一類正負(fù)相間對(duì)偶三角函數(shù)級(jí)數(shù)

張來萍， 及萬(wàn)會(huì)

(銀川能源學(xué)院 基礎(chǔ)部， 寧夏 銀川 750105)

一類正負(fù)相間對(duì)偶三角函數(shù)級(jí)數(shù)

張來萍， 及萬(wàn)會(huì)

(銀川能源學(xué)院 基礎(chǔ)部， 寧夏 銀川 750105)

根據(jù)一個(gè)已知級(jí)數(shù)，利用裂項(xiàng)法得到一些正負(fù)相間二項(xiàng)式系數(shù)倒數(shù)的級(jí)數(shù)，然后利用復(fù)變數(shù)的理論給出系數(shù)為二項(xiàng)式系數(shù)倒數(shù)的正負(fù)相間對(duì)偶三角函數(shù)級(jí)數(shù)封閉形和式.

二項(xiàng)式系數(shù)；裂項(xiàng)；倒數(shù)；級(jí)數(shù)；對(duì)偶；正負(fù)相間；封閉形

定理1 系數(shù)為正負(fù)相間的二項(xiàng)式系數(shù)倒數(shù)的級(jí)數(shù)：

(1)

(2)

(3)

(4)

(5)

定理2 角的偶數(shù)倍正負(fù)相間對(duì)偶三角函數(shù)的級(jí)數(shù)恒等式：

(6)

(7)

以下式中A,B分別表示式(6),(7),

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

定理3 角的奇數(shù)倍正負(fù)相間對(duì)偶三角函數(shù)的級(jí)數(shù)恒等式：

(16)

(17)

以下式中C,D分別表示式(16)和(17),

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

1)對(duì)(1)式左端裂項(xiàng)，

令n-1=m，

由于D1已知，整理得(2)式.

2)設(shè)(2)式右端為D3,對(duì)(1)式左端裂項(xiàng)

令n-2=m，

兩個(gè)分式乘積化成部分分式,得

由于D1，D3已知，整理得(3)式.

3) 設(shè)(3)式右端為D5，對(duì)(1)式左端裂項(xiàng)，

令n-3=m，

兩個(gè)分式乘積化成部分分式，得

由于D1，D3，D5已知，整理得(4)式.

4) 設(shè)(4)式右端為D7,對(duì)(1)式左端裂項(xiàng)，

令n-4=m，

即

4個(gè)分式乘積化成部分分式，計(jì)算得

由于D1，D3，D5，D7已知,整理得 (5)式.

定理2的證明

所以，

于是,

將A，B代入(1)式,得到(6), (7)式.

將x=cost+isint與A,B，D1依次代入(2)，(3)，(4)，(5)式，利用復(fù)數(shù)相等得到(9)～(15)式.

定理3的證明

[1]LEHMERDH.Interestingseriesinvolvingthecentralbinomialcoefficients[J].AmerMatMonthly,1985, 92:449-457．

[2]SURYB,WANGTIANNING,ZHAOFZ.Someidentitiesinvolvingofbinomialcoefficients[J].JournalofIntegralSequences, 2004(7),Article04.2.8．

[3]SOFOA.GeneralpropertiesinvolvingreciprocalofBinomialcoefficients[J].JournalofIntegralSequences, 2006(9),Article.06.4.5．

[4]BORWEINJM,GIRGENSOHNR.Evaluationofbinomialseries[J].AequationensMath,2005,70:25-36．

[5]GRADSHTEYNIS,ZYZHIKIM.Atableofintegral,seriesandproducts[M].7thEd.Beijing：Elsevier，46-47．

[6] 鐘玉泉.復(fù)變函數(shù)論[M].北京:高等教育出版社,2004:88-90.

One Class of Series Alternated with Positive and Negativeof Dual Trigonometric Function

ZHANG Lai-ping, JI Wan-hui

(DepartmentofBasic,YinchuanEnergyCollege,Yinchuan750105,China)

According to a known series, by splitting terms, the reciprocals series alternated with positive and negative of binominal coefficients is obtained. And by complex function, the closed formal is obtained. The closed form of sum of the dual trigonometric function series alternated with positive and negative, whose coefficients involving reciprocals of binominal coefficients, is given.

binominal coefficients； splitting terms； reciprocal; series； dual; alternated with positive and negative；closed form

2014-08-27

銀川能源學(xué)院科學(xué)研究基金項(xiàng)目(2011-37-15)

張來萍(1979—),女，寧夏銀川人，銀川能源學(xué)院基礎(chǔ)部講師.

10.3969/j.issn.1007-0834.2014.04.003

O173

A

1007-0834(2014)04-0011-06