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Remarks on Painlevé’s differential equation P34

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Abstract

This paper is engaged with Painlevé’s differential equation P34:2ww″ = w2 + 2w2(2wz) − α, also known as Ince’s equation XXXIV and closely related to Painlevé’s second differential equation \({{\rm{P}}_\Pi}:\varpi\prime\prime= \alpha + z\varpi + 2{\varpi^3}\). We will show that the transcendental solutions belong to the Yosida class \({\mathfrak{Y}_{{\rm{1,}}{1 \over 2}}}\) and have no deficient rational targets. We will also identify the sub-normal solutions and prove that they are characterised by the fact that their first integrals belong to the class \({\mathfrak{Y}_{{1 \over 2}{\rm{,}}{1 \over 2}}}\).

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References

  1. E. Ciechanowicz and G. Filipuk, Meromorphic solutions of P4,34and their value distribution, Ann. Acad. Sci. Fenn. Math. 41 (2016), 617–638.

    Article  MathSciNet  Google Scholar 

  2. E. Ciechanowicz and G. Filipuk, Transcendental meromorphic solutions of P34and small targets, in Analytic, Algebraic and Geometric Aspects of Differential Equations, Birkhäuser/Springer, Cham, 2017, pp. 307–323.

    Chapter  Google Scholar 

  3. J. Clunie, On integral and meromorphic functions, J. London Math. Soc. 37 (1962), 17–27.

    Article  MathSciNet  Google Scholar 

  4. W. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.

    MATH  Google Scholar 

  5. E. L. Ince, Ordinary Differential Equations, Dover, New York, 1956.

    Google Scholar 

  6. I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin, 1993.

    Book  Google Scholar 

  7. A. Z. Mokhon’ko and V. D. Mokhon’ko, Estimates for the Nevanlinna characteristics of some classes of meromorphic functions and their applications to differential equations, Siberian Math. J. 15 (1974), 921–934.

    Article  Google Scholar 

  8. Y. Ohyama and S. Okumura, A coalescent diagram of the Painleve equations from the viewpoint of isomonodromic deformations, J. Phys. A 39 (2006), 12129–12151.

    Article  MathSciNet  Google Scholar 

  9. K. Okamoto, Polynomial Hamiltonians associated with Painlevé Equations. II. Differential equations satisfied by polynomial Hamiltonians, Proc. Japan Acad. Ser. A Math. Sci. 56 (1980), 367–371.

    Article  MathSciNet  Google Scholar 

  10. X.-C. Pang, Bloch’s principle and normal criterion, Sci. China Ser. A 32 (1989), 782–791.

    MathSciNet  MATH  Google Scholar 

  11. X.-C. Pang, On normal criterion of meromorphic functions, Sci. China Ser. A 33 (1990), 521–527.

    MathSciNet  MATH  Google Scholar 

  12. S. Shimomura, Value distribution of Painlevé transcendents of the first and second kind, J. Anal. Math. 82 (2000), 333–346.

    Article  MathSciNet  Google Scholar 

  13. S. Shimomura, On deficiencies of small functions for Painlevé transcendents of the fourth kind, Ann. Acad. Sci. Fenn. Math. 27 (2002), 109–120.

    MathSciNet  MATH  Google Scholar 

  14. N. Steinmetz, Value distribution of the Painlevé transcendents, Israel J. Math. 128 (2002), 29–52.

    Article  MathSciNet  Google Scholar 

  15. N. Steinmetz, The Yosida class is universal, J. Anal. Math. 117 (2012), 347–364.

    Article  MathSciNet  Google Scholar 

  16. N. Steinmetz, Sub-normal solutions to Painlevé’s second differential equation, Bull. Lond. Math. Soc. 45 (2013), 225–235.

    Article  MathSciNet  Google Scholar 

  17. N. Steinmetz, Complex Riccati differential equations revisited, Ann. Acad. Sci. Fenn. Math. 39 (2014), 503–511.

    Article  MathSciNet  Google Scholar 

  18. N. Steinmetz, First order algebraic differential equations of genus zero, Bull. Lond. Math. Soc. 49 (2017), 391–404.

    Article  MathSciNet  Google Scholar 

  19. N. Steinmetz, Nevanlinna Theory, Normal Families, and Algebraic Differential Equations, Springer, Cham, 2017.

    Book  Google Scholar 

  20. H. Wittich, Neuere Untersuchungen über eindeutige analytische Funktionen, Springer, Berlin–New York, 1968.

    Book  Google Scholar 

  21. K. Yosida, On a class of meromorphic functions, Proc. Phys. Math. Soc. Japan 16 (1934), 227–235.

    MATH  Google Scholar 

  22. L. Zalcman, A heuristic principle in function theory, Amer. Math. Monthly 82 (1975), 813–817.

    Article  MathSciNet  Google Scholar 

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Steinmetz, N. Remarks on Painlevé’s differential equation P34. JAMA 141, 383–395 (2020). https://doi.org/10.1007/s11854-020-0118-3

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  • DOI: https://doi.org/10.1007/s11854-020-0118-3

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