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Multiple gamma functions, multiple sine functions, and Appell’s O-functions
The Ramanujan Journal, 2010zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Determinants of Laplacians and Multiple Gamma Functions
SIAM Journal on Mathematical Analysis, 1988The author reinterpretes the classical formula \(\Gamma(.)=\sqrt{\pi}\) in the form \[ \Gamma(.)=2^{-1/2}(\det \Delta_ 1)^{1/4}, \] where \(\Delta_ 1=-d^ 2/dx^ 2\) denotes the Laplacian on \(S^ 1\). He then introduces so-called multiple Gamma functions \(\Gamma_ n\) for all \(n\geq 0\) and then his main result states that \(\Gamma_ n(.)\) can be ...
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On Riemann's Functional Equation with Multiple Gamma Factors
The Annals of Mathematics, 1958(1.3) (2P7r)-isAl(s)cV(s) = (2P7Tr)4 ( S)M( S) of the following description. We have (1.4) A1(s) = F1' (p fs + Cf), (1.5) A2(S) = ll'+1 F(pS + Ce), where G > 1, H-G > 1, HG ;G. The numbers c1, * C , cH are unrestricted complex constants. The constants p1, ** , PH however are real numbers, and (1.6) PA > ?, h 1, * H (1.7) Ah + Pa + * + PHl= Actually we ...
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Functional Equations With Multiple Gamma Factors and the Average Order of Arithmetical Functions
The Annals of Mathematics, 1962This is a further development of earlier work by the same authors [Ann. Math. (2) 74, 1--23 (1961; Zbl 0107.03702); Acta Arith. 6, 487--503 (1961; Zbl 0101.03703); C. R. Acad. Sci., Paris 251, 1333--1335 (1960; Zbl 0093.05203); erratum, p. 2547]. It is assumed that a functional equation \(\Delta(s)\varphi(s)= \Delta(\delta - s)\psi( \delta - s)\) is ...
Chandrasekharan, K. +1 more
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Multiple Gamma Functions and Their Applications
2014The double Gamma function Γ 2 and the multiple Gamma functions Γ 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, Holder, Alexeiewsky, and Kinkelin. Although these functions did not appear in the tables of the most well-known
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Integrals associated with the multiple gamma function
Integral Transforms and Special Functions, 2014The multiple gamma function Γn(z), defined by a recurrence-functional equation as a generalization of the Euler gamma function, is used in many applications of pure and applied mathematics, and theoretical physics. The theory of the multiple gamma function has been related to certain spectral functions in mathematical physics, to the study of ...
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Local adaptive tone mapping with composite multiple gamma functions
2009 16th IEEE International Conference on Image Processing (ICIP), 2009We describe a high-speed method of correcting and compressing the dynamic range of images that can be operated intuitively and naturally. Adaptive operations are conducted for shadow, middle, and highlight tones in the local areas of images. Although natural image processing can be achieved with default parameters, we can set the parameters for each of
Sohsuke Shimoyama +3 more
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Kummer's formula for multiple gamma functions
2006The multiple gamma function is defined as \(\Gamma_r(x)=\exp(\zeta'_r(0, x))\) where \[ \zeta_r(s, x)=\sum_{n_1,\dots,n_r=0}^\infty(n_1+\dots+n_r+x)^{-s} \] is the multiple Hurwitz zeta-function and the differentiation is in the first variable \(s\). Note that \(\Gamma_1(x)=\Gamma(x)/\sqrt{2\pi}\), where \(\Gamma(x)\) is the Euler gamma function. For \(
Kurokawa Nobushige, Koyama Shin-ya
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An elliptic analogue of the multiple gamma function
Journal of Physics A: Mathematical and General, 2001The author introduces an elliptic analogue of Barnes's multiple gamma function, which generalizes the elliptic gamma function of \textit{S. N. M. Ruijsenaars} [J. Math. Phys. 38, No. 2, 1069-1146 (1997; Zbl 0877.39002)]. The fundamental properties of this function are given.
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