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
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.
- T. Aoki, K. Iwaki, T. Takahashi, Exact WKB analysis of Schrödinger equations with a Stokes curve of loop type, in preparation.
- T. Aoki, T. Kawai, Y. Takei, The Bender-Wu Analysis and the Voros Theory, [in:] Special Functions, Springer, 1991, 1-29.
- 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.
- 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.
- 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.
- T. Aoki, M. Tanda, Parametric Stokes phenomena of the Gauss hypergeometric differential equation with a large parameter, submitted.
- B. Candelpergher, M.A. Coppo, E. Delabaere, La sommation de Ramanujan, L'Enseignement Mathématique 43 (1997), 93-132.
- E. Delabaere, H. Dillinger, F. Pham, Résurgence de Voros et périodes des courbes hyperelliptiques, Ann. Inst. Fourier, Grenoble, 43 (1993), 153-199.
- E. Delabaere, F. Pham, Resurgent methods in semi-classical asymptotics, Ann. Inst. Henri Poincaré 71 (1999), 1-94.
- J. Écalle, Les fonction résurgentes. Tome I., Publications Mathématiques d'Orsay 81, Université de Paris-Sud, Départment de Mathématique, Orsay (1981), 247 pp.
- K. Iwaki, T. Nakanishi, Exact WKB analysis and cluster algebras, J. Phys. A 47 (2014), 474009.
- T. Kawai, Y. Takei, Exact WKB analysis and cluster algebras Algebraic Analysis of Singular Perturbation Theory, Translation of Mathematical Monographs, vol. 227, AMS, 2005.
- 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.
- 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.
- D. Sauzin, Resurgent functions and splitting problems, RIMS Kôkyûroku 1493 (2006), 48-117.
- D. Sauzin, Introduction to 1-summability and resurgence, arXiv:1405.0356 (HAL-00860032), 2014.
- 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.
- M. Tanda, Exact WKB analysis of hypergeometric differential equations, to appear in RIMS Kôkyûroku Bessatsu.
- 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.
- Received: 2014-03-31.
- Revised: 2015-02-03.
- Accepted: 2015-02-05.
- Published online: 2015-04-27.
