Multiple Gamma functions and their applications. (English) Zbl 1323.33002
Milovanović, Gradimir V. (ed.) et al., Analytic number theory, approximation theory, and special functions. In honor of Hari M. Srivastava. New York, NY: Springer (ISBN 978-1-4939-0257-6/hbk; 978-1-4939-0258-3/ebook). 93-129 (2014).
Summary: The double Gamma function \(\Gamma_2\) and the multiple Gamma functions \(\Gamma_n\) were defined and studied systematically by Barnes in about 1900. Before their investigation by Barnes, these functions had been introduced in a different form by, for example, Hölder, Alexeiewsky, and Kinkelin. Although these functions did not appear in the tables of the most well-known special functions, yet the double Gamma function was cited in the exercises by Whittaker and Watson’s book and recorded also by Gradshteyn and Ryzhik’s book. In about the middle of the 1980s, these functions were revived in the study of the determinants of the Laplacians on the \(n\)-dimensional unit sphere \(\mathbf S^n\). Here, in this expository paper, from the middle of the 1980s until today, we aim at giving an eclectic review for recent developments and applications of the simple and multiple Gamma functions.
For the entire collection see [Zbl 1286.00055].
For the entire collection see [Zbl 1286.00055].
MSC:
| 33-02 | Research exposition (monographs, survey articles) pertaining to special functions |
| 33B15 | Gamma, beta and polygamma functions |
Software:
DLMFReferences:
| [1] | Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Tenth Printing, National Bureau of Standards, Applied Mathematics Series, vol. 55. National Bureau of Standards, Washington (1972); Reprinted by Dover Publications, New York (1965) (see also [99]) · Zbl 0171.38503 |
| [2] | Adamchik, V.S.: Polygamma functions of negative order. J. Comput. Appl. Math. 100, 191-199 (1998) · Zbl 0936.33001 · doi:10.1016/S0377-0427(98)00192-7 |
| [3] | Adamchik, V.S.: On the Barnes function. In: Mourrain, B. (ed.) Proceedings of the 2001 International Symposium on Symbolic and Algebraic Computation, pp. 15-20. ACM, New York (2001) (London, Ontario, 22-25 July 2001) · Zbl 1356.33001 |
| [4] | Adamchik, V.S.: The multiple gamma function and its application to computation of series. Ramanujan J. 9, 271-288 (2005) · Zbl 1088.33014 · doi:10.1007/s11139-005-1868-3 |
| [5] | Alexeiewsky, W.P.: Über eine Classe von Funktionen, die der Gammafunktion analog sind. Leipzig Weidmannsche Buchhandlung 46, 268-275 (1894) |
| [6] | Alzer, H.: Inequalities involving \(\Gamma\)(\(x\)) and \(\Gamma\)(1/\(x\)). J. Comput. Appl. Math. 192, 460-480 (2006) · Zbl 1091.33001 · doi:10.1016/j.cam.2005.04.063 |
| [7] | Alzer, H.: Sub-and superadditive property of Euler’s Gamma function. Proc. Amer. Math. Soc. 135, 3641-3648 (2007) · Zbl 1126.33001 · doi:10.1090/S0002-9939-07-09057-0 |
| [8] | Apostol, T.M.: Some series involving the Riemann Zeta function. Proc. Amer. Math. Soc. 5, 239-243 (1954) · Zbl 0055.06903 · doi:10.1090/S0002-9939-1954-0060534-0 |
| [9] | Apostol, T.M.: Introduction to Analytic Number Theory. Springer, New York (1976) · Zbl 0335.10001 |
| [10] | Artin, E.: The Gamma Function. Holt, Rinehart and Winston, New York (1964) · Zbl 0144.06802 |
| [11] | Barnes, E.W.: The theory of the \(G\)-function. Quart. J. Math. 31, 264-314 (1899) · JFM 30.0389.02 |
| [12] | Barnes, E.W.: The genesis of the double Gamma function. Proc. London Math. Soc. (Ser. 1) 31, 358-381 (1900) · JFM 30.0389.03 |
| [13] | Barnes, E.W.: The theory of the double Gamma function. Philos. Trans. Roy. Soc. London Ser. A 196, 265-388 (1901) · doi:10.1098/rsta.1901.0006 |
| [14] | Barnes, E.W.: On the theory of the multiple Gamma functions. Trans. Cambridge Philos. Soc. 19, 374-439 (1904) |
| [15] | Batir, N.: Inequalities for the double Gamma function. J. Math. Anal. Appl. 351, 182-185 (2009) · Zbl 1157.33001 · doi:10.1016/j.jmaa.2008.09.077 |
| [16] | Batir, N., Cancan, M.: A double inequality for the double Gamma function. Internat. J. Math. Anal. 2, 329-335 (2008) · Zbl 1177.33004 |
| [17] | Bendersky, L.: Sur la fonction Gamma généralisée, Acta Math. 61, 263-322 (1933) · JFM 59.0373.02 · doi:10.1007/BF02547794 |
| [18] | 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 · doi:10.1098/rspa.1997.0098 |
| [19] | Campbell, R.: Les Intégrals Eulériennes Et Leurs Applications. Dunod, Paris (1966) |
| [20] | Carrier, G.F., Krook, M., Pearson, C.E.: Functions of a Complex Variable. Hod Books, Ithaca (1983) · Zbl 0548.30001 |
| [21] | Cassou-Noguès, P.: Analogues \(p\)-adiques des fonctions \(\Gamma\)-multiples. In: Journées Arithmétiques de Luminy (International Colloquium of the CNRS, Centre University of Luminy, Luminy, 1978), pp. 43-55. Astérisque, vol. 61. Société mathématique de France, Paris (1979) |
| [22] | Chen, C.-P.: Inequalities associated with Barnes \(G\)-function. Exposition. Math. 29, 119-125 (2011) · Zbl 1216.33004 · doi:10.1016/j.exmath.2010.07.001 |
| [23] | Chen, C.-P.: Glaisher-Kinkelin constant. Integr. Transf. Spec. Funct. 23, 785-792 (2012) (iFirst) · Zbl 1300.11135 |
| [24] | Chen, C.-P., Srivastava, H.M.: Some inequalities and monotonicity properties associated with the Gamma and Psi functions and the Barnes \(G\)-function. Integr. Transf. Spec. Funct. 22, 1-15 (2011) · Zbl 1213.33004 · doi:10.1080/10652469.2010.483899 |
| [25] | Choi, J.: Determinant of Laplacian on \(S\)^{3}. Math. Japon. 40, 155-166 (1994) · Zbl 0806.58053 |
| [26] | Choi, J.: Explicit formulas for the Bernolli polynomials of order \(n\). Indian J. Pure Appl. Math. 27, 667-674 (1996) · Zbl 0860.11009 |
| [27] | Choi, J.: Integral and series representations for the Euler’s constant. In: Proceedings of the Seventh Conference on Real and Complex Analysis, pp. 43-55. Hiroshima University, Japan (2003) |
| [28] | Choi, J.: Some mathematical constants. Appl. Math. Comput. 187, 122-140 (2007) · Zbl 1117.33003 · doi:10.1016/j.amc.2006.08.091 |
| [29] | Choi, J.: A set of mathematical constants arising naturally in the theory of the multiple Gamma functions. Abstract and Applied Analysis, vol. 2012, Article ID 121795, p. 11 (2012) · Zbl 1256.33002 |
| [30] | Choi, J.: Determinants of the Laplacians on the \(n\)-dimensional unit sphere S^{\(n\)} (\(n\)=8,9). · Zbl 1268.11122 |
| [31] | Choi, J., Cho, Y.J., Srivastava, H.M.: Series involving the Zeta function and multiple Gamma functions. Appl. Math. Comput. 159, 509-537 (2004) · Zbl 1061.33001 · doi:10.1016/j.amc.2003.08.134 |
| [32] | Choi, J., Lee, J.: Closed-form evaluation of a class of series associated with the Riemann zeta function. In: Proceedings of the 11th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications, pp. 36-53 (2003) |
| [33] | Choi, J., Lee, J., Srivastava, H.M.: A generalization of Wilf’s formula. Kodai Math. J. 26, 44-48 (2003) · Zbl 1040.11062 · doi:10.2996/kmj/1050496647 |
| [34] | Choi, J., Nash, C.: Integral representations of the Kikelin’s constant \(A\). Math. Japon. 45, 223-230 (1997) · Zbl 0871.33001 |
| [35] | Choi, J., Quine, J.R.: E. W. Barnes’ approach of the multiple Gamma functions. J. Korean Math. Soc. 29, 127-140 (1992) · Zbl 0767.33001 |
| [36] | Choi, J., Seo, T.Y.: The double Gamma function. East Asian Math. J. 13, 159-174 (1997) |
| [37] | Choi, J., Seo, T.Y.: Integral formulas for Euler’s constant. Commun. Korean Math. Soc. 13, 683-689 (1998) · Zbl 1022.11067 |
| [38] | Choi, J., Seo, T.Y.: Identities involving series of the Riemann zeta function. Indian J. Pure Appl. Math. 30, 649-652 (1999) · Zbl 0934.11041 |
| [39] | Choi, J., Srivastava, H.M.: Sums associated with the Zeta function. J. Math. Anal. Appl. 206, 103-120 (1997) · Zbl 0869.11067 · doi:10.1006/jmaa.1997.5198 |
| [40] | Choi, J., Srivastava, H.M.: Certain classes of series involving the Zeta function. J. Math. Anal. Appl. 231, 91-117 (1999) · Zbl 0932.11054 · doi:10.1006/jmaa.1998.6216 |
| [41] | Choi, J., Srivastava, H.M.: An application of the theory of the double Gamma function. Kyushu J. Math. 53, 209-222 (1999) · Zbl 1013.11053 · doi:10.2206/kyushujm.53.209 |
| [42] | Choi, J., Srivastava, H.M.: Certain classes of series associated with the Zeta function and multiple Gamma functions. J. Comput. Appl. Math. 118, 87-109 (2000) · Zbl 0969.11030 · doi:10.1016/S0377-0427(00)00311-3 |
| [43] | Choi, J., Srivastava, H.M.: A certain class of series associated with the Zeta function. Integr. Transf. Spec. Funct. 12, 237-250 (2001) · Zbl 1023.11043 · doi:10.1080/10652460108819348 |
| [44] | Choi, J., Srivastava, H.M.: A certain family of series associated with the Zeta and related functions. Hiroshima Math. J. 32, 417-429 (2002) · Zbl 1160.11339 |
| [45] | Choi, J., Srivastava, H.M.: A family of log-Gamma integrals and associated results. J. Math. Anal. Appl. 303, 436-449 (2005) · Zbl 1064.33003 · doi:10.1016/j.jmaa.2004.08.043 |
| [46] | Choi, J., Srivastava, H.M.: Certain families of series associated with the Hurwitz-Lerch Zeta function. Appl. Math. Comput. 170, 399-409 (2005) · Zbl 1082.11052 · doi:10.1016/j.amc.2004.12.004 |
| [47] | Choi, J., Srivastava, H.M.: A note on a multiplication formula for the multiple Gamma function \(\Gamma\)_{\(n\)}. Italian J. Pure Appl. Math. 23, 179-188 (2008) · Zbl 1180.33002 |
| [48] | Choi, J., Srivastava, H.M.: Some applications of the Gamma and Polygamma functions involving convolutions of the Rayleigh functions, multiple Euler sums and log-sine integrals. Math. Nachr. 282, 1709-1723 (2009) · Zbl 1214.33002 · doi:10.1002/mana.200710032 |
| [49] | Choi, J., Srivastava, H.M.: Integral representations for the Gamma function, the Beta function, and the double Gamma function. Integr. Transf. Spec. Funct. 20, 859-869 (2009) · Zbl 1242.33002 · doi:10.1080/10652460902943519 |
| [50] | Choi, J., Srivastava, H.M.: Integral representations for the Euler-Mascheroni constant \(\gamma\). Integr. Transf. Spec. Funct. 21, 675-690 (2010) · Zbl 1203.11085 |
| [51] | Choi, J., Srivastava, H.M.: Mathieu series and associated sums involving the Zeta functions. Comput. Math. Appl. 59, 861-867 (2010) · Zbl 1189.33036 · doi:10.1016/j.camwa.2009.10.008 |
| [52] | Choi, J., Srivastava, H.M.: Asymptotic formulas for the triple Gamma function \(\Gamma\)_{3} by means of its integral representation. Appl. Math. Comput. 218, 2631-2640 (2011) · Zbl 1241.33003 · doi:10.1016/j.amc.2011.08.002 |
| [53] | Choi, J., Srivastava, H.M.: The multiple Hurwitz Zeta function and the multiple Hurwitz-Euler Eta function. Taiwanese J. Math. 15, 501-522 (2011) · Zbl 1273.11133 |
| [54] | Choi, J., Srivastava, H.M.: Some two-sided inequalities for multiple Gamma functions and related results. Appl. Math. Comput. 219, 10343-10354 (2013) · Zbl 1293.33001 · doi:10.1016/j.amc.2013.04.006 |
| [55] | Choi, J., Srivastava, H.M.: Series involving the Zeta functions and a family of generalized Goldbach-Euler series. Accepted for publication at Amer. Math. Monthly (2013) |
| [56] | 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 · doi:10.1017/S0004972700014210 |
| [57] | 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 · doi:10.1017/S0004972700014210 |
| [58] | Choi, J., Srivastava, H.M., Zhang, N.-Y.: Integrals involving a function associated with the Euler-Maclaurin summation formula. Appl. Math. Comput. 93, 101-116 (1998) · Zbl 0939.33002 · doi:10.1016/S0096-3003(97)10070-4 |
| [59] | Choi, J., Srivastava, H.M., Adamchik, V.S.: Multiple Gamma and related functions. Appl. Math. Comput. 134, 515-533 (2003) · Zbl 1026.33003 · doi:10.1016/S0096-3003(01)00301-0 |
| [60] | Conway, J.B.: Functions of One Complex Variable, 2nd edn. Springer, New York (1978) · doi:10.1007/978-1-4612-6313-5 |
| [61] | D’Hoker, E., Phong, D.H.: On determinant of Laplacians on Riemann surface. Comm. Math. Phys. 104, 537-545 (1986) · Zbl 0599.30073 · doi:10.1007/BF01211063 |
| [62] | D’Hoker, E., Phong, D.H.: Multiloop amplitudes for the bosonic polyakov string. Nucl. Phys. B 269, 204-234 (1986) |
| [63] | Dittrich, W., Reuter, M.: Effective QCD-Lagrangian with \(\xi\)-function regularization. Phys. Lett. B 128, 321-326 (1983) · doi:10.1016/0370-2693(83)90268-X |
| [64] | Dufresnoy, J., Pisot, C.: Sur la relation fonctionnelle · Zbl 0122.09802 |
| [65] | Edwards, J.: A treatise on the integral calculus with applications: Examples and problems, Vol. 1, 2. Chelsea Publishing Company, New York (1954) |
| [66] | Elizalde, E.: Derivative of the generalized Riemann Zeta function \(\zeta\)(\(z\),\(q\)) at · Zbl 0603.10039 |
| [67] | Elizalde, E.: An asymptotic expansion for the first derivative of the generalized Riemann Zeta function. Math. Comput. 47, 347-350 (1986) · Zbl 0603.10040 · doi:10.1090/S0025-5718-1986-0842140-X |
| [68] | Elizalde, E., Romeo, A.: An integral involving the generalized Zeta function. Internat. J. Math. Math. Sci. 13, 453-460 (1990) · Zbl 0708.11039 · doi:10.1155/S0161171290000679 |
| [69] | Elizalde, E., Odintsov, S.D., Romeo, A., Bytsenko, A.A., Zerbini, S.: Zeta Regularization Techniques with Applications. World Scientific Publishing Company, Singapore (1994) · Zbl 1050.81500 · doi:10.1142/9789812779342 |
| [70] | Euler, L.: Comm. Acad. Petropol. 7, 156 (1734-1735) |
| [71] | Ferreira, C.: An asymptotic expansion of the double Gamma function. J. Approx. Theory 111, 298-314 (2001) · Zbl 0983.41020 · doi:10.1006/jath.2001.3578 |
| [72] | Finch, S.R.: Mathematical Constants. Cambridge University Press, Cambridge (2003) · Zbl 1054.00001 |
| [73] | Friedman, E., Ruijsenaars, S.: Shintani-Barnes Zeta and Gamma functions. Adv. Math. 187, 362-395 (2004) · Zbl 1112.11042 · doi:10.1016/j.aim.2003.07.020 |
| [74] | Glaisher, J.W.L.: On the history of Euler’s constant. Messenger Math. 1, 25-30 (1872) |
| [75] | Glaisher, J.W.L.: On the product · JFM 09.0190.01 |
| [76] | Gosper, Jr., R.W.: |
| [77] | Gradshteyn, I.S., Ryzhik, I.M. (eds.): Tables of Integrals, Series, and Products (Corrected and Enlarged edition prepared by A. Jeffrey), 6th edn. Academic, New York (2000) · Zbl 0981.65001 |
| [78] | Hardy, G.H.: Divergent Series. Clarendon (Oxford University) Press, Oxford (1949); 2nd (Textually Unaltered) edn, Chelsea Publishing Company, New York (1991) · Zbl 0032.05801 |
| [79] | Havil, J.: Gamma (Exploring Euler’s Constant). Princeton University Press, Princeton (2003) · Zbl 1023.11001 |
| [80] | Hölder, O.: Über eine Transcendente Funktion, vol. 1886, pp. 514-522. Dieterichsche Verlags-Buchhandlung, Göttingen (1886) · JFM 18.0376.01 |
| [81] | Hölder, O.: Über eine von Abel untersuchte Transzendente und eine merkwürdige Funktionalbeziehung. In: Berichte über die Verhandlungen der Saechsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse, vol. 80, pp. 312-325 (1928) · JFM 54.0390.02 |
| [82] | Kanemitsu, S., Kumagai, H., Yoshimoto, M.: Sums involving the Hurwitz zeta function. Ramanujan J. 5, 5-19 (2001) · Zbl 0989.11043 · doi:10.1023/A:1011496709753 |
| [83] | Kinkelin, V.H.: Über eine mit der Gamma Funktion verwandte Transcendente und deren Anwendung auf die Integralrechnung. J. Reine Angew. Math. 57, 122-158 (1860) · doi:10.1515/crll.1860.57.122 |
| [84] | Knopp, K.: Theory and Application of Infinite Series. Hafner Publishing Company, New York (1951) (2nd English edn., translated from the 2nd German edn. revised in accordance with the 4th German edn. by R.C.H. Young) · Zbl 0042.29203 |
| [85] | Knuth, D.E.: Euler’s constant to 1271 places. Math. Comput. 16, 275-281 (1962) · Zbl 0117.10801 |
| [86] | Koumandos, S.: On Ruijsenaars’ asymptotic expansion of the logarithm of the double gamma function. J. Math. Anal. Appl. 341, 1125-1132 (2008) · Zbl 1140.33007 · doi:10.1016/j.jmaa.2007.11.021 |
| [87] | Koumandos, S., Pedersen, H.L.: Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler’s gamma function. J. Math. Anal. Appl. 355, 33-40 (2009) · Zbl 1169.33001 · doi:10.1016/j.jmaa.2009.01.042 |
| [88] | Koumandos, S., Pedersen, H.L.: Absolutely monotonic functions related to Euler’s gamma function and Barnes’ double and triple gamma function. Monatsh. Math. 163, 51-69 (2011) · Zbl 1241.33004 · doi:10.1007/s00605-010-0197-9 |
| [89] | Kumagai, H.: The determinant of the Laplacian on the \(n\)-sphere. Acta Arith. 91, 199-208 (1999) · Zbl 0946.11021 |
| [90] | Lewin, L.: Polylogarithms and Associated Functions. Elsevier (North-Holland), New York (1981) · Zbl 0465.33001 |
| [91] | Matsumoto, K.: Asymptotic series for double Zeta and double Gamma functions of Barnes. RIMS Kōkyūroku 958, 162-165 (1996) · Zbl 1044.11614 |
| [92] | Matsumoto, K.: Asymptotic series for double Zeta, double Gamma, and Hecke \(L\)-functions. Math. Proc. Cambridge Philos. Soc. 123, 385-405 (1998); Corrigendum and Addendum: Math. Proc. Cambridge Philos. Soc. 132, 377-384 (2002) · Zbl 0903.11021 |
| [93] | Melzak, Z.A.: Companion to Concrete Mathematics: Mathematical Techniques and Various Applications, vol. I. Wiley, New York (1973) · Zbl 0254.26003 |
| [94] | Mortici, C.: New improvements of the Stirling formula. Appl. Math. Comput. 217, 699-704 (2010) · Zbl 1202.33004 · doi:10.1016/j.amc.2010.06.007 |
| [95] | Nielsen, N.: Handbuch der Theorie der Gammafunktion. Druck und Verlag von B.G. Teubner, Leipzig (1906); Reprinted by Chelsea Publishing Company, New York (1965) |
| [96] | Nishizawa, M.: On a \(q\)-analogue of the multiple Gamma functions. Lett. Math. Phys. 37, 201-209 (1996) · Zbl 0872.33011 · doi:10.1007/BF00416023 |
| [97] | Nishizawa, M.: Multiple Gamma function, its \(q\)- and elliptic analogue. Rocky Mountain J. Math. 32, 793-811 (2002) · Zbl 1042.33003 · doi:10.1216/rmjm/1030539697 |
| [98] | Olver, F.W.J.: Asymptotics and Special Functions. A.K. Peters, Wellesley, Massachusetts (1997) · Zbl 0982.41018 |
| [99] | Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions [With 1 CD-ROM (Windows, Macintosh and UNIX)]. U.S. Department of Commerce, National Institute of Standards and Technology, Washington (2010); Cambridge University Press, Cambridge (2010) (see also [1]) · Zbl 1198.00002 |
| [100] | Osgood, B., Phillips, R., Sarnak, P.: Extremals of determinants of Laplacians. J. Funct. Anal. 80, 148-211 (1988) · Zbl 0653.53022 · doi:10.1016/0022-1236(88)90070-5 |
| [101] | Qi, F., Cui, R.-Q., Chen, C.-P., Guo, B.-N.: Some completely monotonic functions involving Polygamma functions and an application. J. Math. Anal. Appl. 310, 303-308 (2005) · Zbl 1074.33005 · doi:10.1016/j.jmaa.2005.02.016 |
| [102] | Qi, F., Chen, S.-X., Cheung, W.-S.: Logarithmically completely monotonic functions concerning gamma and digamma functions. Integr. Transf. Spec. Funct. 18, 435-443 (2007) · Zbl 1120.33003 · doi:10.1080/10652460701318418 |
| [103] | Quine, J.R., Choi, J.: Zeta regularized products and functional determinants on spheres. Rocky Mountain J. Math. 26, 719-729 (1996) · Zbl 0864.47024 · doi:10.1216/rmjm/1181072081 |
| [104] | Quine, J.R., Heydari, S.H., Song, R.Y.: Zeta regularized products. Trans. Amer. Math. Soc. 338, 213-231 (1993) · Zbl 0774.30030 · doi:10.1090/S0002-9947-1993-1100699-1 |
| [105] | Ramanujan, S.: A series for Euler’s constant \(\gamma\). Messenger Math. 46, 73-80 (1916/1917) |
| [106] | Ruijsenaars, S.: First order analytic difference equations and integrable quantum systems. J. Math. Phys. 38, 1069-1146 (1997) · Zbl 0877.39002 · doi:10.1063/1.531809 |
| [107] | Ruijsenaars, S.: On Barnes’ multiple zeta and gamma functions. Adv. Math. 156, 107-132 (2000) · Zbl 0966.33013 · doi:10.1006/aima.2000.1946 |
| [108] | Sarnak, P.: Determinants of Laplacians. Comm. Math. Phys. 110, 113-120 (1987) · Zbl 0618.10023 · doi:10.1007/BF01209019 |
| [109] | Shallit, J.D., Zikan, K.: A theorem of Goldbach. Amer. Math. Monthly 93, 402-403 (1986) · doi:10.2307/2323614 |
| [110] | Shintani, T.: A proof of the classical Kronecker limit formula. Tokyo J. Math. 3, 191-199 (1980) · Zbl 0462.10014 · doi:10.3836/tjm/1270472992 |
| [111] | Srivastava, H.M.: A unified presentation of certain classes of series of the Riemann Zeta function. Riv. Mat. Univ. Parma Ser. 4 14, 1-23 (1988) · Zbl 0659.10047 |
| [112] | Srivastava, H.M.: Sums of certain series of the Riemann Zeta function. J. Math. Anal. Appl. 134, 129-140 (1988) · Zbl 0632.10040 · doi:10.1016/0022-247X(88)90013-3 |
| [113] | Srivastava, H.M.: Some rapidly converging series for \(\zeta\)(2\(n\) + 1). Proc. Amer. Math. Soc. 127, 385-396 (1999) · Zbl 0903.11020 · doi:10.1090/S0002-9939-99-04945-X |
| [114] | Srivastava, H.M., Choi, J.: Series Associated with the Zeta and Related Functions. Kluwer Academic Publishers, Dordrecht (2001) · Zbl 1014.33001 · doi:10.1007/978-94-015-9672-5 |
| [115] | Srivastava, H.M., Choi, J.: Zeta and \(q\)-Zeta Functions and Associated Series and Integrals. Elsevier Science Publishers, Amsterdam (2012) · Zbl 1239.33002 |
| [116] | Steiner, F.: On Selberg’s Zeta function for compact Riemann surfaces. Phys. Lett. B 188, 447-454 (1987) · doi:10.1016/0370-2693(87)91646-7 |
| [117] | Terras, A.: Harmonic Analysis on Symmetric Spaces and Applications. vol. I, Springer, Berlin (1985) · Zbl 0574.10029 · doi:10.1007/978-1-4612-5128-6 |
| [118] | Vardi, I.: Determinants of Laplacians and multiple Gamma functions. SIAM J. Math. Anal. 19, 493-507 (1988) · Zbl 0641.33003 · doi:10.1137/0519035 |
| [119] | Vignéras, M.-F.: L’équation fonctionnelle de la fonction zêta de Selberg du groupe moudulaire \(PSL\)(2,\(Z\)). In: Journées Arithmétiques de Luminy (International Colloquium of the CNRS, Centre University of Luminy Luminy, 1978), pp. 235-249. Astérisque, vol. 61. Société mathématique de France, Paris (1979) |
| [120] | Voros, A.: The Hadamard factorization of the Selberg Zeta function on a compact Riemann surface. Phys. Lett. B 180, 245-246 (1986) · doi:10.1016/0370-2693(86)90303-5 |
| [121] | Voros, A.: Special functions, spectral functions and the Selberg Zeta function. Comm. Math. Phys. 110, 439-465 (1987) · Zbl 0631.10025 · doi:10.1007/BF01212422 |
| [122] | Wade, W.R.: An Introduction to Analysis, 3rd edn. Pearson Prentice Hall, Upper Saddle River (2004) |
| [123] | Walfisz, A.: Weylsche Exponentialsummen in der Neueren Zahlentheorie, pp. 114-115. B.G. Teubner, Leipzig (1963) · Zbl 0146.06003 |
| [124] | Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions, 4th edn. Cambridge University Press, Cambridge (1963) · Zbl 0951.30002 |
| [125] | Wilf, H.S.: Problem 10588. Amer. Math. Monthly 104, 456 (1997) · doi:10.2307/2974795 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
