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

Abstract

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.