楊存典,劉端森(商洛學(xué)院數(shù)學(xué)與計(jì)算機(jī)科學(xué)學(xué)院,陜西商洛　726000)

高階Bernoulli多項(xiàng)式和高階Euler多項(xiàng)式的組合恒等式

楊存典,劉端森
(商洛學(xué)院數(shù)學(xué)與計(jì)算機(jī)科學(xué)學(xué)院,陜西商洛726000)

研究了高階Bernoulli多項(xiàng)式和高階Euler多項(xiàng)式的關(guān)系,并得到了高階Bernoulli多項(xiàng)式和高階Euler多項(xiàng)式的表達(dá)式及關(guān)系式.運(yùn)用Bernoulli多項(xiàng)式和Euler多項(xiàng)式的基本性質(zhì)以及初等方法,對(duì)經(jīng)典Bernoulli多項(xiàng)式和Euler多項(xiàng)式的恒等式進(jìn)行了推廣.

高階Bernoulli多項(xiàng)式;高階Euler多項(xiàng)式;恒等式

引用格式:Yang Cundian,Liu Duansen．Combinatorial Identity of HighGorder Bernoulli Polynomial and HighGorG der Euler Polynomial[J]．Journal of Gansu Sciences,2016,28(2):7G9．[楊存典,劉端森．高階Bernoulli多項(xiàng)式和高階Euler多項(xiàng)式的組合恒等式[J]．甘肅科學(xué)學(xué)報(bào),2016,28(2):7G9．]

眾所周知,多項(xiàng)式和特殊函數(shù)的研究在數(shù)論、組合數(shù)學(xué)、特殊函數(shù)中的應(yīng)用非常廣泛[1],引起了許多專家和學(xué)者的興趣.特別是經(jīng)典Bernoulli多項(xiàng)式、Euler多項(xiàng)式、Bernoulli數(shù)和Euler數(shù)的研究更是成果斐然. Apostol在文獻(xiàn)[2]中研究LipschitzGLerch Zeta函數(shù)時(shí)將Bernoulli多項(xiàng)式做了有益的推廣.雒秋明等在文獻(xiàn)[3]中將Euler多項(xiàng)式也做了類似的定義.文獻(xiàn)[4G8]中作者給出了Bernoulli多項(xiàng)式和Euler多項(xiàng)式的一些恒等式.研究將在高階Bernoulli多項(xiàng)式和高階Euler多項(xiàng)式定義的基礎(chǔ)上,得出一些有趣的組合恒等式.

1　定義及引理

定義3第二類Stirling數(shù)S( n,k)由下列生成函數(shù)定義:

引理1若n是非負(fù)整數(shù),λ是復(fù)數(shù)且R(λ)＞0,則[4]

引理2若n是非負(fù)整數(shù),α,λ是實(shí)數(shù)或復(fù)數(shù),R(λ)＞0,則[4]

這里F[ a,b;c;t]表示Gauss超幾何函數(shù)[1].

2　主要結(jié)論

定理1若n是非負(fù)整數(shù),α,λ是實(shí)數(shù)或復(fù)數(shù),R(λ)＞0,則有

定理2若n是非負(fù)整數(shù),α,λ是實(shí)數(shù)或復(fù)數(shù),R(λ)＞0,有加法公式

在式(4)和式(5)中,取λ＝1,α＝1,即得經(jīng)典加法公式.

定理3若n是非負(fù)整數(shù),α,λ是實(shí)數(shù)或復(fù)數(shù),R(λ)＞0,有差分公式

定理4若n是非負(fù)整數(shù),α,λ是實(shí)數(shù)或復(fù)數(shù),R(λ)＞0,有組合式

3　定理的證明

對(duì)式(1)兩邊關(guān)于t求n階導(dǎo)數(shù)并使用Leibniz法則,得

定理1

由式(1)可得Bn(α)( x＋y,λ)的生成函數(shù)是

比較兩邊系數(shù)即得定理2的式(4).同理可得定理2的式(5).

由定義可得

由定義1得

由定義2得

比較式(12)兩邊的系數(shù)即得式(9).

[1]王竹溪,郭敦仁．特殊函數(shù)概論[M]．北京:北京大學(xué)出版社,2004．

[2]Apostol Tom M．On the Lerch Zeta Funtion[J]．Pacific J．Maths,1951,1(1):161G167．

[3]Luo Qiuming．On the ApostolGBernoulli Polynomials[J]．Central European J．Maths．,2004,2(4):509G515．

[4]雒秋明,劉愛啟．高階Euler多項(xiàng)式的推廣及其應(yīng)用[J]．?dāng)?shù)學(xué)雜志,2006,26(5):574G578．

[5]巫朝霞,何園．關(guān)于Bernoulli多項(xiàng)式和Euler多項(xiàng)式的一個(gè)注記[J]．內(nèi)蒙古師范大學(xué)學(xué)報(bào):自然科學(xué)漢文版,2012,54(6):604G606．[6]雒秋明．Bernoulli多項(xiàng)式和Euler多項(xiàng)式的關(guān)系[J]．?dāng)?shù)學(xué)的實(shí)踐與認(rèn)識(shí),2003,33(3):119G122．

[7]李志榮．廣義m階Bernoulli數(shù)和廣義m階Euler數(shù)的計(jì)算公式[J]．?dāng)?shù)學(xué)的實(shí)踐與認(rèn)識(shí),2007,37(10):167G172．

[8]劉國棟．廣義n階BernoulliGEuler多項(xiàng)式[J]．?dāng)?shù)學(xué)的實(shí)踐與認(rèn)識(shí),1999,29(3):5G10．

Combinatorial Identity of HighGorder Bernoulli Polynomial and HighGorder Euler Polynomial

Yang Cundian,Liu Duansen
(Department of Mathematics and Computational Science,Shangluo University,Shangluo 726000,China)

The relationship between the highGorder Bernoulli polynomial and highGorder Euler polynomial was researched,and the expression and the relation of the highGorder Bernoulli polynomial and highGorder Euler polynomial were acquired．Further,the identity of the classical Bernoulli polynomial and Euler polyG nomial was popularized by using the basic properties of the Bernoulli polynomial and the Euler polynomial and the elementary method．

HighGorder Bernoulli polynomial;HighGorder Euler polynomial;Identity

O156．4

A

1004G0366(2016)02G0007G03

10．16468/j．cnkii．ssn1004G0366．2016．02．002．

2015G03G23;

2015G05G21．

國家自然科學(xué)基金項(xiàng)目(10671155)．

楊存典(1965G),男,陜西山陽人,教授,研究方向?yàn)閿?shù)論及特殊函數(shù)應(yīng)用．EGmail:slycd＠126．com．