Mathematical Notes

, Volume 98, Issue 5–6, pp 714–724 | Cite as

On a remarkable sequence of Bessel matrices

  • V. M. Bukhshtaber
  • S. I. Tertychnyi


A sequence of matrices whose elements are modified Bessel functions of the first kind is considered. Such a sequence arises when studying certain ordinary linear homogeneous second-order differential equations belonging to the family of double confluent Heun equations. The conjecture that these matrices are nonsingular is discussed together with its application to the problem of the existence of solutions analytic at the singular point of the equation referred to above.


modified Bessel function of the first kind double confluent Heun equation Bessel matrix Josephson junction Laurent series Hessenberg matrix 


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  1. 1.
    Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Ed. by M. Abramowitz and I. Stegun (Dover Publications, New York, 1972; Nauka, Moscow, 1979).zbMATHGoogle Scholar
  2. 2.
    V.M. Bukhshtaber and S. I. Tertychnyi, “Holomorphic solutions of double confluentHeun equation associated with the RSJ-model of the Josephson junction” Teoret.Mat. Fiz. 182 (3), 373–404 (2015).MathSciNetCrossRefGoogle Scholar
  3. 3.
    G. Watson, A Treatise on the Theory of Bessel Functions (Cambridge Univ. Press, Cambridge, 1945; Inostr. Lit., Moscow, 1949), Vol. 1.Google Scholar
  4. 4.
    D. Schmidt and G. Wolf, “Double confluent Heun equation,” in Heun’s Differential Equations, Oxford Sci. Publ., Ed. by A. Ronveaux (Oxford Univ. Press, Oxford, 1995), Part C.Google Scholar
  5. 5.
    S. Slavyanov and V. Lai, Special Function Unified Theory Based on the Analysis of Singularities (Izd. “Nevskii Dialekt”, St. Petersburg., 2002) [in Russian].Google Scholar
  6. 6.
    V. M. Bukhshtaber and S. I. Tertychnyi, “Explicit solution family for the equation of the resistively shunted Josephson junction model,” Teoret. Mat. Fiz. 176 (2), 163–188 (2013) [Theoret. and Math. Phys. 176 (2), 965–986 (2013)]; The Modelling a Josephson Junction and Heun Polynomials, arXiv: Scholar
  7. 7.
    A. I.Markushevich, Theory of Analytic Functions (Nauka, Moscow, 1967), Vol. 1.Google Scholar
  8. 8.
    R. A. Horn and Ch. R. Johnson, Matrix Analysis (Cambridge Univ. Press, London, 1986; Mir, Moscow, 1989).zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia
  2. 2.All-Russia Scientific Research Institute of Physical and Radio Engineering MeasurementsMendeleevoRussia

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