zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Multiple gamma and related functions. (English) Zbl 1026.33003
The multiple Barnes function $G_n(z)$ obeys the functional equation $$G_{n+1}(z+1)=G_n(z)G_{n+1}(z),$$ with initial conditions $G_1(z)=\Gamma(z)$, $G_n(1)=1.$ In this interesting article, a large number of properties of $G_2(z)$, $G_3(z)$ and some related functions are surveyed, and some new formulas are given.

33B15Gamma, beta and polygamma functions
33C99Hypergeometric functions
Full Text: DOI
[1] Adamchik, V. S.: Polygamma functions of negative order. J. comput. Appl. math. 100, 191-199 (1998) · Zbl 0936.33001
[2] V.S. Adamchik, Integral representations for the Barnes function, 2001 (Preprint)
[3] V.S. Adamchik, On the Barnes function, in: B. Mourrain (Ed.), Proceedings of the 2001 International Symposium on Symbolic and Algebraic Computation, London, July 22--25, 2001, ACM Press, New York, 2001, pp. 15--20
[4] Adamchik, V. S.; Srivastava, H. M.: Some series of the zeta and related functions. Analysis 18, 131-144 (1998) · Zbl 0919.11056
[5] Alexeiewsky, W. P.: Über eine classe von funktionen, die der gammafunktion analog sind. Leipzig weidmannsche buchhandlung 46, 268-275 (1894)
[6] Ayoub, R.: Euler and the zeta function. Amer. math. Monthly 81, 1067-1086 (1974) · Zbl 0293.10001
[7] Barnes, E. W.: The theory of the G-function. Quart. J. Math. 31, 264-314 (1899) · Zbl 30.0389.02
[8] Barnes, E. W.: On the theory of the multiple gamma function. Trans. Cambridge philos. Soc. 19, 374-439 (1904) · Zbl 35.0462.01
[9] Bendersky, L.: Sur la fonction gamma généralisée. Acta math. 61, 263-322 (1933) · Zbl 59.0373.02
[10] Chen, M. -P.; Srivastava, H. M.: Some families of series representations for the Riemann ${\zeta}$(3). Resultate math. 33, 179-197 (1998) · Zbl 0908.11039
[11] Choi, J.: Determinant of Laplacian on S3. Math. japon. 40, 155-166 (1994) · Zbl 0806.58053
[12] Choi, J.: Explicit formulas for Bernoulli polynomials of order n. Indian J. Pure appl. Math. 27, 667-674 (1996) · Zbl 0860.11009
[13] Choi, J.; Nash, C.: Integral representations of the kinkelin’s constant A. Math. japon. 45, 223-230 (1997) · Zbl 0871.33001
[14] Choi, J.; Seo, T. Y.: The double gamma function. East asian math. J. 13, 159-174 (1997)
[15] Choi, J.; Srivastava, H. M.: Sums associated with the zeta function. J. math. Anal. appl. 206, 103-120 (1997) · Zbl 0869.11067
[16] Choi, J.; Srivastava, H. M.: Certain classes of series involving the zeta function. J. math. Anal. appl. 231, 91-117 (1999) · Zbl 0932.11054
[17] 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
[18] Choi, J.; Srivastava, H. M.: Certain classes of series associated the zeta function and multiple gamma functions. J. comput. Appl. math. 118, 87-109 (2000) · Zbl 0969.11030
[19] 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
[20] Cvijović, D.; Klinowski, J.: Closed-form summation of some trigonometric series. Math. comput. 64, 205-210 (1995) · Zbl 0824.42002
[21] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G.: Higher transcendental functions. (1953) · Zbl 0051.30303
[22] Glaisher, J. W. L.: On the product $11{\cdot}22{\cdot}$33\cdotsnn. Messenger math. 7, 43-47 (1877)
[23] Jr., R. W. Gosper: \intn/4m/$6ln{\Gamma}(z)$dz. Fields inst. Comm. 14, 71-76 (1997)
[24] Gradshteyn, I. S.; Ryzhik, I. M.: Table of integrals, series, and products. (1980) · Zbl 0521.33001
[25] Grosjean, C. C.: Formulae concerning the computation of the clausen integral cl2({$\theta$}). J. comput. Appl. math. 11, 331-342 (1984) · Zbl 0553.33006
[26] Hansen, E. R.: A table of series and products. (1975) · Zbl 0438.00001
[27] Kanemitsu, S.; Kumagai, H.; Yoshimoto, M.: Sums involving the Hurwitz zeta function. Ramanujan J. 5, 5-19 (2001) · Zbl 0989.11043
[28] Lewin, L.: Polylogarithms and associated functions. (1981) · Zbl 0465.33001
[29] A.P. Prudnikov, Yu.A. Bryčkov, O.I. Maričev, Integrals and Series (Elementary Functions), Nauka, Moscow, 1981 (in Russian); see also Integrals and Series, Vol. I: Elementary Functions (Translated from the Russian by N.M. Queen), Gordon and Breach, New York, 1986
[30] Quine, J. R.; Choi, J.: Zeta regularized products and functional determinants on spheres. Rocky mountain J. Math. 26, 719-729 (1996) · Zbl 0864.47024
[31] Spiegel, M. R.: Mathematical handbook. (1968)
[32] Srivastava, H. M.: A unified presentation of certain classes of series of the Riemann zeta function. Riv. mat. Univ. parma 4, No. 14, 1-23 (1988) · Zbl 0659.10047
[33] Srivastava, H. M.: Some rapidly converging series for ${\zeta}$(2n+1). Proc. amer. Math. soc. 127, 385-396 (1999) · Zbl 0903.11020
[34] Srivastava, H. M.: A note on the closed-form summation of some trigonometric series. Kobe J. Math. 16, 177-182 (1999) · Zbl 0958.65147
[35] Srivastava, H. M.: Some simple algorithms for the evaluations and representations of the Riemann zeta function at positive integer arguments. J. math. Anal. appl. 246, 331-351 (2000) · Zbl 0957.11036
[36] Srivastava, H. M.; Choi, J.: Series associated with the zeta and related functions. (2001) · Zbl 1014.33001
[37] Srivastava, H. M.; Glasser, M. L.; Adamchik, V. S.: Some definite integrals associated with the Riemann zeta function. Z. anal. Anwendungen 19, 831-846 (2000) · Zbl 0979.11042
[38] Titchmarsh, E. C.: The theory of the Riemann zeta-function. (1951) · Zbl 0042.07901
[39] Vardi, I.: Determinants of Laplacians and multiple gamma functions. SIAM J. Math. anal. 19, 493-507 (1988) · Zbl 0641.33003
[40] Vignéras, M. -F.: L’équation fonctionnelle de la fonction Zêta de Selberg du groupe moudulaire $PSL(2,Z)$. Astérisque 61, 235-249 (1979)
[41] Voros, A.: Special functions, spectral functions and the Selberg zeta function. Comm. math. Phys. 110, 439-465 (1987) · Zbl 0631.10025
[42] 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. (1963) · Zbl 0951.30002