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TMF, 1999, Volume 119, Number 1, Pages 3–19 (Mi tmf723)  

This article is cited in 5 scientific papers (total in 5 papers)

Structural theory of special functions

S. Yu. Slavyanov

Saint-Petersburg State University

Abstract: A block diagram is suggested for classifying differential equations whose solutions are special functions of mathematical physics. Three classes of these equations are identified: the hypergeometric, Heun, and Painlevé classes. The constituent types of equations are listed for each class. The confluence processes that transform one type into another are described. The interrelations between the equations belonging to different classes are indicated. For example, the Painlevé-class equations are equations of classical motion for Hamiltonians corresponding to Heun-class equations, and linearizing the Painlevé-class equations leads to hypergeometric-class equations. The “confluence principle” is stated, and an example of its application is given.

DOI: https://doi.org/10.4213/tmf723

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English version:
Theoretical and Mathematical Physics, 1999, 119:1, 393–406

Bibliographic databases:

Received: 09.07.1998
Revised: 23.11.1998

Citation: S. Yu. Slavyanov, “Structural theory of special functions”, TMF, 119:1 (1999), 3–19; Theoret. and Math. Phys., 119:1 (1999), 393–406

Citation in format AMSBIB
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\pages 3--19
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\jour Theoret. and Math. Phys.
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\vol 119
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\pages 393--406
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. S. Yu. Slavyanov, “Isomonodromic deformations of Heun and Painlevé equations”, Theoret. and Math. Phys., 123:3 (2000), 744–753  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. Tarasov, VF, “The Heun-Schrodinger radial equation with two auxiliary parameters for H-like atoms”, Modern Physics Letters B, 16:25 (2002), 937  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    3. B. I. Suleimanov, ““Quantizations” of the second Painlevé equation and the problem of the equivalence of its $L$$A$ pairs”, Theoret. and Math. Phys., 156:3 (2008), 1280–1291  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. A. Mirjalili, M. Taki, “Noncommutative correction to the Cornell potential in heavy-quarkonium atoms”, Theoret. and Math. Phys., 186:2 (2016), 280–285  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    5. B. I. Suleimanov, “Quantum aspects of the integrability of the third Painlevé equation and a non-stationary time Schrödinger equation with the Morse potential”, Ufa Math. J., 8:3 (2016), 136–154  mathnet  crossref  mathscinet  isi  elib
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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