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An asymptotic expansion of the double gamma function. (English) Zbl 0983.41020

The Barnes double gamma function \(G(z)\) satisfies the recursion \(G(z+1)=\Gamma(z)G(z)\). This paper gives a new integral representation for \(\log G(z)\) and an asymptotic expansion for this function in negative powers of \(z\) for large complex \(z\) (\(z\) not negative real). An error bound for the remainder at any order is supplied, and numerical experiments show the accuracy of the expansion and of the bounds.

MSC:

41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
33B15 Gamma, beta and polygamma functions
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References:

[1] Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions (1970), Dover: Dover New York · Zbl 0515.33001
[2] Barnes, E. W., The theory of the G-function, Quart. J. Math., 31, 264-314 (1899) · JFM 30.0389.02
[3] Barnes, E. W., Genesis of the double gamma function, Proc. London Math. Soc., 31, 358-381 (1900) · JFM 30.0389.03
[4] Barnes, E. W., The theory of the double gamma function, Philos. Trans. Roy. Soc. London Ser. A, 196, 265-388 (1901) · JFM 32.0442.02
[5] Basor, E. L.; Forrester, P. J., Formulas for the evaluation of Toeplitz determinants with rational generating functions, Math. Nachr., 170, 5-18 (1994) · Zbl 0813.15003
[6] Billingham, J.; King, A. C., Uniform asymptotic expansions for the Barnes double gamma function, Proc. Roy. Soc. London Ser. A, 453, 1817-1829 (1997) · Zbl 0926.33001
[7] Cassou-Nogués, P., Analogues \(p\)-adiques des fonctions \(Γ\)-multiples, Journées Arithmétiques de Luminy, Colloq. Internat. CNRS, Centre Univ. Luminy, Luminy, 1978. Journées Arithmétiques de Luminy, Colloq. Internat. CNRS, Centre Univ. Luminy, Luminy, 1978, Astérisque, 61 (1979), Soc. Math. France: Soc. Math. France Paris, p. 43-55 · Zbl 0425.12018
[8] Choi, J., Determinant of laplacians on \(S^3\), Math. Japon., 40, 155-166 (1994) · Zbl 0806.58053
[9] Choi, J.; Srivastava, H. M., Summs associated with the zeta function, J. Math. Anal. Appl., 206, 103-120 (1997) · Zbl 0869.11067
[10] Choi, J.; Srivastava, H. M.; Quine, J. R., Some series involving the zeta function, Bull. Austral. Math. Soc., 51, 383-393 (1995) · Zbl 0830.11030
[11] Choi, J.; Srivastava, H. M., Certain classes of series involving the zeta function, J. Math. Anal. Appl., 231, 91-117 (1999) · Zbl 0932.11054
[12] Gradshteyn, I. S.; Ryzhik, I. M., Tables of Integrals, Series and Products (1980), Academic Press: Academic Press New York · Zbl 0521.33001
[13] Matsumoto, K., Asymptotic series for double zeta and double gamma functions of Barnes, RIMS Kokyuroku., 958, 162-165 (1996) · Zbl 1044.11614
[14] Matsumoto, K., Asymptotic series for double zeta, double gamma, and Hecke L-functions, Math. Proc. Camb. Philos. Soc., 123, 385-405 (1998) · Zbl 0903.11021
[15] Olver, F. W.J., Special Functions: Asymptotics and Special Functions (1974), Academic Press: Academic Press New York · Zbl 0303.41035
[16] Osgood, B.; Phillips, R.; Sarnak, P., Extremal of determinants of Laplacians, J. Funct. Anal., 80, 148-211 (1988) · Zbl 0653.53022
[17] Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I., Integrals and Series (1990), Gordon and Breach: Gordon and Breach London · Zbl 0967.00503
[18] Quine, J. R.; Choi, J., Zeta regularized products and functional determinants on spheres, Rocky Mountain J. Math., 26, 719-729 (1996) · Zbl 0864.47024
[19] Sarnak, P., Determinants of Laplacians, Commun. Math. Phys., 110, 113-120 (1987) · Zbl 0618.10023
[20] Shintani, T., A proof of the classical Kronecker limit formula, Tokyo J. Math., 3, 191-199 (1980) · Zbl 0462.10014
[21] Srivastava, H. M., A unified presentation of certain classes of series of the Riemann zeta function, Riv. Mat. Univ. Parma (4), 14, 1-23 (1988) · Zbl 0659.10047
[22] Temme, N. M., Special Functions: An Introduction to the Classical Functions of Mathematical Physics (1996), Wiley: Wiley New York · Zbl 0863.33002
[23] Vardi, I., Determinants of Laplacians and multiple gamma functions, SIAM J. Math. Anal., 19, 493-507 (1988) · Zbl 0641.33003
[24] Voros, A., Spectral functions, special functions and the Selberg zeta function, Commun. Math. Phys., 110, 439-465 (1987) · Zbl 0631.10025
[25] Whittaker, E. T.; Watson, G. N., A Course of Mothern Analysis (1964), Cambridge Univ. Press: Cambridge Univ. Press Cambridge
[26] Wong, R., Asymptotic Approximations of Integrals (1989), Academic Press: Academic Press New York · Zbl 0679.41001
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