The Experts below are selected from a list of 81 Experts worldwide ranked by ideXlab platform
Michitomo Nishizawa - One of the best experts on this subject based on the ideXlab platform.
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infinite product representations fot Multiple Gamma Function
arXiv: Classical Analysis and ODEs, 2004Co-Authors: Michitomo NishizawaAbstract:Two kinds of infinite product representations for Vign\'eras Multiple Gamma Function are presented. As an application of these formulas, a multiplication formula for the Function is derived.
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Multiple Gamma Function its q and elliptic analogue
Rocky Mountain Journal of Mathematics, 2002Co-Authors: Michitomo NishizawaAbstract:Vigneras's Multiple Gamma Function is introduced as a Function satisfying a generalization of the Bohr-Mollerup theorem. An infinite product representation and an asymptotic expansion of the Function are given. Furthermore, its q- and elliptic analogue are introduced as relevant with the defining relations of q-Gamma Function and of elliptic Gamma Function.
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an elliptic analogue of the Multiple Gamma Function
Journal of Physics A, 2001Co-Authors: Michitomo NishizawaAbstract:A hierarchy of Functions including the elliptic Gamma Function is introduced. It can be interpreted as an elliptic analogue of the Multiple Gamma Function and its trigonometric limit coincides with a q-analogue of the Multiple Gamma Function. Some properties of the Functions are considered.
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the Multiple Gamma Function and its q analogue
arXiv: Quantum Algebra, 1996Co-Authors: Kimio Ueno, Michitomo NishizawaAbstract:We give an asymptotic expansion (the higher Stirling formula) and an infinite product representation (the Weierstrass product formula) of the Vign\'{e}ras Multiple Gamma Function by considering the classical limit of the Multiple q-Gamma Function.
Hiroyuki Yoshida - One of the best experts on this subject based on the ideXlab platform.
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On the p -adic absolute CM-period symbol
Algebra and Number Theory, 2020Co-Authors: Tomokazu Kashio, Hiroyuki YoshidaAbstract:The absolute CM-period symbol was defined by the second author using the Multiple Gamma Function. Conjecturally it coincides with Shimura’s period symbol up to multiplication by algebraic numbers. In this paper we define a p-adic analogue of the absolute CM-period symbol using the p-adic Multiple Gamma Function studied by the first author. We present an explicit conjecture with solid evidence.
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Absolute CM-Periods
2020Co-Authors: Hiroyuki YoshidaAbstract:Introduction Multiple Gamma Function and its generalizations The Stark-Shintani conjecture Absolute CM-periods Explicit cone decompositions and applications Applications of a limit formula of Kronecker's type Eisenstein series on $GL(2)$ On higher derivatives of $L$-Functions Transcendental property of CM-periods References Index.
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On p-adic absolute CM-periods I
American Journal of Mathematics, 2008Co-Authors: Tomokazu Kashio, Hiroyuki YoshidaAbstract:The second author defined the absolute CM-period symbol using the Multiple Gamma Function. This symbol is conjectured to give the same value as Shimura's period symbol up to multiplication by algebraic numbers. In this paper, we define a p-adic analogue of the absolute CM-period symbol, using the p-adic Multiple Gamma Function studied by the first author. In the completely split case, we present a conjecture which predicts the exact value of this symbol as well as supporting evidences.
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On absolute CM-periods, II
American Journal of Mathematics, 1998Co-Authors: Hiroyuki YoshidaAbstract:For a CM-fieldK, Shimura defined the period symbolpK by factorizing periods of abelian varieties with complex multiplication. We define the absolute period symbolgK using division values of the Multiple Gamma Function and conjecture that pK coincides with gK up to the multiplication by algebraic numbers. Taking the action of Gal(Q Q) into account, we present a refined version of this conjecture. We show that these conjectures are consistently formulated and discuss various numerical examples which support our conjectures strongly. In our previous paper (Y2), we formulated a conjecture which gives an expres- sion of the derivatives of Artin L-Functions at s = 0 by CM-periods. However we could not express CM-periods themselves by such a conjecture. (This point will be shown explicitly by an example in 7. See also the discussion in the beginning of (Y2), 2.) In the present paper, we shall give a conjecture which expresses CM-periods by the values of the Multiple Gamma Function at division points, and present various numerical examples which support it. Let us explain our ideas and the contents of this paper more precisely. Let K be a CM-field, JK be the set of all isomorphisms of K into C and IK be the free abelian group generated by JK. For every , IK, Shimura defined (S2), (S3) the CM-period pK( , ) C , which is uniquely determined mod Q.T he fundamental properties of the period symbol pK will be reviewed in 1. For a, b C, let us write a b if b =0a nda b Q.U singpK, we can write the Chowla-Selberg formula as
Nobushige Kurokawa - One of the best experts on this subject based on the ideXlab platform.
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On q-Basic Multiple Gamma Functions
International Journal of Mathematics, 2020Co-Authors: Nobushige Kurokawa, Masato WakayamaAbstract:We define a q-analogue of the basic Multiple Gamma Function [Formula: see text] introduced in [17] which differs from the one defined by Barnes [1] via the zeta regularized product. We call it a q-basic Multiple Gamma Function. Using this q-basic Multiple Gamma Function we introduce a q-analogue of the Multiple sine Function of order m + 1. We study properties of such Functions from the periodicity, differential-difference equations and duplication formulas, etc. points of view.
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Values of absolute tensor products
Proceedings of the Japan Academy Series A Mathematical Sciences, 2005Co-Authors: Nobushige KurokawaAbstract:We study values of absolute tensor products (Multiple zeta Functions) at integral arguments. We obtain a simple formula for the absolute value of the double sine Function. We express values of the Multiple Gamma Function related to the Functional equation.
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On q-Basic Multiple Gamma Functions
International Journal of Mathematics, 2003Co-Authors: Nobushige Kurokawa, Masato WakayamaAbstract:We define a q-analogue of the basic Multiple Gamma Function introduced in [17] which differs from the one defined by Barnes [1] via the zeta regularized product. We call it a q-basic Multiple Gamma Function. Using this q-basic Multiple Gamma Function we introduce a q-analogue of the Multiple sine Function of order m + 1. We study properties of such Functions from the periodicity, differential-difference equations and duplication formulas, etc. points of view.
Victor S Adamchik - One of the best experts on this subject based on the ideXlab platform.
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integrals associated with the Multiple Gamma Function
Integral Transforms and Special Functions, 2014Co-Authors: Victor S AdamchikAbstract:The 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 Functional determinants of Laplacians of the n-sphere, to Hecke L-Functions, to the Selberg zeta Function, and to the random matrix theory. There is a wide class of definite integrals and infinite sums appearing in statistical physics (the Potts model) and the lattice theory, which can be computed by means of the Γn(z) Function. This paper presents new integral representations for the Multiple Gamma Function and other mathematical Functions and constants.
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Multiple Gamma Function and its application to computation of series and products
2011Co-Authors: Victor S AdamchikAbstract:The Multiple Gamma Function Γn, defined by a recurrence-Functional equation as a generalization of the Euler Gamma Function, was originally introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of Conrey, Katz and Sarnak, interest in the Barnes Function has been revived. This paper discusses some theoretical aspects of the Γn Function and their applications to summation of series and infinite products.
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The Multiple Gamma Function and Its Application to Computation of Series
The Ramanujan Journal, 2005Co-Authors: Victor S AdamchikAbstract:The Multiple Gamma Function Γ_ n , defined by a recurrence-Functional equation as a generalization of the Euler Gamma Function, was originally introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of Conrey, Katz and Sarnak, interest in the Multiple Gamma Function has been revived. This paper discusses some theoretical aspects of the Γ_ n Function and their applications to summation of series and infinite products.
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Contributions to the Theory of the Barnes Function
arXiv: Classical Analysis and ODEs, 2003Co-Authors: Victor S AdamchikAbstract:This paper presents a family of new integral representations and asymptotic series of the Multiple Gamma Function. The numerical schemes for high-precision computation of the Barnes Gamma Function and Glaisher’s constant are also discussed.
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Multiple Gamma Function and its application to computation of series
arXiv: Classical Analysis and ODEs, 2003Co-Authors: Victor S AdamchikAbstract:The Multiple Gamma Function $\Gamma_n$, defined by a recurrence-Functional equation as a generalization of the Euler Gamma Function, was originally introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of Conrey, Katz and Sarnak, interest in the Multiple Gamma Function has been revived. This paper discusses some theoretical aspects of the $\Gamma_n$ Function and their applications to summation of series and infinite products.
Tomokazu Kashio - One of the best experts on this subject based on the ideXlab platform.
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On the p -adic absolute CM-period symbol
Algebra and Number Theory, 2020Co-Authors: Tomokazu Kashio, Hiroyuki YoshidaAbstract:The absolute CM-period symbol was defined by the second author using the Multiple Gamma Function. Conjecturally it coincides with Shimura’s period symbol up to multiplication by algebraic numbers. In this paper we define a p-adic analogue of the absolute CM-period symbol using the p-adic Multiple Gamma Function studied by the first author. We present an explicit conjecture with solid evidence.
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on the algebraicity of some products of special values of barnes Multiple Gamma Function
arXiv: Number Theory, 2015Co-Authors: Tomokazu KashioAbstract:We consider partial zeta Functions $\zeta(s,c)$ associated with ray classes $c$'s of a totally real field. Stark's conjecture implies that an appropriate product of $\exp(\zeta'(0,c))$'s is an algebraic number which is called a Stark unit. Shintani gave an explicit formula for $\exp(\zeta'(0,c))$ in terms of Barnes' Multiple Gamma Function. Yoshida ``decomposed'' Shintani's formula: he defined the symbol $X(c,\iota)$ satisfying that $\exp(\zeta'(0,c))=\prod_{\iota} \exp(X(c,\iota))$ where $\iota$ runs over all real embeddings of $F$. Hence we can decompose a Stark unit into a product of $[F:\mathbb Q]$ terms. The main result is to show that $([F:\mathbb Q]-1)$ of them are algebraic numbers. We also study a relation between Yoshida's conjecture on CM-periods and Stark's conjecture.
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On p-adic absolute CM-periods I
American Journal of Mathematics, 2008Co-Authors: Tomokazu Kashio, Hiroyuki YoshidaAbstract:The second author defined the absolute CM-period symbol using the Multiple Gamma Function. This symbol is conjectured to give the same value as Shimura's period symbol up to multiplication by algebraic numbers. In this paper, we define a p-adic analogue of the absolute CM-period symbol, using the p-adic Multiple Gamma Function studied by the first author. In the completely split case, we present a conjecture which predicts the exact value of this symbol as well as supporting evidences.