Opuscula Math. 35, no. 5 (2015), 803-823
http://dx.doi.org/10.7494/OpMath.2015.35.5.803

Opuscula Mathematica

# Alien derivatives of the WKB solutions of the Gauss hypergeometric differential equation with a large parameter

Mika Tanda

Abstract. We compute alien derivatives of the WKB solutions of the Gauss hypergeometric differential equation with a large parameter and discuss the singularity structures of the Borel transforms of the WKB solution expressed in terms of its alien derivatives.

Keywords: hypergeometric differential equation, WKB solution, Voros coefficient, alien derivative, Stokes curve, fixed singularity.

Mathematics Subject Classification: 33C05, 34M40, 34M60.

Full text (pdf)

1. T. Aoki, K. Iwaki, T. Takahashi, Exact WKB analysis of Schrödinger equations with a Stokes curve of loop type, in preparation.
2. T. Aoki, T. Kawai, Y. Takei, The Bender-Wu Analysis and the Voros Theory, [in:] Special Functions, Springer, 1991, 1-29.
3. T. Aoki, T. Iizuka, Classification of Stokes graphs of second order Fuchsian differential equations of genus two, Publ. RIMS, Kyoto Univ. 43 (2007), 241-276.
4. T. Aoki, M. Tanda, Characterization of Stokes graphs and Voros coefficients of hypergeometric differential equations with a large parameter, RIMS Kôkyûroku Bessatsu B40 (2013), 147-162.
5. T. Aoki, M. Tanda, Borel sums of Voros coefficients of hypergeometric differential equations with a large parameter, RIMS Kôkyûroku, Kyoto Uni. 1861 (2013), 17-24.
6. T. Aoki, M. Tanda, Parametric Stokes phenomena of the Gauss hypergeometric differential equation with a large parameter, submitted.
7. B. Candelpergher, M.A. Coppo, E. Delabaere, La sommation de Ramanujan, L'Enseignement Mathématique 43 (1997), 93-132.
8. E. Delabaere, H. Dillinger, F. Pham, Résurgence de Voros et périodes des courbes hyperelliptiques, Ann. Inst. Fourier, Grenoble, 43 (1993), 153-199.
9. E. Delabaere, F. Pham, Resurgent methods in semi-classical asymptotics, Ann. Inst. Henri Poincaré 71 (1999), 1-94.
10. J. Écalle, Les fonction résurgentes. Tome I., Publications Mathématiques d'Orsay 81, Université de Paris-Sud, Départment de Mathématique, Orsay (1981), 247 pp.
11. K. Iwaki, T. Nakanishi, Exact WKB analysis and cluster algebras, J. Phys. A 47 (2014), 474009.
12. T. Kawai, Y. Takei, Exact WKB analysis and cluster algebras Algebraic Analysis of Singular Perturbation Theory, Translation of Mathematical Monographs, vol. 227, AMS, 2005.
13. T. Koike, R. Schäfke, On the Borel summability of WKB solutions of Schrödinger equations with polynomial potential and its application, to appear in RIMS Kôkyûroku Bessatsu.
14. T. Koike, Y. Takei, On the Voros coefficient for the Whittaker equation with a large parameter - Some progress around Sato's conjecture in exact WKB analysis, Publ. RIMS, Kyoto Univ. 47 (2011), 375-396.
15. D. Sauzin, Resurgent functions and splitting problems, RIMS Kôkyûroku 1493 (2006), 48-117.
16. D. Sauzin, Introduction to 1-summability and resurgence, arXiv:1405.0356 (HAL-00860032), 2014.
17. Y. Takei, Sato's conjecture for the Weber equation and transformation theory for Schrödinger equations with a merging pair of turning points, RIMS Kôkyûroku Bessatsu B10 (2008), 205-224.
18. M. Tanda, Exact WKB analysis of hypergeometric differential equations, to appear in RIMS Kôkyûroku Bessatsu.
19. M. Tanda, Parametric Stokes phenomena of the Gauss hypergeometric differential equation with a large parameter, Doctoral Thesis, Kinki University, 2014.
• Mika Tanda
• Interdisciplinary Graduate School of Science and Engineering, Kinki University, Higashi-Osaka, Osaka 577-8502, Japan
• Communicated by Yoshishige Haraoka.
• Revised: 2015-02-03.
• Accepted: 2015-02-05.
• Published online: 2015-04-27.