高阶多元Norlund Euler-Bernoulli多项式

给出了高阶多元Nrlund Euler多项式和高阶多元Nrlund Bernoulli多项式的定义,讨论了它们的一些重要性质,建立了一些包含递归序列和上述多项式的恒等式.     

高阶多元Noerlund Euler—Bernoulli多项式

给出了高阶多元Noerlund Euler多项式和高阶多元Noerlund Bernoulli多项式的定义，讨论了它们的一些重要性质，建立了一些包含递归序列和上述多项式的恒等式。     

递归序列与高阶多元Euler-Bernoulli多项式

引言 关于递归序列与Euler-Bernoulli数和多项式、递归序列与高阶Euler-Bernoulli数和多项式的关系问题的研究一直是国内外许多学者感兴趣的课题,并有了许多研究成果(见[1]～[7]).     

高阶Bernoulli多项式和高阶Euler多项式的关系

利用发生函数的方法，讨论了高阶Bernoulli数和高阶Euler数，高阶Bernoulli多项式和高阶Euler多项式之间的关系，得到了经典Bernoulli数和Euler数，经典Bernoulli多项式和Euler多项式之间的新型关系。     

广义n阶Euler-Bernoulli多项式

本文得到了广义n阶Euler数和广义n阶Bernoulli数,广义n阶Euler多项式和广义n阶Bernoulli多项式的关系式。     

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.     

高阶多元Euler多项多和高阶多元Bernoulli多项式

本给出了高阶多元Ｅｕｌｅｒ数和多项式与同阶多元Ｂｅｒｎｏｕｌｌｉ数和多项式的定义，讨论了它们的一些重要性质，得到了高阶多元Ｅｕｌｅｒ多项式（数）和高阶多元Ｂｅｒｎｏｕｌｌｉ多项式（数）的关系式。     

高阶多元Nrlund Euler-Bernoulli多项式

给出了高阶多元N rlundEuler多项式和高阶多元N rlundBernoulli多项式的定义,讨论了它们的一些重要性质,建立了一些包含递归序列和上述多项式的恒等式·     

渐近于Hermite多项式的双正交系统

本文利用渐近于Gauss函数的函数类?,给出渐近于Hermite正交多项式的一类Appell多项式的构造方法,使得该序列与?的n阶导数之间构成了一组双正交系统.利用此结果,本文得到多种正交多项式和组合多项式的渐近性质.特别地,由N阶B样条所生成的Appell多项式序列恰为N阶Bernoulli多项式.从而,Bernoulli多项式与B样条的导函数之间构成了一组双正交系统,且标准化之后的Bernoulli多项式的渐近形式为Hermite多项式.由二项分布所生成的Appell序列为Euler多项式,从而,Euler多项式与二项分布的导函数之间构成一组双正交系统,且标准化之后的Euler多项式渐近于Hermite多项式.本文给出Appell序列的生成函数满足的尺度方程的充要条件,给出渐近于Hermite多项式的函数列的判定定理.应用该定理,验证广义Buchholz多项式、广义Laguerre多项式和广义Ultraspherical(Gegenbauer)多项式渐近于Hermite多项式的性质,从而验证超几何多项式的Askey格式的成立.     

关于勒让德多项式导数的正交性

本文证明了勒让德多项式 Pn( x)的 k阶导数 P( k)n ( x)是 [-1 ,1 ]上关于权函数 ρ( x) =( 1 -x2 ) k的正交多项式 ,推广了 [1 ]的结果 .     

Reciprocal relations of Bernoulli and Euler numbers/polynomials

By means of the symmetric summation theorem on polynomial differences due to Chu and Magli [Summation formulae on reciprocal sequences. European J Combin. 2007;28(3):921–930], we examine Bernoulli and Euler polynomials of higher order. Several reciprocal relations on Bernoulli and Euler numbers and polynomials are established, including some recent ones obtained by Agoh Shortened recurrence relations for generalized Bernoulli numbers and polynomials. J Number Theory. 2017;176:149–173.     

An identity of symmetry for the Bernoulli polynomials

The main purpose of this paper is to prove an identity of symmetry for the higher order Bernoulli polynomials. It turns out that the recurrence relation and multiplication theorem for the Bernoulli polynomials which discussed in [F.T. Howard, Application of a recurrence for the Bernoulli numbers, J. Number Theory 52 (1995) 157-172], as well as a relation of symmetry between the power sum polynomials and the Bernoulli numbers developed in [H.J.H. Tuenter, A symmetry of power sum polynomials and Bernoulli numbers, Amer. Math. Monthly 108 (2001) 258-261], are all special cases of our results.     

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.     

Extended Zeilberger's algorithm for identities on Bernoulli and Euler polynomials

We present a computer algebra approach to proving identities on Bernoulli polynomials and Euler polynomials by using the extended Zeilberger's algorithm given by Chen, Hou and Mu. The key idea is to use the contour integral definitions of the Bernoulli and Euler numbers to establish recurrence relations on the integrands. Such recurrence relations have certain parameter free properties which lead to the required identities without computing the integrals. Furthermore two new identities on Bernoulli numbers are derived.     

Padé approximation and Apostol-Bernoulli and Apostol-Euler polynomials

Using the Padé approximation of the exponential function, we obtain recurrence relations between Apostol-Bernoulli and between Apostol-Euler polynomials. As applications, we derive some new lacunary recurrence relations for Bernoulli and Euler polynomials with gap of length 4 and lacunary relations for Bernoulli and Euler numbers with gap of length 6.     

Euler多项式的若干对称恒等式

Using the generating functions, we prove some symmetry identities for the Euler polynomials and higher order Euler polynomials, which generalize the multiplication theorem for the Euler polynomials. Also we obtain some relations between the Bernoulli polynomials, Euler polynomials, power sum, alternating sum and Genocchi numbers.     

New characterization of Appell polynomials

We prove characterizations of Appell polynomials by means of symmetric property. For these polynomials, we establish a simple linear expression in terms of Bernoulli and Euler polynomials. As applications, we give interesting examples. In addition, from our study, we obtain Fourier expansions of Appell polynomials. This result recovers Fourier expansions known for Bernoulli and Euler polynomials and obtains the Fourier expansions for higher order Bernoulli–Euler's one.