MR3221837 Hisashi ANDO, Mike HAY, Kenji KAJIWARA and Tetsu MASUDA Hisashi ANDO, Mike HAY, Kenji KAJIWARA and Tetsu MASUDA An Explicit Formula for the Discrete Power Function Associated with Circle Patterns of Schramm Type Funkcialaj Ekvacioj. Serio Internacia 57 2014 1--41 http://fe.math.kobe-u.ac.jp/FE/FullPapers/57-1/57_1.pdf http://www.ams.org/mathscinet-getitem?mr=MR3221837 We present an explicit formula for the discrete power function introduced by Bobenko, which is expressed in terms of the hypergeometric $\tau$ functions for the sixth Painlevé equation. The original definition of the discrete power function imposes strict conditions on the domain and the value of the exponent. However, we show that one can extend the value of the exponent to arbitrary complex numbers except even integers and the domain to a discrete analogue of the Riemann surface. Moreover, we show that the discrete power function is an immersion when the real part of the exponent is equal to one. Discrete conformal mapping, Hypergeometric function, Painlevé VI equation, Circle patterns, Discrete differential geometry. 33C05, 33E17, 34M55, 52C26, 53A30. 57-1 2014 An Explicit Formula for the Discrete Power Function Associated with Circle Patterns of Schramm Type Hisashi ANDO, Mike HAY, Kenji KAJIWARA and Tetsu MASUDA Hisashi ANDO, Mike HAY, Kenji KAJIWARA and Tetsu MASUDA 1 Abramowitz, M.; Stegun, I. A. (Eds.) Handbook of mathematical functions with formulas, graphs, and mathematical tables Dover Publications, Inc., New York 1992 MR1225604
2 Agafonov, S. I. Imbedded circle patterns with the combinatorics of the square grid and discrete Painlevé equations Discrete Comput. Geom. 29 2003 305-319 MR1957234
3 Agafonov, S. I. Discrete Riccati equation, hypergeometric functions and circle patterns of Schramm type Glasg. Math. J. 47 2005 1-16 MR2237186
4 Agafonov, S. I.; Bobenko, A. I. Discrete $Z^\gamma$ and Painlevé equations Internat. Math. Res. Notices 2000 2000 165-193 MR1747617
5 Agafonov, S. I.; Bobenko, A. I. Hexagonal circle patterns with constant intersection angles and discrete Painlevé and Riccati equations J. Math. Phys. 44 2003 3455-3469 MR2006759
6 Beardon, A. F.; Stephenson, K. The uniformization theorem for circle packing Indiana Univ. Math. J. 39 1990 1383-1425 MR1087197
7 Bobenko, A. I. Discrete conformal maps and surfaces London Math. Soc. Lecture Note Ser., 255, Cambridge Univ. Press, Cambridge 1999 pp. 97-108 MR1705222
8 Bobenko, A. I.; Hoffmann, T. Hexagonal circle patterns and integrable systems: patterns with constant angles Duke Math. J. 116 2003 525-566 MR1958097
9 Bobenko, A. I.; Hoffmann, T.; Suris, Y. B. Hexagonal circle patterns and integrable systems: patterns with the multi-ratio property and Lax equations on the regular triangular lattice Int. Math. Res. Not. 2002 2002 111-164 MR1878802
10 Bobenko, A. I.; Pinkall, U. Discrete isothermic surfaces J. Reine Angew. Math. 475 1996 187-208 MR1396732
11 Bobenko, A. I.; Pinkall, U. Discretization of surfaces and integrable systems Oxford Lecture Ser. Math. Appl., 16, Oxford Univ. Press, New York 1999 pp. 3-58 MR1676682
12 Hay, M.; Kajiwara, K.; Masuda, T. Bilinearization and special solutions to the discrete Schwarzian KdV equation J. Math-for-Ind. 3A 2011 53-62 MR2788707
13 Masuda, T. Classical transcendental solutions of the Painlevé equations and their degeneration Tohoku Math. J. 56 2004 467-490 MR2097156
14 Nijhoff, F. W.; Ramani, A.; Grammaticos, B.; Ohta, Y. On discrete Painlevé equations associated with the lattice KdV systems and the Painlevé VI equation Stud. Appl. Math. 106 2001 261-314 MR1819383
15 Rodin, B. Schwarz's lemma for circle packings Invent. Math. 89 1987 271-289 MR0894380
16 Schramm, O. Circle patterns with the combinatorics of the square grid Duke Math. J. 86 1997 347-389 MR1430437
17 Stephenson, K. Introduction to circle packing Cambridge University Press, Cambridge 2005 MR2131318 18 Thurston, W. P., The finite Riemann mapping theorem, Invited address, International Symposium in Celebration of the Proof of the Bieberbach Conjecture (Purdue University, 1985)