On Transformations of Bessel Functions

Allahverdyan, A. A.
Vladikavkaz Mathematical Journal 2019. Vol. 21. Issue 3.

Abstract:
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 \(B_1\) and \(B_2\). 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 \(A\psi=\lambda\psi\) of the Euler operators \(A=e^{2t}\left(D_t^2+a_1D_t+a_2\right)\) with constant coefficients \(a_1\) and \(a_2\). 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 \(z=kx\), \(k^2=\lambda\), \(x=e^{-t}\). It is also shown that these formal series in inverse powers of the spectral parameter \(k=\sqrt \lambda\) are convergent if the rational solutions of the corresponding Bessel-Riccati equation from Theorem 2 are exist.

Keywords: Bessel functions, invertible Darboux transforms, continued fractions, Euler operator, Riccati equation.

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