<p>We study the properties of the family of functions of complex variable encountered in finding the coefficients of expansion in the trigonometric system of functions of the solutions of Helmholtz equation in a cylindrical coordinate system in the form of homogeneous polynomials in two biorthogonal systems of functions. We construct associated functions biorthogonal to these functions on closed curves of the complex plane and establish sufficient conditions for the expansions of analytic functions in series in the analyzed system of functions. The application of biorthogonal systems of functions to the construction of solutions of some boundary-value problems for the Helmholtz equation in cylindrical coordinate systems is discussed.</p>