Results 1 - 10 of 280
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[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] 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] 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] 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] 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] 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.
[en] Starting from equations obeyed by functions involving the first or the second derivatives of the biconfluent Heun function, we construct two expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta functions. The first series applies single Beta functions as expansion functions, while the second one involves a combination of two Beta functions. The coefficients of expansions obey four- and five-term recurrence relations, respectively. It is shown that the proposed technique is potent to produce series solutions in terms of other special functions. Two examples of such expansions in terms of the incomplete Gamma-functions are presented
[en] This paper is devoted to extend the notion of almost convergence and its statistical forms with respect to the difference operator involving (p, q)-gamma function and an increasing sequence of positive numbers. We firstly introduce some new concepts of almost -statistical convergence, statistical almost -convergence and strong almost -convergence. Moreover, we present some inclusion relations between these newly proposed methods and give some counterexamples to show that these are non-trivial generalizations of existing literature on this topic. We then prove a Korovkin type approximation theorem for functions of two variables through statistically almost -convergence and also present an illustrative example via bivariate non-tensor type Meyer–König and Zeller generalization of Bernstein power series. Furthermore, we estimate the rate of almost convergence of approximating linear operators by means of the modulus of continuity and derive some Voronovskaja type results by using the generalized Meyer–König and Zeller operators. Finally, some computational and geometrical interpretations for the convergence of operators to a function are presented.