Restricted partition functions as Bernoulli and Eulerian polynomials of higher order

Explicit expressions for restricted partition function W(s,d^m) and its quasiperiodic components W_j(s,d^m) (called Sylvester waves) for a set of positive integers d^m = {d₁, d₂, ..., d_m} are derived. The formulas are represented in a form of a finite sum over Bernoulli and Eulerian polynomials of higher order with periodic coefficients. A novel recursive relation for the Sylvester waves is established. Application to counting algebraically independent homogeneous polynomial invariants of finite groups is discussed. 2000 Mathematics Subject Classification Primary—11P81; Secondary—11B68, 11B37 The research was supported in part (LGF) by the Gileadi Fellowship program of the Ministry of Absorption of the State of Israel.     

Several formulas for special values of the Bell polynomials of the second kind and applications

In the paper, the authors establish several explicit formulas for special values of the Bell polynomials of the second kind, connect these formulas with the Bessel polynomials, and apply these formulas to give new expressions for the Catalan numbers and to compute arbitrary higher order derivatives of elementary functions such as the since, cosine, exponential, logarithm, arcsine, and arccosine of the square root for the variable.     

On multivariate logarithmic polynomials and their properties

In the paper, the author introduces a new notion “multivariate logarithmic polynomial”, establishes two recurrence relations, an explicit formula, and an identity for multivariate logarithmic polynomials by virtue of the Faà di Bruno formula and two identities for the Bell polynomials of the second kind in terms of the Stirling numbers of the first and second kinds, and constructs some determinantal inequalities, some product inequalities, and logarithmic convexity for multivariate logarithmic polynomials by virtue of some properties of completely monotonic functions.     

Necessary conditions for reversed Dickson polynomials of the second kind to be permutational

In this paper, we present several necessary conditions for the reversed Dickson polynomial $E_n(1,x)$ of the second kind to be a permutation of ${\mathbb{F}}_q$. In particular, we give explicit evaluation of the sum ${\sum }_{a\in {\mathbb{F}}_q}{E}_n(1,a)$.     

Some properties and an application of multivariate exponential polynomials

In the paper, the authors introduce a notion “multivariate exponential polynomials” which generalize exponential numbers and polynomials, establish explicit formulas, inversion formulas, and recurrence relations for multivariate exponential polynomials in terms of the Stirling numbers of the first and second kinds with the help of the Faà di Bruno formula, two identities for the Bell polynomials of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds, construct some determinantal inequalities and product inequalities for multivariate exponential polynomials with the aid of some properties of completely monotonic functions and other known results, derive the logarithmic convexity and logarithmic concavity for multivariate exponential polynomials, and finally find an application of multivariate exponential polynomials to white noise distribution theory by confirming that multivariate exponential polynomials satisfy conditions for sequences required in white noise distribution theory.     

联系Bernoulli数和第二类Stirling数的一个恒等式

利用指数型生成函数建立起联系Bernoulli数和第二类Stirling数的一个有趣的恒等式．     

A new summation identity for the Srivastava-Singhal polynomials

In his recent investigations involving differential operators for some generalizations of the classical Laguerre polynomials, H. Bavinck [J. Phys. A Math. Gen. 29 (1996) L277-L279] encountered and proved a certain summation identity for the classical Laguerre polynomials. The main object of this sequel to Bavinck's work is to prove a generalization of this summation identity for the Srivastava-Singhal polynomials. The demonstration, which is presented here in the general case, differs markedly from the earlier proof given for the known special case. It is also indicated how the general summation identity can be applied to derive the corresponding result for one class of the Konhauser biorthogonal polynomials.     

A new result for hypergeometric polynomials

In some recent investigations involving differential operators for generalized Laguerre polynomials, Herman Bavinck (1996) encountered and proved a certain summation formula for the classical Laguerre polynomials. The main object of this sequel to Bavinck's work is to prove a generalization of this summation formula for a class of hypergeometric polynomials. The demonstration, which is presented here in the general case, differs markedly from the earlier proof given for the known special case. The general summation formula is also applied to derive the corresponding result for the classical Jacobi polynomials.

     

Construction a new generating function of Bernstein type polynomials

Main purpose of this paper is to reconstruct generating function of the Bernstein type polynomials. Some properties of this generating functions are given. By applying this generating function, not only derivative of these polynomials but also recurrence relations of these polynomials are found. Interpolation function of these polynomials is also constructed by Mellin transformation. This function interpolates these polynomials at negative integers which are given explicitly. Moreover, relations between these polynomials, the Stirling numbers of the second kind and Bernoulli polynomials of higher order are given. Furthermore some remarks associated with the Bezier curves are given.     

Annihilating polynomials for quadratic forms and Stirling numbers of the second kind

We present a set of generators of the full annihilator ideal for the Witt ring of an arbitrary field of characteristic unequal to two satisfying a non‐vanishing condition on the powers of the fundamental ideal in the torsion part of the Witt ring. This settles a conjecture of Ongenae and Van Geel. This result could only be proved by first obtaining a new lower bound on the 2‐adic valuation of Stirling numbers of the second kind. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Polynomial interpolation and sums of powers of integers

In this note, we revisit the problem of polynomial interpolation and explicitly construct two polynomials in n of degree k + 1, P_k(n) and Q_k(n), such that P_k(n) = Q_k(n) = f_k(n) for n = 1, 2,…?, k, where f_k(1), f_k(2),…?, f_k(k) are k arbitrarily chosen (real or complex) values. Then, we focus on the case that f_k(n) is given by the sum of powers of the first n positive integers S_k(n) = 1^k + 2^k + ??? + n^k, and show that S_k(n) admits the polynomial representations S_k(n) = P_k(n) and S_k(n) = Q_k(n) for all n = 1, 2,…?, and k ≥ 1, where the first representation involves the Eulerian numbers, and the second one the Stirling numbers of the second kind. Finally, we consider yet another polynomial formula for S_k(n) alternative to the well-known formula of Bernoulli.     

Impulsive BVPs with nonlinear boundary conditions for the second order differential equations without growth restrictions

The paper deals with the impulsive nonlinear boundary value problem

     

Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials

The main object of this paper is to give analogous definitions of Apostol type (see [T.M. Apostol, On the Lerch Zeta function, Pacific J. Math. 1 (1951) 161-167] and [H.M. Srivastava, Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Cambridge Philos. Soc. 129 (2000) 77-84]) for the so-called Apostol-Bernoulli numbers and polynomials of higher order. We establish their elementary properties, derive several explicit representations for them in terms of the Gaussian hypergeometric function and the Hurwitz (or generalized) Zeta function, and deduce their special cases and applications which are shown here to lead to the corresponding results for the classical Bernoulli numbers and polynomials of higher order.     

高阶Bernoulli多项式和高阶Euler多项式的新计算公式

使用发生函数方法,利用两种第一类Stirling数给出高阶Bernoulli多项式和高阶Euler多项式的简捷计算公式.     

一类具非线性边界条件的高阶方程的奇摄动问题

通过引入伸展变量和非常规的渐近序列{∈}）,运用合成展开法,对一类具非线性边界条件的非线性高阶微分方程的奇摄动问题构造了形式渐近解,再运用微分不等式理论证明了原问题解的存在性及所得渐近近似式的一致有效性．     

慢增长系数微分方程的解及其导数的不动点

运用值分布理论研究了高阶慢增长系数线性微分方程的解及其导数的不动点问题.当存在某个系数对方程的解的性质起主要支配作用时,得到了方程解及其导数的不动点收敛指数的精确估计,推广了有关文献中的结论.     

Solvability and iterative algorithms for a higher order nonlinear neutral delay differential equation

The paper is concerned with the higher order nonlinear neutral delay differential equation

     

Maximal σ‐polynomials of connected 3‐chromatic graphs

In the set of graphs of order n and chromatic number k the following partial order relation is defined. One says that a graph G is less than a graph H if c_i(G) ≤ c_i(H) holds for every i, k ≤ i ≤ n and at least one inequality is strict, where c_i(G) denotes the number of i‐color partitions of G. In this paper the first ? n/2 ? levels of the diagram of the partially ordered set of connected 3‐chromatic graphs of order n are described. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 210–222, 2003     

Differential equations and recursion relations for Laguerre functions on symmetric cones

We obtain the differential equation and recurrence relations satisfied by the Laguerre functions on an arbitrary symmetric cone .