On transformations of Bessel functions

Elementary Darboux transformations of Bessel functions are discussed. In Theorem 1 we present an improved version of a general factorization approach which goes back to E. Schrodinger, in terms of the two interrelated linear differential substitutions B1 and B2. The main Theorem 2 deals with the Bessel-Riccati equations. The elementary Darboux transformations are reduced to fraction-rational ones. It is shown that a fixed point of the latter generates the rational in x solutions of Bessel-Riccati equations introduced by Theorem 2. It should be noted that Bessel functions are considered as eigenfunctions Aψ=λψ of the Euler operators A=e2t(D2t+a1Dt+a2) with constant coefficients a1 and a2. This enables one (Lemma 3) to build up asymptotic solutions of the Bessel-Riccati equations in the form of series in inverse powers of the parameter z=kx, k2=λ, x=e-t. It is also shown that these formal series in inverse powers of the spectral parameter k=λ--√ are convergent if the rational solutions of the corresponding Bessel-Riccati equation from Theorem 2 are exist.