Abstract
This paper is engaged with Painlevé’s differential equation P34:2ww″ = w″2 + 2w2(2w − z) − 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
E. Ciechanowicz and G. Filipuk, Meromorphic solutions of P4,34and their value distribution, Ann. Acad. Sci. Fenn. Math. 41 (2016), 617–638.
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.
J. Clunie, On integral and meromorphic functions, J. London Math. Soc. 37 (1962), 17–27.
W. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.
E. L. Ince, Ordinary Differential Equations, Dover, New York, 1956.
I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin, 1993.
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.
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.
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.
X.-C. Pang, Bloch’s principle and normal criterion, Sci. China Ser. A 32 (1989), 782–791.
X.-C. Pang, On normal criterion of meromorphic functions, Sci. China Ser. A 33 (1990), 521–527.
S. Shimomura, Value distribution of Painlevé transcendents of the first and second kind, J. Anal. Math. 82 (2000), 333–346.
S. Shimomura, On deficiencies of small functions for Painlevé transcendents of the fourth kind, Ann. Acad. Sci. Fenn. Math. 27 (2002), 109–120.
N. Steinmetz, Value distribution of the Painlevé transcendents, Israel J. Math. 128 (2002), 29–52.
N. Steinmetz, The Yosida class is universal, J. Anal. Math. 117 (2012), 347–364.
N. Steinmetz, Sub-normal solutions to Painlevé’s second differential equation, Bull. Lond. Math. Soc. 45 (2013), 225–235.
N. Steinmetz, Complex Riccati differential equations revisited, Ann. Acad. Sci. Fenn. Math. 39 (2014), 503–511.
N. Steinmetz, First order algebraic differential equations of genus zero, Bull. Lond. Math. Soc. 49 (2017), 391–404.
N. Steinmetz, Nevanlinna Theory, Normal Families, and Algebraic Differential Equations, Springer, Cham, 2017.
H. Wittich, Neuere Untersuchungen über eindeutige analytische Funktionen, Springer, Berlin–New York, 1968.
K. Yosida, On a class of meromorphic functions, Proc. Phys. Math. Soc. Japan 16 (1934), 227–235.
L. Zalcman, A heuristic principle in function theory, Amer. Math. Monthly 82 (1975), 813–817.
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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