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 Vladikavkaz. Mat. Zh., 2019, Volume 21, Number 3, Pages 5–13 (Mi vmj695)

On transformations of Bessel functions

A. A. Allahverdyan

Adyghe State University, 208 Pervomayskaya St., Maikop 385000, Russia

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. Schrödinger, 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 $A=e^{2t}(D_t^2+a_1D_t+a_2)$ 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.

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

DOI: https://doi.org/10.23671/VNC.2019.3.36456

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UDC: 517.95
MSC: 34K08

Citation: A. A. Allahverdyan, “On transformations of Bessel functions”, Vladikavkaz. Mat. Zh., 21:3 (2019), 5–13

Citation in format AMSBIB
\Bibitem{All19} \by A.~A.~Allahverdyan \paper On transformations of Bessel functions \jour Vladikavkaz. Mat. Zh. \yr 2019 \vol 21 \issue 3 \pages 5--13 \mathnet{http://mi.mathnet.ru/vmj695} \crossref{https://doi.org/10.23671/VNC.2019.3.36456} \elib{https://elibrary.ru/item.asp?id=40874231} 

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This publication is cited in the following articles:
1. A. A. Allahverdyan, A. B. Shabat, “Products of eigenfunctions and Wronskians”, Ufa Math. J., 12:2 (2020), 3–9
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