SOLUTION OF A SECOND-ORDER POINCARE-PERRON-TYPE EQUATIONAND DIFFERENTIAL EQUATIONS THAT CAN BE REDUCED TO IT

摘要

We give an analytic solution of a second-order difference Poincare–Perron-type equation. This enables usto construct a solution of the differential equation t2(A1t2 + Bit + Ci )u" + t(A2t2 + B2t + C2)u' + (A3t2 + B3t + C3)u = 0 in explicit form. A solution of this equation is expressed in terms of two hypergeometric functions andone new special function. As a separate case, we obtain an explicit solution of the Heun equation anddetermine its polynomial solutions.