ISSN 1683-3414 (Print)   •   ISSN 1814-0807 (Online)
   Log in


Address: Vatutina st. 53, Vladikavkaz,
362025, RNO-A, Russia
Phone: (8672)23-00-54





DOI: 10.23671/VNC.2019.3.36456

On Transformations of Bessel Functions

Allahverdyan, A. A.
Vladikavkaz Mathematical Journal 2019. Vol. 21. Issue 3.
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.
Language: Russian Download the full text  
For citation: Allahverdyan, A. A. On Transformations of Bessel Functions, Vladikavkaz Math. J., 2019, vol. 21, no. 3, pp. 5-13 (in Russian). DOI 10.23671/VNC.2019.3.36456
+ References

← Contents of issue
  | Home | Editorial board | Publication ethics | Peer review guidelines | Current | Archive | Rules for authors | Send an article |  
© 1999-2023 ёжный математический институт