Some mathematical constants. (English) Zbl 1117.33003
This expository paper describes the appearance of various mathematical constants related to the Gamma function and multiple Gamma functions. The most famous of these is Euler’s constant \(\gamma\), and the author lists many integral and series representations for \(\gamma\), as well as infinite product formulas containing \(\gamma\). Among other constants treated are the Glaisher-Kinkelin constant \(A\), the Bendersky-Adamchik constants \(B\) and \(C\), and related families of constants.
Reviewer: Tom M. Apostol (Pasadena)
MSC:
| 33B15 | Gamma, beta and polygamma functions |
Keywords:
Euler-Mascheroni constant; Glaisher-Kinkelin constant; Bendersky-Adamchik constants; determinants of Laplacians; Euler-Maclaurin summation formula; gamma and multiple gamma functions; Riemann and Hurwitz zeta functions; Stirling’s formulaReferences:
| [1] | M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Applied Math. Ser. 55, National Bureau of Standards, Washington, DC, 1964; reprinted by Dover Publications, New York, 1965.; M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Applied Math. Ser. 55, National Bureau of Standards, Washington, DC, 1964; reprinted by Dover Publications, New York, 1965. · Zbl 0171.38503 |
| [2] | Adamchik, V. S., Polygamma functions of negative order, J. Comput. Appl. Math., 100, 191-199 (1998) · Zbl 0936.33001 |
| [3] | Adamchik, V. S., On the Barnes function, (Proceedings of the 2001 International Symposium on Symbolic and Algebraic Computation (London, Ontario; July 22-25, 2001) (2001), ACM Press: ACM Press New York) · Zbl 1356.33001 |
| [4] | Adamchik, V. S., The multiple Gamma function and its application to computation of series, The Ramanujan J., 9, 271-288 (2005) · Zbl 1088.33014 |
| [5] | Alexeiewsky, W. P., Uber eine Classe von Funktionen, die der Gammafunktion analog sind, Leipzig Weidmannsche Buchhandlung, 46, 268-275 (1894) |
| [6] | Anglesio, J., Problems and solutions, Amer. Math. Monthly, 103, 427 (1996) |
| [7] | Anglesio, J., Problems and solutions, Amer. Math. Monthly, 103, 903 (1996) |
| [8] | Apostol, T. M., Some series involving the Riemann Zeta function, Proc. Amer. Math. Soc., 5, 239-243 (1954) · Zbl 0055.06903 |
| [9] | Barnes, E. W., The theory of the G-function, Quart. J. Math., 31, 264-314 (1899) · JFM 30.0389.02 |
| [10] | Barnes, E. W., Genesis of the double Gamma function, Proc. London Math. Soc., 31, 358-381 (1900) · JFM 30.0389.03 |
| [11] | 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 |
| [12] | Barnes, E. W., On the theory of the multiple Gamma functions, Trans. Cambridge Philos. Soc., 19, 374-439 (1904) · JFM 35.0462.01 |
| [13] | Bendersky, L., Sur la fonction Gamma généralisée, Acta Math., 61, 263-322 (1933) · Zbl 0008.07001 |
| [14] | Campbell, R., Les Intégrals Eulériennes Et Leurs Applications (1966), Dunod · Zbl 0174.36201 |
| [15] | Carrier, G. F.; Krook, M.; Pearson, C. E., Functions of a Complex Variable (1983), Hod Books: Hod Books Ithaca, New York · Zbl 0146.29801 |
| [16] | P. Cassou-Noguès, Analogues p-adiques des fonctions Γ;-multiples in “Journées Arithmétiques de Luminy” (Colloq. Internat. CNRS, Centre Univ. Luminy, Luminy, 1978), pp. 43-55, Ast ’erisque 61 (1979) Soc. Math. France, Paris.; P. Cassou-Noguès, Analogues p-adiques des fonctions Γ;-multiples in “Journées Arithmétiques de Luminy” (Colloq. Internat. CNRS, Centre Univ. Luminy, Luminy, 1978), pp. 43-55, Ast ’erisque 61 (1979) Soc. Math. France, Paris. · Zbl 0425.12018 |
| [17] | Choi, J., Determinant of Laplacian on \(S^3\), Math. Japon., 40, 155-166 (1994) · Zbl 0806.58053 |
| [18] | Choi, J., Explicit formulas for the Bernolli polynomials of order n, Indian J. Pure Appl. Math., 27, 667-674 (1996) · Zbl 0860.11009 |
| [19] | J. Choi, Integral and series representations for the Euler’s constant, in: Proceedings of the Seventh Conference on Real and Complex Analysis, Hiroshima University, Japan, 2003, pp. 43-55.; J. Choi, Integral and series representations for the Euler’s constant, in: Proceedings of the Seventh Conference on Real and Complex Analysis, Hiroshima University, Japan, 2003, pp. 43-55. |
| [20] | 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 |
| [21] | J. Choi, J. Lee, 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, 2003, pp. 36-53.; J. Choi, J. Lee, 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, 2003, pp. 36-53. |
| [22] | Choi, J.; Lee, J.; Srivastava, H. M., A generalization of Wilf’s formula, Kodai Math. J., 26, 44-48 (2003) · Zbl 1040.11062 |
| [23] | Choi, J.; Nash, C., Integral representations of the Kikelin’s constant A, Math. Japon., 45, 223-230 (1997) · Zbl 0871.33001 |
| [24] | 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 |
| [25] | Choi, J.; Seo, T. Y., The double Gamma function, East Asian Math. J., 13, 159-174 (1997) |
| [26] | Choi, J.; Seo, T. Y., Integral formulas for Euler’s constant, Comm. Korean Math. Soc., 13, 683-689 (1998) · Zbl 1022.11067 |
| [27] | 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 |
| [28] | Choi, J.; Srivastava, H. M., Sums associated with the Zeta function, J. Math. Anal. Appl., 206, 103-120 (1997) · Zbl 0869.11067 |
| [29] | Choi, J.; Srivastava, H. M., Certain classes of series involving the Zeta function, J. Math. Anal. Appl., 231, 91-117 (1999) · Zbl 0932.11054 |
| [30] | 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 |
| [31] | 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 |
| [32] | Choi, J.; Srivastava, H. M., A certain class of series associated with the Zeta function, Integral Transform. Spec. Funct., 12, 237-250 (2001) · Zbl 1023.11043 |
| [33] | 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 |
| [34] | 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 |
| [35] | 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 |
| [36] | Choi, J.; Srivastava, H. M.; Adamchik, V. S., Multiple Gamma and related functions, Appl. Math. Comput., 134, 515-533 (2003) · Zbl 1026.33003 |
| [37] | 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 |
| [38] | 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 |
| [39] | Conway, J. B., Functions of One Complex Variable (1978), Springer-Verlag: Springer-Verlag New York, Heidelberg and Berlin |
| [40] | D’Hoker, E.; Phong, D. H., On determinant of Laplacians on Riemann surface, Comm. Math. Phys., 104, 537-545 (1986) · Zbl 0599.30073 |
| [41] | Dittrich, W.; Reuter, M., Effective QCD-Lagrangian with \(ξ\)-function regularization, Phys. Lett. B, 128, 321-326 (1983) |
| [42] | Dufresnoy, J.; Pisot, Ch., Sur la relation fonctionnelle \(f(x+1)\)−\(f(x)=ϕ(x)\), Bull. Soc. Math. Belg., 15, 259-270 (1963) · Zbl 0122.09802 |
| [43] | Edwards, J., A Treatise on the Integral Calculus with Applications: Examples and Problems, vols. 1 and 2 (1954), Chelsea Publishing Company: Chelsea Publishing Company New York |
| [44] | Elizalde, E., Derivative of the generalized Riemann Zeta function \(ζ(z,q)\) at \(z\)=−1, J. Phys. A: Math. Gen., 18, 1637-1640 (1985) · Zbl 0603.10039 |
| [45] | Elizalde, E., An asymptotic expansion for the first derivative of the generalized Riemann Zeta function, Math. Comput., 47, 347-350 (1986) · Zbl 0603.10040 |
| [46] | Elizalde, E.; Odintsov, S. D.; Romeo, A.; Bytsenko, A. A.; Zerbini, S., Zeta Regularization Techniques with Applications (1994), World Scientific Publishing Company: World Scientific Publishing Company Singapore, New Jersey, London, and Hong Kong · Zbl 1050.81500 |
| [47] | Elizalde, E.; Romeo, A., An integral involving the generalized Zeta function, Int. J. Math. Math. Sci., 13, 453-460 (1990) · Zbl 0708.11039 |
| [48] | Euler, L., Comm. Acad. Petropol., 7, 156 (1734-1735) |
| [49] | Finch, S. R., Mathematical Constants (2003), Cambridge University Press: Cambridge University Press Cambridge, New York, Port Melbourne, Madrid, Cape Town · Zbl 1054.00001 |
| [50] | Friedman, E.; Ruijsenaars, S., Shintani-Barnes zeta and gamma functions, Adv. Math., 187, 362-395 (2004) · Zbl 1112.11042 |
| [51] | Glaisher, J. W.L., On the history of Euler’s constant, Messenger Math., 1, 25-30 (1872) · JFM 05.0144.01 |
| [52] | Glaisher, J. W.L., On the product \(1^12^2\)⋯\(n^n\), Messenger Math., 7, 43-47 (1877) · JFM 09.0116.02 |
| [53] | Gosper, R. W., \( \int_{n / 4}^{m / 6} \ln \Gamma(s) d s\), Fields Inst. Comm., 14, 71-76 (1997) · Zbl 0870.33001 |
| [54] | Gradshteyn, I. S.; Ryzhik, I. M., Tables of Integrals, Series, and Products (1980), Academic Press: Academic Press New York, Corrected and enlarged ed. prepared by A. Jeffrey · Zbl 0521.33001 |
| [55] | Hardy, G. H., Divergent Series (1949), Clarendon (Oxford Univ.) Press: Clarendon (Oxford Univ.) Press Oxford-London-New York · Zbl 0032.05801 |
| [56] | Havil, J., Gamma (Exploring Euler’s Constant) (2003), Princeton University Press: Princeton University Press Princeton and Oxford · Zbl 1023.11001 |
| [57] | Hölder, O., Über eine Transcendente Funktion (1886), Dieterichsche Verlags-Buchhandlung: Dieterichsche Verlags-Buchhandlung Göttingen, pp. 514-522 · JFM 18.0376.01 |
| [58] | Kanemitsu, S.; Kumagai, H.; Yoshimoto, M., Sums involving the Hurwitz zeta function, Ramanujan J., 5, 5-19 (2001) · Zbl 0989.11043 |
| [59] | 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) · ERAM 057.1509cj |
| [60] | Knopp, K., Theory and Application of Infinite Series (1951), Hafner Publ. Company: Hafner Publ. Company New York, (2nd English ed., translated from the 2nd German ed. revised in accordance with the 4th German Ed. by R.C.H. Young) · JFM 54.0222.09 |
| [61] | Knuth, D. E., Euler’s constant to 1271 places, Math. Comput., 16, 275-281 (1962) · Zbl 0117.10801 |
| [62] | Kumagai, H., The determinant of the Laplacian on the \(n\)-sphere, Acta Arith., 91, 199-208 (1999) · Zbl 0946.11021 |
| [63] | Lewin, L., Polylogarithms and Associated Functions (1981), Elsevier (North-Holland): Elsevier (North-Holland) New York, London, and Amsterdam · Zbl 0465.33001 |
| [64] | Matsumoto, K., Asymptotic series for double Zeta, double Gamma, and Hecke L-functions, Math. Proc. Cambridge Philos. Soc., 123, 385-405 (1998) · Zbl 0903.11021 |
| [65] | Osgood, B.; Phillips, R.; Sarnak, P., Extremals of determinants of Laplacians, J. Funct. Anal., 80, 148-211 (1988) · Zbl 0653.53022 |
| [66] | Quine, J. R.; Choi, J., Zeta regularized products and functional determinants on spheres, Rocky Mountain J. Math., 26, 719-729 (1996) · Zbl 0864.47024 |
| [67] | Ramanujan, S., A series for Euler’s constant \(γ\), Messenger Math., 46, 73-80 (1916-1917) · JFM 46.0340.01 |
| [68] | Ruijsenaars, S., First order analytic difference equations and integrable quantum systems, J. Math. Phys., 38, 1069-1146 (1997) · Zbl 0877.39002 |
| [69] | Ruijsenaars, S., On Barnes’ multiple zeta and gamma functions, Adv. Math., 156, 107-132 (2000) · Zbl 0966.33013 |
| [70] | Sarnak, P., Determinants of Laplacians, Comm. Math. Phys., 110, 113-120 (1987) · Zbl 0618.10023 |
| [71] | Shintani, T., A proof of the classical Kronecker limit formula, Tokyo J. Math., 3, 191-199 (1980) · Zbl 0462.10014 |
| [72] | 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 |
| [73] | Srivastava, H. M.; Choi, J., Series Associated with the Zeta and Related Functions (2001), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, Boston and London · Zbl 1014.33001 |
| [74] | Steiner, F., On Selberg’s Zeta function for compact Riemann surfaces, Phys. Lett. B, 188, 447-454 (1987) |
| [75] | Vardi, I., Determinants of Laplacians and multiple Gamma functions, SIAM J. Math. Anal., 19, 493-507 (1988) · Zbl 0641.33003 |
| [76] | M.-F. Vignéras, L’Équation fonctionnelle de la fonction zêta de Selberg du groupe moudulaire PSL(2,Z), “Journées Arithmétiques de Luminy” (Collq. Internat. CNRS, hfill Centre Univ. Luminy, 1978), Ast ’erisque 61 (1979) 235-249.; M.-F. Vignéras, L’Équation fonctionnelle de la fonction zêta de Selberg du groupe moudulaire PSL(2,Z), “Journées Arithmétiques de Luminy” (Collq. Internat. CNRS, hfill Centre Univ. Luminy, 1978), Ast ’erisque 61 (1979) 235-249. · Zbl 0401.10036 |
| [77] | Voros, A., The Hadamard factorization of the Selberg Zeta function on a compact Riemann surface, Phys. Lett. B, 180, 245-246 (1986) |
| [78] | Voros, A., Special functions, spectral functions and the Selberg Zeta function, Comm. Math. Phys., 110, 439-465 (1987) · Zbl 0631.10025 |
| [79] | A. Walfisz, Weylsche exponentialsummen in der neueren Zahlentheorie, Leipzig: B.G. Teubner (1963) 114-115.; A. Walfisz, Weylsche exponentialsummen in der neueren Zahlentheorie, Leipzig: B.G. Teubner (1963) 114-115. · Zbl 0146.06003 |
| [80] | Whittaker, E. T.; Watson, G. N., A Course of Modern Analysis (1963), Cambridge University Press: Cambridge University Press Cambridge, London and New York · Zbl 0108.26903 |
| [81] | Wilf, H. S., Problem 10588, Amer. Math. Monthly, 104, 456 (1997) |
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