Generalized Bernoulli Polynomials and Numbers,Revisited

We describe with some new details the connection between generalized Bernoulli polynomials, Bernoulli polynomials and generalized Bernoulli numbers (Norlund polynomials). A new recursive and explicit formulae for these polynomials are derived.     

A Unified Calculus Using the Generalized Bernoulli Polynomials

We introduce an α-calculus with the help of the generalized Bernoulli polynomials. The parameter α is the order of a Bessel function of the first kind. The differential α-calculus can be put in a general context where the concept of supporting function is an important tool for practical purposes. Our somewhat more restrictive point of view has the advantage of permitting a consistent definition of an α-integral with several interesting properties. It results in the possibility of expressing a remainder, in the aforementioned context, in a completely new form in our case.     

广义m阶Bernoulli数和广义m阶Euler数的计算公式

使用发生函数方法,利用第一类Stirling数和第二类Stirling数分别给出广义m阶Bernoulli数和广义m阶Euler数的计算公式.     

The Value of Bernoulli Polynomials at Rational Numbers

We give a short elementary proof of a result of Almkvist andMeurman [1] on an integrality property of the values taken bythe Bernoulli polynomials at a rational number. We use a lemmaon the divisibility properties of certain binomial coefficientswhich seems to be of independent interest.     

Chebyshev Polynomials and Generalized Complex Numbers

The generalized complex numbers can be realized in terms of 2 × 2 or higher-order matrices and can be exploited to get different ways of looking at the trigonometric functions. Since Chebyshev polynomials are linked to the power of matrices and to trigonometric functions, we take the quite natural step to discuss them in the context of the theory of generalized complex numbers.We also briefly discuss the two-variable Chebyshev polynomials and their link with the third-order Hermite polynomials.     

Central Factorial Numbers and Values of Bernoulli and Euler Polynomials at Rationals

The nth order derivatives of tan x and sec x may be represented by polynomials P _n (u) and Q _n (u) in u = tan x, which are known as the derivative polynomials for the tangent and secant and have occurred in diverse contexts. In this paper, explicit representations of P _n (u) and Q _n (u) are derived in terms of the central factorial numbers of the second kind, and the values of the Bernoulli and Euler polynomials at rationals are expressed by means of these polynomials.     

关于与广义Fibonacci,Lucas数有关的一些级数的注记

求出了一些与广义Fibonacci,Lucas数有关的一些倒数级数的值。     

Explicit Formulas Associated with Some Families of Generalized Bernoulli and Euler Polynomials

In this paper, we propose and derive several new explicit formulas of the generalized Bernoulli and Euler polynomials in terms of the generalized Stirling numbers of the second kind. A study of some families of the modified generalized Euler polynomials yields an interesting algorithm for calculating the generalized Euler polynomials.     

关于勒让德多项式的一个注记

本文利用勒让德多项式的性质证明了其导数多项式是［－１，１］上关于权１－ｘ２的正交多项式．     

关于Faber多项式的一个注记

Faber多项式通常被用来研究单叶函数的性质,同时也提供了用多项式逼近区域内的解析函数的一个有价值的工具.本文给出一个关于∑类单叶函数的Faber多项式递推公式,并由此得到计算Faber系数的新公式,在一定情形下,它比已有的公式更为简单.     

关于Bernstein多项式的一个注记

本给出了Z.Ditzian定理的一个简短证明。     

Bernoulli多项式和Euler多项式的关系

本文给出了 Bernoulli- Euler数之间的关系和 Bernoulli- Euler多项式之间的关系 ,从而深化和补充了有关文献中的相关结果 .     

广义中心阶乘数与高阶Nrlund Euler-Bernoulli多项式

本文讨论了广义中心阶乘数的性质,刻画了广义中心阶乘数与高阶Ｅｕｌｅｒ－Ｂｅｒｎｏｕｌｌｉ数和多项式的关系,建立了一些包含 Ｎｏｒｌｕｎｄ Ｅｕｌｅｒ－Ｂｅｒｎｏｕｌｌｉ多项式恒等式,推广了 Ｄｉｌｃｈｅｒ Ｋ．［１］,Ｚｈａｎｇ Ｗｅｎｐｅｎｇ［２］和 Ｚｅｉｔｌｉｎ　Ｄａｖｉｄ［３］的结果．     

Diophantine Equations and Bernoulli Polynomials

Given m, n 2, we prove that, for sufficiently large y, the sum 1 ⁿ +···+ y ⁿ is not a product of m consecutive integers. We also prove that for m n we have 1 ^m +···+ x ^m 1 ⁿ +···+ y ⁿ , provided x, y are sufficiently large. Among other auxiliary facts, we show that Bernoulli polynomials of odd index are indecomposable, and those of even index are almost indecomposable, a result of independent interest.     

关于高阶Genocchi数和广义Lucas多项式的恒等式

利用组合数学的方法,得到了一些包含高阶Genocchi数和广义Lucas多项式的恒等式,并且由此建立了Fibonacci数与Riemann Zeta函数的关系式.     

A New Formula for the Bernoulli Polynomials

In this note we show that a seemingly new class of Stirling-type pairs can be applied to produce a new representation of the Bernoulli polynomials at positive rational arguments. A class of generalized harmonic numbers is also investigated, and we point out that these give a new relation for the so-called harmonic polynomials.     

退化高阶伯努利多项式与广义等幂和多项式的对称等式

研究了退化伯努利多项式与广义等幂和多项式的对称关系,获得了关于多个退化高阶伯努利多项式与广义等幂和多项式的若干对称关系.     

A Note on Some Link Polynomials

Hass and Wajnryb [2] have given a set of sufficient conditionsfor a recursive definition over crossing changes in link diagramsto provide a well-defined link invariant in a similar way to[3]. They also provide a number of examples satisfying theseconditions, including a number with non-linear recursive definitions.I here show that all of these examples are in fact special casesof the known invariant P(l, m) of [3, 1]. In particular, thereexists no linear recursive definition providing an invariantnot covered by P(l, m).