Theq-gamma function forx<0

F. H. Jackson defined aq analogue of the gamma function which extends theq-factorial (n!) _q =1(1+q)(1+q+q ²)...(1+q+q ²+...+q ^n–1) to positivex. Askey studied this function and obtained analogues of most of the classical facts about the gamma function, for 0<q<1. He proved an analogue of the Bohr-Mollerup theorem, which states that a logarithmically convex function satisfyingf(1)=1 andf(x+1)=[(q ^x –1)/(q–1)]f(x) is in fact theq-gamma function He also studied the behavior of _q asq changes and showed that asq1^–, theq-gamma function becomes the ordinary gamma function forx>0.I proved many of these results forq>1. The current paper contains a study of the behavior of _q (x) forx<0 and allq>0. In addition to some basic properties of _q , we will study the behavior of the sequence {x _n (q)} of critical points asn orq changes.     

The local Hölder function of a continuous function

This work focuses on the local Hölder exponent as a measure of the regularity of a function around a given point. We investigate in detail the structure and the main properties of the local Hölder function (i.e., the function that associates to each point its local Hölder exponent). We prove that it is possible to construct a continuous function with prescribed local and pointwise Hölder functions outside a set of Hausdorff dimension 0.     

M‐constants in Orlicz–Lorentz function spaces

In this paper some lower and upper estimates of M‐constants for Orlicz–Lorentz function spaces for both, the Luxemburg and the Amemiya norms, are given. Since degenerated Orlicz functions φ and degenerated weighted sequences ω are also admitted, this investigations concern the most possible wide class of Orlicz–Lorentz function spaces. M‐constants were defined in 1969 by E. A. Lifshits, and used by many authors in the study of lattice structures on Banach spaces, as well as in the fixed point theory.     

Symmetry of two‐parameter homogeneous transition function

In this paper, we study a very interesting problem, that is, symmetry of the two parameters s,t>0 for a two‐parameter homogeneous transition function and obtain some important and interesting results.     

Well-posed reduction formulas for the <Emphasis Type="Italic">q</Emphasis>-Kampé-de-Fériet function

By using the limiting case of Watson’s q-Whipple transformation as n → ∞, we investigate the transformations of the nonterminating q-Kampé-de-Fériet series. Further, new formulas for the transformations and well-posed reduction formulas are established for the basic Clausen hypergeometric series. Several remarkable formulas are also found for new function classes beyond the q-Kampé-de-Fériet function.     

A simplification of Laplace’s method: Applications to the Gamma function and Gauss hypergeometric function

The main difficulties in the Laplace’s method of asymptotic expansions of integrals are originated by a change of variables. We propose a variant of the method which avoids that change of variables and simplifies the computations. On the one hand, the calculation of the coefficients of the asymptotic expansion is remarkably simpler. On the other hand, the asymptotic sequence is as simple as in the standard Laplace’s method: inverse powers of the asymptotic variable. New asymptotic expansions of the Gamma function Γ(z) for large z and the Gauss hypergeometric function ₂F₁(a,b,c;z) for large b and c are given as illustrations. An explicit formula for the coefficients of the classical Stirling expansion of Γ(z) is also given.     

Some properties of compatibility function based on <Emphasis Type="Italic">T</Emphasis>-operation

In this paper, the concept of compatibility function is proposed for fuzzy value with respect to a kind of the binomial operations which is called t-operation. Besides the continuity of compatibility function on three kinds of metric spaces are discussed, respectively.     

Example of a function of two variables that cannot be an <Emphasis Type="Italic">R</Emphasis>-function

We note that the definition of R-functions depends on the choice of a certain surjection and pose the problem of the construction of a function of two variables that is not an R-function for any choice of a surjective mapping. It is shown that the function x ₁ x ₂ − 1 possesses this property. We prove a theorem according to which, in the case of finite sets, every mapping is an R-mapping for a proper choice of a surjection.     

Quadratic Hermite–Padé polynomials associated with the exponential function

The asymptotic behavior of quadratic Hermite–Padé polynomials associated with the exponential function is studied for n→∞. These polynomials are defined by the relation

(*)

p_n(z)+q_n(z)e^z+r_n(z)e^2z=O(z³ⁿ⁺²) as z→0,

where O(·) denotes Landau's symbol. In the investigation analytic expressions are proved for the asymptotics of the polynomials, for the asymptotics of the remainder term in (*), and also for the arcs on which the zeros of the polynomials and of the remainder term cluster if the independent variable z is rescaled in an appropriate way. The asymptotic expressions are defined with the help of an algebraic function of third degree and its associated Riemann surface. Among other possible applications, the results form the basis for the investigation of the convergence of quadratic Hermite–Padé approximants, which will be done in a follow-up paper.     

On the spectral function of the Poisson–Voronoi cells

Denote by (t)=∑_n1e^−λ_nt, t>0, the spectral function related to the Dirichlet Laplacian for the typical cell of a standard Poisson–Voronoi tessellation in . We show that the expectation E(t), t>0, is a functional of the convex hull of a standard d-dimensional Brownian bridge. This enables us to study the asymptotic behaviour of E(t), when t→0⁺,+∞. In particular, we prove that the law of the first eigenvalue λ₁ of satisfies the asymptotic relation lnP{λ₁t}−2^dω_dj_(d−2)/2^d·t^−d/2 when t→0⁺, where ω_d and j_(d−2)/2 are respectively the Lebesgue measure of the unit ball in and the first zero of the Bessel function J_(d−2)/2.     

The <Emphasis Type="Italic">p</Emphasis>-Daugavet property for function spaces

A natural extension of the Daugavet property for p-convex Banach function spaces and related classes is analysed. As an application, we extend the arguments given in the setting of the Daugavet property to show that no reflexive space falls into this class.     

A Padé family of iterations for the matrix sign function and related problems

In this paper we consider the Pad'e family of iterations for computing the matrix sign function and the Padé family of iterations for computing the matrix p‐sector function. We prove that all the iterations of the Padé family for the matrix sign function have a common convergence region. It completes a similar result of Kenney and Laub for half of the Padé family. We show that the iterations of the Padé family for the matrix p‐sector function are well defined in an analogous common region, depending on p. For this purpose we proved that the Padé approximants to the function (1?z)^?σ, 0<σ<1, are a quotient of hypergeometric functions whose poles we have localized. Furthermore we proved that the coefficients of the power expansion of a certain analytic function form a positive sequence and in a special case this sequence has the log‐concavity property. Copyright © 2011 John Wiley & Sons, Ltd.     

Continuity of the scattering function and Levinson‐type formula of Klein–Gordon equation

We considered the inverse problem of scattering theory for a boundary value problem on the half line generated by Klein–Gordon differential equation with a nonlinear spectral parameter‐dependent boundary condition. We defined the scattering data, and we proved the continuity of the scattering function S(λ); in a special case, the relation for the difference of the logarithm of the scattering function, which is called the Levinson‐type formula, was obtained. Copyright © 2012 John Wiley & Sons, Ltd.     

On a reducibility of the Kampé de Fériet function

The main objective of this research note is to provide an interesting result for the reducibility of the Kampé de Fériet function. The result is derived with the help of two results for the terminating ₃F₂ series very recently obtained by Rakha et al. A few interesting special cases have also been given. Copyright © 2014 John Wiley & Sons, Ltd.     

A weighted estimate for the square function on the unit ball in ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

We show that the Luzin area integral or the square function on the unit ball of ℂ ⁿ , regarded as an operator in the weighted space L ²(w) has a linear bound in terms of the invariant A ₂ characteristic of the weight. We show a dimension-free estimate for the “area-integral” associated with the weighted L ²(w) norm of the square function. We prove the equivalence of the classical and the invariant A ₂ classes.     

Convexity andq-gamma function

Inequalities and convexity properties known for the gamma function are extended to theq-gamma function, 0<q<1. Applying some classical inequalities for convex functions, we deduce monotonicity results for several functions involving theq-gamma function. Further applications to the probability theory are given.     

Positivity of Szegö's rational function

We consider the problem of deciding whether a given rational function has a power series expansion with all its coefficients positive. Introducing an elementary transformation that preserves such positivity we are able to provide an elementary proof for the positivity of Szegö's function

which has been at the historical root of this subject starting with Szegö. We then demonstrate how to apply the transformation to prove a 4-dimensional generalization of the above function, and close with discussing the set of parameters (a,b) such that

has positive coefficients.     

The hardy-littlewood maximal function of a sobolev function

We prove that the Hardy-Littlewood maximal operator is bounded in the Sobolev spaceW ^1,p (R ⁿ ) for 1<p≤∞. As an application we study a weak type inequality for the Sobolev capacity. We also prove that the Hardy-Littlewood maximal function of a Sobolev function is quasi-continuous.     

On <Emphasis Type="Italic">g</Emphasis>-functions for Laguerre function expansions of Hermite type

We examine weighted L ^p boundedness of g-functions based on semigroups related to multi-dimensional Laguerre function expansions of Hermite type. A technique of vector-valued Calderón–Zygmund operators is used.     

B-splines and Hermite–Padé approximants to the exponential function

This paper is the continuation of a work initiated in [P. Sablonnière, An algorithm for the computation of Hermite–Padé approximations to the exponential function: divided differences and Hermite–Padé forms. Numer. Algorithms 33 (2003) 443–452] about the computation of Hermite–Padé forms (HPF) and associated Hermite–Padé approximants (HPA) to the exponential function. We present an alternative algorithm for their computation, based on the representation of HPF in terms of integral remainders with B-splines as Peano kernels. Using the good properties of discrete B-splines, this algorithm gives rise to a great variety of representations of HPF of higher orders in terms of HPF of lower orders, and in particular of classical Padé forms. We give some examples illustrating this algorithm, in particular, another way of constructing quadratic HPF already described by different authors. Finally, we briefly study a family of cubic HPF.