  Elementary transformations of equations $A\psi=\lambda\psi$ are considered.
The invertibility condition (Theorem 1) is established and similar
transformations of Riccati equations in the case of second order differential
operator $A$ are constructed (Theorem 2). Applications to continuous fractions
for Bessel functions and Chebyshev polynomials are established. It is shown
particularly that the elementary solutions of Bessel equations are related to a
fixed point transformations of Riccati equations.
