Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 002, 23 pages      arXiv:2104.13751      https://doi.org/10.3842/SIGMA.2022.002

Voros Coefficients at the Origin and at the Infinity of the Generalized Hypergeometric Differential Equations with a Large Parameter

Takashi Aoki a and Shofu Uchida b
a) Department of Mathematics, Kindai University, Higashi-Osaka 577-8502, Japan
b) Graduate School of Science and Engineering, Kindai University, Higashi-Osaka 577-8502, Japan

Received July 20, 2021, in final form December 30, 2021; Published online January 03, 2022

Abstract
Voros coefficients of the generalized hypergeometric differential equations with a large parameter are defined and their explicit forms are given for the origin and for the infinity. It is shown that they are Borel summable in some specified regions in the space of parameters and their Borel sums in the regions are given.

Key words: exact WKB analysis; Voros coefficients; generalized hypergeometric differential equations.

pdf (486 kb)   tex (33 kb)  

References

  1. Aoki T., Symbols and formal symbols of pseudodifferential operators, in Group Representations and Systems of Differential Equations (Tokyo, 1982), Adv. Stud. Pure Math., Vol. 4, North-Holland, Amsterdam, 1984, 181-208.
  2. Aoki T., Kawai T., Koike T., Takei Y., On the exact WKB analysis of microdifferential operators of WKB type, Ann. Inst. Fourier (Grenoble) 54 (2004), 1393-1421.
  3. Aoki T., Kawai T., Koike T., Takei Y., On the exact WKB analysis of operators admitting infinitely many phases, Adv. Math. 181 (2004), 165-189.
  4. Aoki T., Takahashi T., Tanda M., Exact WKB analysis of confluent hypergeometric differential equations with a large parameter, in Exponential Analysis of Differential Equations and Related Topics, RIMS Kôkyûroku Bessatsu, Vol. B52, Res. Inst. Math. Sci. (RIMS), Kyoto, 2014, 165-174.
  5. Aoki T., Takahashi T., Tanda M., Relation between the hypergeometric function and WKB solutions, in Microlocal Analysis and Singular Perturbation Theory, RIMS Kôkyûroku Bessatsu, Vol. B61, Res. Inst. Math. Sci. (RIMS), Kyoto, 2017, 1-7.
  6. Aoki T., Takahashi T., Tanda M., The hypergeometric function, the confluent hypergeometric function and WKB solutions, J. Math. Soc. Japan 73 (2021), 1019-1062.
  7. Aoki T., Takahashi T., Tanda M., Voros coefficients of the Gauss hypergeometric differential equation with a large parameter, Integral Transforms Spec. Funct. 32 (2021), 336-345.
  8. Aoki T., Tanda M., Parametric Stokes phenomena of the Gauss hypergeometric differential equation with a large parameter, J. Math. Soc. Japan 68 (2016), 1099-1132.
  9. Barnes E.W., A new development of the theory of the hypergeometric functions, Proc. London Math. Soc. 6 (1908), 141-177.
  10. Beukers F., Heckman G., Monodromy for the hypergeometric function $_nF_{n-1}$, Invent. Math. 95 (1989), 325-354.
  11. Delabaere E., Dillinger H., Pham F., Résurgence de Voros et périodes des courbes hyperelliptiques, Ann. Inst. Fourier (Grenoble) 43 (1993), 163-199.
  12. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions, Vol. I, McGraw-Hill Book Company, Inc., New York, 1953.
  13. Forster O., Lectures on Riemann surfaces, Graduate Texts in Mathematics, Vol. 81, Springer-Verlag, New York - Berlin, 1981.
  14. Graham R.L., Knuth D.E., Patashnik O., Concrete mathematics: a foundation for computer science, 2nd ed., Addison-Wesley Publishing Company, Reading, MA, 1994.
  15. Hollands L., Kidwai O., Higher length-twist coordinates, generalized Heun's opers, and twisted superpotentials, Adv. Theor. Math. Phys. 22 (2018), 1713-1822, arXiv:1710.04438.
  16. Honda N., Kawai T., Takei Y., Virtual turning points, SpringerBriefs in Mathematical Physics, Vol. 4, Springer, Tokyo, 2015.
  17. Iwaki K., Koike T., On the computation of Voros coefficients via middle convolutions, in Exponential Analysis of Differential Equations and Related Topics, RIMS Kôkyûroku Bessatsu, Vol. B52, Res. Inst. Math. Sci. (RIMS), Kyoto, 2014, 55-70.
  18. Iwaki K., Koike T., Takei Y., Voros coefficients for the hypergeometric differential equations and Eynard-Orantin's topological recursion: Part II: For confluent family of hypergeometric equations, J. Integrable Syst. 4 (2019), xyz004, 46 pages, arXiv:1810.02946.
  19. Iwaki K., Nakanishi T., Exact WKB analysis and cluster algebras, J. Phys. A: Math. Theor. 47 (2014), 474009, 98 pages, arXiv:1401.7094.
  20. Kawai T., Takei Y., Algebraic analysis of singular perturbation theory, Translations of Mathematical Monographs, Vol. 227, Amer. Math. Soc., Providence, RI, 2005.
  21. Kishioka H., Noumi M., Mellin transform and connection problem, RIMS Kôkyûroku, Vol. 1662, Res. Inst. Math. Sci. (RIMS), Kyoto, 2009, 231-260.
  22. Koike T., Takei Y., On the Voros coefficient for the Whittaker equation with a large parameter - some progress around Sato's conjecture in exact WKB analysis, Publ. Res. Inst. Math. Sci. 47 (2011), 375-395.
  23. Levelt A.H.M., Hypergeometric functions. I, Indag. Math. 64 (1961), 361-372.
  24. Olver F.W.J., Lozier D.W., Boisvert R.F., Clark C.W. (Editors), NIST handbook of mathematical functions, Cambridge University Press, Cambridge, 2010.
  25. Shen H., Silverstone H.J., Observations on the JWKB treatment of the quadratic barrier, in Algebraic Analysis of Differential Equations from Microlocal Analysis to Exponential Asymptotics, Springer, Tokyo, 2008, 237-250.
  26. Smith F.C., Relations among the fundamental solutions of the generalized hypergeometric equation when $p=q+1$. I. Non-logarithmic cases, Bull. Amer. Math. Soc. 44 (1938), 429-433.
  27. Takei Y., Sato's conjecture for the Weber equation and transformation theory for Schrödinger equations with a merging pair of turning points, in Differential Equations and Exact WKB Analysis, RIMS Kôkyûroku Bessatsu, Vol. B10, Res. Inst. Math. Sci. (RIMS), Kyoto, 2008, 205-224.
  28. Voros A., The return of the quartic oscillator: the complex WKB method, Ann. Inst. H. Poincaré Sect. A (N.S.) 39 (1983), 211-338.

Previous article  Next article  Contents of Volume 18 (2022)