Geodesic
One-parameter meromorphic solution of the degenerate third Painlevé equation with formal monodromy parameter $a=\pm\mathrm{i}/2$ vanishing at the origin
Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 29, Tome 520 (2023), pp. 189-226 
A. V. Kitaev; A. Vartanian. One-parameter meromorphic solution of the degenerate third Painlevé equation with formal monodromy parameter $a=\pm\mathrm{i}/2$ vanishing at the origin. Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 29, Tome 520 (2023), pp. 189-226. http://geodesic.mathdoc.fr/item/ZNSL_2023_520_a7/
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     author = {A. V. Kitaev and A. Vartanian},
     title = {One-parameter meromorphic solution of the degenerate third {Painlev\'e} equation with formal monodromy parameter $a=\pm\mathrm{i}/2$ vanishing at the origin},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {189--226},
     year = {2023},
     volume = {520},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2023_520_a7/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We prove that there exists a one-parameter family of meromorphic solutions $u(\tau)$ vanishing at $\tau=0$ of the degenerate third Painlevé equation, \begin{equation*} u^{\prime \prime}(\tau) = \frac{(u^{\prime}(\tau))^{2}}{u(\tau)} - \frac{u^{\prime}(\tau)}{\tau} + \frac{1}{\tau} \left(-8 \varepsilon (u(\tau))^{2} + 2ab \right) + \frac{b^{2}}{u(\tau)},\ \varepsilon=\pm1,\ \varepsilon b>0, \end{equation*} for formal monodromy parameter $a=\pm\mathrm{i}/2$. We study number-theoretic properties of the coefficients of the Taylor-series expansion of $u(\tau)$ at $\tau=0$ and its asymptotic behaviour as $\tau\to+\infty$. These asymptotics are visualized for generic initial data.

[1] G. W. Adamson, Sequence A128135 in The On-Line Encyclopedia of Integer Sequences, 2007 https://oeis.org

[2] A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher transcendental functions, v. 1, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953 (Based, in part, on notes left by Harry Bateman) | MR

[3] G. Eremin, Legendre's formula and p-adic analysis, arXiv: 1907.11902

[4] R. Garnier, “Sur des équations différentielles du troisième ordre dont l'intégrale générale est uniforme et sur une classe d'équations nouvelles d'ordre supérieur dont l'intégrale générale a ses points critiques fixes”, Annales scientifiques de l'École Normale Supérieure, Série 3, 29 (1912), 1-126 | MR

[5] V. I. Gromak, “The solutions of Painlevé's third equation”, Differencial'nye Uravnenija, 9:11 (1973), 2082–2083 | MR | Zbl

[6] E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944 | MR | Zbl

[7] A. V. Kitaev, “Meromorphic solution of the degenerate third Painlevé equation vanishing at the origin”, SIGMA, 15 (2019), 046, 53 pp. | MR | Zbl

[8] A. V. Kitaev, A. H. Vartanian, “Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation: I”, Inverse Problems, 20 (2004), 1165–1206 | DOI | MR | Zbl

[9] A. V. Kitaev, A. Vartanian, “Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation: II”, Inverse Problems, 26 (2010), 105010, 58 pp. | DOI | MR | Zbl

[10] A. V. Kitaev, A. Vartanian, “Asymptotics of integrals of some functions related to the degenerate third Painlevé equation”, J. Math. Sci., 242 (2019), 715–721 | DOI | MR | Zbl

[11] A. V. Kitaev, A. Vartanian, Algebroid solutions of the degenerate third Painlevé equation for vanishing formal monodromy parameter, arXiv: 2304.05671 | MR

[12] N. J. A. Sloane, A. Wilks, Sequence A005187 in The On-Line Encyclopedia of Integer Sequences, 1991, 1999 https://oeis.org

[13] N. J. A. Sloane, Sequence A004134 in The On-Line Encyclopedia of Integer Sequences https://oeis.org | MR

[14] N. J. A. Sloane, Sequences A085058, A001511 in The On-Line Encyclopedia of Integer Sequences, 2003 https://oeis.org | MR

[15] B. I. Suleimanov, “Effect of a small dispersion on self-focusing in a spatially one-dimensional case”, JETP Letters, 106 (2017), 400–405 | DOI