Hypersingular integro-differential equation with recurrent relations in coefficients

A new hypersingular integro-differential equation is considered on a closed curve located the complex plane. The refers to linear equations with variable coefficients of special kind. characteristic feature presence constant multipliers in coefficients, given by some recurrent relations. first reduced solving Riemann boundary value problem original curve. class functions established for problem, after which this solved. Next, it necessary solve two differential arbitrary order analytical different regions corresponding fundamental systems solutions are found, method variation constants used solution. Restrictions imposed obtained achieve their analyticity. As result, all resulting solvability conditions written explicitly. solution can be Solved example.