction of a fundamental system of solutions of a linear finite-order difference equation. We obtain expressions in which all elements of this system are expressed via one of its elements and find a particular solution of an inhomogeneous equation.

### Solution of a second-order Poincaré-Perron-type equation and differential equations that can be reduced to it

Ukr. Mat. Zh. - 2008. - 60, № 7. - pp. 900–917

The analytical solution of the second-order difference Poincare–Perron equation is presented. This enables us to construct in the explicit form a solution of the differential equation $$t^2(A_1t^2 + B_1t + C_1)u'' + t(A_2t^2 + B_2t + C_2)u' + (A_3t^2 + B_3t + C_3)u = 0 $$ The solution of the equation is represented in terms of two hypergeometric functions and one new special function. As a separate case, the explicit solution of the Heun equation is obtained, and polynomial solutions of this equation are found.

### Factorization of matrices of permutation type

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1473–1478

By using the methods of the theory of algebraic functions, we present an explicit construction of the canonical factorization of matrices of permutation type given on an open contour.

### Partial indices of a matrix Riemann problem on the torus

Dmitrieva I. Yu., Kruglov V. E.

Ukr. Mat. Zh. - 1984. - 36, № 2. - pp. 247 - 252

### The number of linearly independent functions which are multiples of a given divisor, and the vanishing of Riemann Θ-function

Ukr. Mat. Zh. - 1975. - 27, № 1. - pp. 101–107

### Analog of the Cauchy kernel and the Riemann boundary problem of a three-sheeted surface of genus two

Ukr. Mat. Zh. - 1972. - 24, № 3. - pp. 351—366