Results 1 - 10 of 288
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[en] The method of brackets is a method for the evaluation of definite integrals based on a small number of rules. This is employed here for the evaluation of Mellin-Barnes integral. The fundamental idea is to transform these integral representations into a bracket series to obtain their values. The expansion of the gamma function in such a series constitute the main part of this new application. The power and flexibility of this procedure is illustrated with a variety of examples.
[en] In the paper, the authors investigate properties, including the monotonicity, logarithmic concavity, concavity, and inequalities, of a sequence arising from geometric probability for pairs of hyperplanes intersecting with a convex body.
[en] We obtain a new solution to the star–triangle relation for an Ising-type model with two kinds of spin variables at each lattice site, taking continuous real values and arbitrary integer values, respectively. The Boltzmann weights are manifestly real and positive. They are expressed through the Euler gamma function and depend on sums and differences of spins at the ends of an edge of the lattice. (paper)
[en] In this paper, the authors construct a new approximation of the factorial function and a double inequality for the gamma function which are based on Windschitl’s formula. Moreover, for demonstrating the superiority of our new series over Windschitl’s formula, Nemes’ formula and Mortici’s formula, some numerical computations are given.
[en] Several determinants with gamma functions as elements are evaluated. These kinds of determinants are encountered, for example, in the computation of the probability density of the determinant of random matrices. The s-shifted factorial is defined as a generalization for non-negative integers of the power function, the rising factorial (or Pochammer's symbol) and the falling factorial. It is a special case of a polynomial sequence of the binomial type studied in combinatorics theory. In terms of the gamma function, an extension is defined for negative integers and even complex values. Properties, mainly composition laws and binomial formulae, are given. They are used to evaluate families of generalized Vandermonde determinants with s-shifted factorials as elements, instead of power functions
[en] We prove a recently conjectured star–star relation, which plays the role of an integrability condition for a class of 2D Ising-type models with multicomponent continuous spin variables. Namely, we reduce this relation to an identity for elliptic gamma functions, previously obtained by Rains. (fast track communication)
[en] We study a problem of finding good approximations to Euler's constant γ=lim→∞ Sn, where Sn = Σk=Ln (1)/k-log(n+1), by linear forms in logarithms and harmonic numbers. In 1995, C. Elsner showed that slow convergence of the sequence Sn can be significantly improved if Sn is replaced by linear combinations of Sn with integer coefficients. In this paper, considering more general linear transformations of the sequence Sn we establish new accelerating convergence formulae for γ. Our estimates sharpen and generalize recent Elsner's, Rivoal's and author's results. (author)
[en] In this paper, we proceed to study properties of Mellin–Barnes (MB) transforms of Usyukina–Davydychev (UD) functions. In our previous papers (Allendes et al., 2013 , Kniehl et al., 2013 ), we showed that multi-fold Mellin–Barnes (MB) transforms of Usyukina–Davydychev (UD) functions may be reduced to two-fold MB transforms and that the higher-order UD functions may be obtained in terms of a differential operator by applying it to a slightly modified first UD function. The result is valid in dimensions, and its analog in dimensions exits, too (Gonzalez and Kondrashuk, 2013 ). In Allendes et al. (2013) , the chain of recurrence relations for analytically regularized UD functions was obtained implicitly by comparing the left-hand side and the right-hand side of the diagrammatic relations between the diagrams with different loop orders. In turn, these diagrammatic relations were obtained using the method of loop reduction for the triangle ladder diagrams proposed in 1983 by Belokurov and Usyukina. Here, we reproduce these recurrence relations by calculating explicitly, via Barnes lemmas, the contour integrals produced by the left-hand sides of the diagrammatic relations. In this a way, we explicitly calculate a family of multi-fold contour integrals of certain ratios of Euler gamma functions. We make a conjecture that similar results for the contour integrals are valid for a wider family of smooth functions, which includes the MB transforms of UD functions.
[en] The problem of reconstructing the γ-multiplicity distribution from the measured γ-fold distribution in a fusion reaction is revisited. The known moments of the γ-multiplicity distribution are used to reconstruct the distribution itself, utilising the fact that γ-multiplicity is a discrete variable. It is shown that the computational challenges encountered in this method can be overcome by switching to multiple precision calculation.