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1997 | 40 | 1 | 429-441
Tytuł artykułu

The multiple gamma function and its q-analogue

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We give an asymptotic expansion (the higher Stirling formula) and an infinite product representation (the Weierstrass product formula) of the Vignéras multiple gamma function by considering the classical limit of the multiple q-gamma function.
Słowa kluczowe
Rocznik
Tom
40
Numer
1
Strony
429-441
Opis fizyczny
Daty
wydano
1997
Twórcy
autor
  • Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169, Japan
  • Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169, Japan
Bibliografia
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  • [2] E. W. Barnes, The theory of G-function, Quart. J. Math. 31 (1899), pp. 264-314.
  • [3] E. W. Barnes, Genesis of the double gamma function, Proc. London. Math. Soc. 31 (1900), pp. 358-381.
  • [4] E. W. Barnes, The theory of the double gamma function, Phil. Trans. Royal Soc. (A) 196 (1900), pp. 265-388.
  • [5] E. W. Barnes, On the theory of the multiple gamma functions, Trans. Cambridge Phil. Soc. 19 (1904), pp. 374-425.
  • [6] J. Dufresnoy et C. Pisot, Sur la relation fonctionnelle f(x+1)-f(x)=ϕ(x), Bull. Soc. Math. Belgique. 15 (1963), pp. 259-270.
  • [7] G. H. Hardy, On the expression of the double zeta-function and double gamma function in terms of elliptic functions, Trans. Cambridge. Phil. Soc. 20 (1905), pp. 395-427.
  • [8] G. H. Hardy, On double Fourier series and especially these which represent the double zeta-function and incommensurable parameters, Quart. J. Math. 37, (1906), pp. 53-79.
  • [9] F. H. Jackson, A generalization of the functions Γ(n) and $x^n$, Proc. Roy. Soc. London. 74 (1904), pp. 64-72.
  • [10] F. H. Jackson, The basic gamma function and the elliptic functions, Proc. Roy. Soc. London. A 76 (1905), pp. 127-144.
  • [11] T. Koornwinder, Jacobi function as limit cases of q-ultraspherical polynomial, J. Math. Anal. and Appl 148 (1990), pp. 44-54.
  • [12] N. Kurokawa, Multiple sine functions and Selberg zeta functions, Proc. Japan. Acad. 67 A (1991), pp. 61-64.
  • [13] N. Kurokawa, Multiple zeta functions; an example, Adv. Studies. Pure. Math. 21 (1992), pp. 219-226.
  • [14] N. Kurokawa, Gamma factors and Plancherel measures, Proc. Japan. Acad. 68 A (1992), pp. 256-260.
  • [15] N. Kurokawa, On a q-analogues of multiple sine functions, RIMS. kokyuroku 843. (1992), pp. 1-10
  • [16] N. Kurokawa, Lectures delivered at Tokyo Institute of Technology, 1993.
  • [17] Yu. Manin, Lectures on Zeta Functions and Motives, Asterisque. 228 (1995), pp. 121-163.
  • [18] D. S. Moak, The q-analogue of Stirling Formula, Rocky Mountain J. Math,14 (1984),pp. 403-413.
  • [19] M. Nishizawa, On a q-analogue of the multiple gamma functions, to appear in Lett. Math. Phys. q-alg/9505086.
  • [20] T. Shintani, On a Kronecker limit formula for real quadratic fields, J. Fac. Sci. Univ. Tokyo Sect. 1A. Vol 24 (1977), pp 167-199.
  • [21] T. Shintani, A proof of Classical Kronecker limit formula, Tokyo J. Math. Vol.3 (1980), pp 191-199.
  • [22] K. Ueno and M. Nishizawa, Quantum groups and zeta-functions in: J. Lukierski, Z.Popowicz and J.Sobczyk (eds.) 'Quantum Groups: Formalism and Applications' Proceedings of the XXX-th Karpacz Winter School. pp. 115-126 Polish Scientific Publishers PWN. hep-th/9408143.
  • [23] K. Ueno and M. Nishizawa, in preparation.
  • [24] I. Vardi, Determinants of Laplacians and multiple gamma functions, SIAM. J. Math. Anal 19 (1988), pp. 493-507.
  • [25] M. F. Vignéras, L'équation fonctionnelle de la fonction zeta de Selberg de groupe modulaire PSL(2,Z), Asterisque. 61 (1979), pp. 235-249.
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  • [27] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Fourth edition, Cambrige Univ. Press.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-bcpv40z1p429bwm
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