Polynomials Bn Research Articles

A Sufficient and Necessary Condition for the Power-Exponential Function 1+1xαx to Be a Bernstein Function and Related nth Derivatives

In the paper, the authors find a sufficient and necessary condition for the power-exponential function 1+1xαx to be a Bernstein function, derive closed-form formulas for the nth derivatives of the power-exponential functions 1+1xαx and (1+x)α/x, and present a closed-form formula of the partial Bell polynomials Bn,k(H0(x),H1(x),⋯,Hn−k(x)) for n≥k≥0, where Hk(x)=∫0∞eu−1−ueuuk−1e−xudu for k≥0 are completely monotonic on (0,∞).

A Note on Explicit Formulas for Bernoulli Polynomials

For r 2 { 1; - 1; 1 2 } , we prove several explicit formulas for the n-th Bernoulli polynomial Bn (x), in which Bn (x) is equal to a linear combination of the polynomials xn, (x + r)n ; : : : ; (x + rm)n, where m is any fixed positive integer greater than or equal to n

Asymptotic normality of coefficients of some polynomials related to Dowling lattices

Recently, we introduced two sequences of polynomials (Bn(x, y, z)) and (Fn(x, y, z)), which unify many familiar polynomials related to Dowling lattices, such as the Bell polynomials, the Dowling polynomials, the ordered Bell polynomials, the r-Bell polynomials and the r-Dowling polynomials. In this paper, we show the asymptotic normality of coefficients of Bn(x, y, z) and Fn(x, y, z). As applications, we obtain the asymptotic normality of coefficients of some polynomials related to Dowling lattices in a unified approach.

On arithmetic properties of binary partition polynomials

Let the polynomial bn(t) be defined as the n-th coefficient in the power series expansion (in variable x) of the functionB(t,x)=∏n=0∞11−tx2n=∑n=0∞bn(t)xn. The polynomial bn(t)=∑i=0na(i,n)ti has the following combinatorial interpretation: the i-th coefficient a(i,n) counts the number of representations of n as sums of exactly i powers of 2. The bn(t) can be seen as a polynomial analogue of the number bn(1), which counts the number of binary partitions of n. In the present paper we obtain several results concerning arithmetic properties of the polynomials bn(t) as well as its coefficients. Moreover, we show an interesting connection between coefficients of bn(t) and the number counting so called s-partitions, i.e., the representations of n as sums of numbers of the form 2k−1.

On a product of certain primes

We study the properties of the product, which runs over the primes,pn=∏sp(n)≥pp(n≥1), where sp(n) denotes the sum of the base-p digits of n. One important property is the fact that pn equals the denominator of the Bernoulli polynomial Bn(x)−Bn, where we provide a short p-adic proof. Moreover, we consider the decomposition pn=pn−⋅pn+, where pn+ contains only those primes p>n. Let ω(⋅) denote the number of prime divisors. We show that ω(pn+)<n, while we raise the explicit conjecture thatω(pn+)∼κnlog⁡nas n→∞ with a certain constant κ>1, supported by several computations.

Analytic summability of real and complex functions

Gamma-type functions satisfying the functional equation f(x+1) = g(x)f(x) and limit summability of real and complex functions were introduced by Webster (1997) and Hooshmand (2001). However, some important special functions are not limit summable, and so other types of such summability are needed. In this paper, by using Bernoulli numbers and polynomials Bn(z), we define the notions of analytic summability and analytic summand function of complex or real functions, and prove several criteria for analytic summability of holomorphic functions on an open domain D. As consequences of our results, we give some criteria for absolute convergence of the functional series $$\sum\nolimits_{n = 0}^\infty {{c_n}\sigma \left( {{Z^n}} \right)} ,where\sigma \left( {{Z^n}} \right) = {S_n}\left( z \right) = \frac{{{B_{n + 1}}\left( {z + 1} \right) - {B_{n + 1}}\left( 1 \right)}}{{n + 1}}$$ . Finally, we state some open problems.

Remarks on Bell and higher order Bell polynomials and numbers

We recover a recurrence relation for representing in an easy form the coefficients An,k of the Bell polynomials, which are known in literature as the partial Bell polynomials. Several applications in the framework of classical calculus are derived, avoiding the use of operational techniques. Furthermore, we generalize this result to the coefficients An,k[2] of the second-order Bell polynomials, i.e. of the Bell polynomials relevant to nth derivative of a composite function of the type f(g(h(t))). The second-order Bell polynomials Bn[2] and the relevant Bell numbers bn[2] are introduced. Further extension of the nth derivative of M-nested functions is also touched on.

On the q-Bernstein polynomials of the logarithmic function in the case q &gt; 1

Abstract The q-Bernstein basis used to construct the q-Bernstein polynomials is an extension of the Bernstein basis related to the q-binomial probability distribution. This distribution plays a profound role in the q-boson operator calculus. In the case q &gt; 1, q-Bernstein basic polynomials on [0, 1] combine the fast increase in magnitude with sign oscillations. This seriously complicates the study of q-Bernstein polynomials in the case of q &gt; 1. The aim of this paper is to present new results related to the q-Bernstein polynomials Bn , q of discontinuous functions in the case q &gt; 1. The behavior of polynomials Bn , q (f; x) for functions f possessing a logarithmic singularity at 0 has been examined.

On the Convergence of Bernstein-Stancu Polynomials in the Variation Seminorm

ABSTRACTThis article studies the variation detracting property and rate of approximation of the Bernstein-Stancu polynomials in the space of functions of bounded variation with respect to the variation seminorm. Moreover, we will present Voronovskaya-type theorems for Bernstein-Stancu polynomials Bn, α, βf and for the first derivative of these polynomials. Finally we include some graphical examples.

Formula omitted]-poly-Bernoulli numbers and [formula omitted]-poly-Cauchy numbers with a parameter by Jackson’s integrals

We define q-poly-Bernoulli polynomials Bn,ρ,q(k)(z) with a parameter ρ, q-poly-Cauchy polynomials of the first kind cn,ρ,q(k)(z) and of the second kind ĉn,ρ,q(k)(z) with a parameter ρ by Jackson’s integrals, which generalize the previously known numbers and polynomials, including poly-Bernoulli numbers Bn(k) and the poly-Cauchy numbers of the first kind cn(k) and of the second kind ĉn(k). We investigate their properties connected with usual Stirling numbers and weighted Stirling numbers. We also give the relations between generalized poly-Bernoulli polynomials and two kinds of generalized poly-Cauchy polynomials.

Bounds for the logarithm of the Euler gamma function and its derivatives

We consider differences between log⁡Γ(x) and truncations of certain classical asymptotic expansions in inverse powers of x−λ whose coefficients are expressed in terms of Bernoulli polynomials Bn(λ), and we obtain conditions under which these differences are strictly completely monotonic. In the symmetric cases λ=0 and λ=1/2, we recover results of Sonin, Nörlund and Alzer. Also we show how to derive these asymptotic expansions using the functional equation of the logarithmic derivative of the Euler gamma function, the representation of 1/x as a difference F(x+1)−F(x), and a backward induction.

Multivariate Stirling polynomials of the first and second kind

Two doubly indexed families of homogeneous and isobaric polynomials in several indeterminates are considered: the (partial) exponential Bell polynomials Bn,k and a new family Sn,k∈Z[X1,…,Xn−k+1] such that X1−(2n−1)Sn,k and Bn,k obey an inversion law which generalizes that of the Stirling numbers of the first and second kind. Both polynomial families appear as Lie coefficients in expansions of certain derivatives of higher order. Substituting Dj(φ) (the jth derivative of a fixed function φ) in place of the indeterminates Xj shows that both Sn,k and Bn,k are differential polynomials depending on φ and on its inverse φ¯, respectively. Some new light is shed thereby on Comtet’s solution of the Lagrange inversion problem in terms of the Bell polynomials. According to Haiman and Schmitt that solution is essentially the antipode on the Faà di Bruno Hopf algebra. It can be represented by X1−(2n−1)Sn,1. Moreover, a general expansion formula that holds for the whole family Sn,k (1≤k≤n) is established together with a closed expression for the coefficients of Sn,k. Several important properties of the Stirling numbers are demonstrated to be special cases of relations between the corresponding polynomials. As a non-trivial example, a Schlömilch-type formula is derived expressing Sn,k in terms of the Bell polynomials Bn,k, and vice versa.

Cvijovic's addition formula for the Lah numbers

Recently, Cvijovic found a new addition formula for the partial Bell polynomials Bn,k. In this note, we consider the unsigned Lah numbers L(n, k) and derive algebraically Cvijovic’s addition formula for L(n, k). Also, we obtain a couple of vertical recurrence relations for L(n, k). Mathematics Subject Classification: 11B83, 11B75, 11B37

Uniform convergence of the p-Bieberbach polynomials in domains with zero angles

Let G ⊂ ℂ be a simply connected domain whose boundary L := ∂G is a Jordan curve and 0 ∈ G. Let w = φ(z) be the conformal mapping of G onto the disk B(0, r0) := {w : |w| 0: $$\iint_G {|\phi '(z) - P'_n (z)|^p d\sigma _z \to \min }$$ in the class of all polynomials Pn(z) of degree not exceeding n with Pn(0) = 0, Pn′ (0) = 1. The solution to this extremal problem is called the p-Bieberbach polynomial of degree n for the pair (G, 0). We study the uniform convergence of the p-Bieberbach polynomials Bn,p(z) to the φ(z) on \(\bar G\) with interior and exterior zero angles determined depending on the properties of boundary arcs and the degree of their “touch”.

On the q-Bernstein polynomials of rational functions with real poles

The paper aims to investigate the convergence of the q-Bernstein polynomials Bn,q(f;x) attached to rational functions in the case q>1. The problem reduces to that for the partial fractions (x−α)−j, j∈N. The already available results deal with cases, where either the pole α is simple or α≠q−m, m∈N0. Consequently, the present work is focused on the polynomials Bn,q(f;x) for the functions of the form f(x)=(x−q−m)−j with j⩾2. For such functions, it is proved that the interval of convergence of {Bn,q(f;x)} depends not only on the location, but also on the multiplicity of the pole – a phenomenon which has not been considered previously.

The q-Bernstein polynomials of the Cauchy kernel with a pole on [formula omitted] in the case [formula omitted

The problem to describe the Bernstein polynomials of unbounded functions goes back to Lorentz. The aim of this paper is to investigate the convergence properties of the q-Bernstein polynomials Bn,q(f;x) of the Cauchy kernel 1x-α with a pole α∈[0,1] for q>1. The previously obtained results allow one to describe these properties when a pole is different from q-m for some m∈0,1,2,…. In this context, the focus of the paper is on the behavior of polynomials Bn,q(f;x) for the functions of the form fm(x)=1/(x-q-m),x≠q-m and fm(q-m)=a,a∈R. Here, the problem is examined both theoretically and numerically in detail.

On affinity relating two positive measures and the connection coefficients between polynomials orthogonalized by these measures

We consider two positive, normalized measures dA(x) and dB(x) related by the relationship dA(x)=Cx+DdB(x) or by dA(x)=Cx2+EdB(x) and dB(x) is symmetric. We show that then the polynomial sequences an(x),bn(x) orthogonal with respect to these measures are related by the relationship an(x)=bn(x)+κnbn-1(x) or by an(x)=bn(x)+λnbn-2(x) for some sequences κn and λn. We present several examples illustrating this fact and also present some attempts for extensions and generalizations. We also give some universal identities involving polynomials bn(x) and the sequence κn that have a form of the Fourier series expansion of the Radon–Nikodym derivative of one measure with respect to the other.

A note on the Bernoulli polynomials with the variable [x

In this paper, we introduce the Bernoulli polynomials with the variable [x]q and we get polynomials related to the Bernoulli polynomials with the variable [x]q and the Bernoulli numbers as coefficients. By the p-adic integral on Zp, we define Bernoulli polynomials and we also get the some interesting properties. Finally, we observe an interesting phenomenon of ‘scattering’ of the zeros of the Bernoulli polynomials Bn,q(x) in complex plane.

Tensor product q-Bernstein polynomials

An affine de Casteljau type algorithm to compute q-Bernstein Bezier curves is introduced and its intermediate points are obtained explicitly in two ways. Furthermore we define a tensor product patch, based on this algorithm, depending on two parameters. Degree elevation procedure is studied. The matrix representation of tensor product patch is given and we find the transformation matrix between a classical tensor product Bezier patch and a tensor product q-Bernstein Bezier patch. Finally, q-Bernstein polynomials Bn,m(f;x,y) for a function f(x,y), (x,y)∈[0,1]×[0,1] are defined and fundamental properties are discussed.

HYPERGEOMETRIC BERNOULLI POLYNOMIALS AND APPELL SEQUENCES

There are two analytic approaches to Bernoulli polynomials Bn(x): either by way of the generating function zexz/(ez - 1) = ∑ Bn(x)zn/n! or as an Appell sequence with zero mean. In this article, we discuss a generalization of Bernoulli polynomials defined by the generating function zN exz/(ez - TN-1(z)), where TN(z) denotes the Nth Maclaurin polynomial of ez, and establish an equivalent definition in terms of Appell sequences with zero moments in complete analogy to their classical counterpart. The zero-moment condition is further shown to generalize to Bernoulli polynomials generated by the confluent hypergeometric series.