關(guān)于Lucas多項(xiàng)式平方和的恒等式

楊瑞妮,董曉茹

(西北大學(xué)數(shù)學(xué)系,陜西西安 710127)

關(guān)于Lucas多項(xiàng)式平方和的恒等式

楊瑞妮,董曉茹

(西北大學(xué)數(shù)學(xué)系,陜西西安 710127)

利用廣義Lucas多項(xiàng)式Ln(x,y)的性質(zhì),通過構(gòu)造組合和式Tn(x,y;tx2),結(jié)合Bernou lli多項(xiàng)式的生成函數(shù)和Euler多項(xiàng)式的生成函數(shù),采用分析學(xué)中的方法,得到兩個有關(guān)L2n(x,y)的恒等式.并從這一結(jié)果出發(fā),得到了兩個推論,推廣了相關(guān)文獻(xiàn)的一些結(jié)果.

Lucas多項(xiàng)式;Bernoulli多項(xiàng)式;Euler多項(xiàng)式

DO I：10.3969/j.issn.1008-5513.2013.04.014

1 引言及其結(jié)論

2 引理及其證明

3 定理的證明

兩個定理得證.

類似地,利用以上的方法和步驟還可以對有關(guān)Lucas多項(xiàng)式的更高次恒等式進(jìn)行推導(dǎo),進(jìn)而獲得更為一般的推廣結(jié)果.巧妙的構(gòu)造法和分析學(xué)中的方法值得在以后的研究中將其推廣,應(yīng)用到更多的多項(xiàng)式中去,例如切比雪夫多項(xiàng)式[6].

[1]Claudio de Jesus Pita Ruiz Velasco.A Note on Fibonacci&Lucas and Bernoulli&Eu ler Polynom ials[J]. Journal of Integer Sequences,2012,15:1-17.

[2]Ireland K,Rosen M.A classical introduction tomodern number theory[M].znd ed.New York:Sp ring-Verlag, 1990.

[3]Tom M Apostol.Introduction to Analytic Number Theory[M].New York:Spring-Verlag,1976.

[4]Sun Zhiwei,Pan Hao.Identities concerning Bernou lliand Eu ler polynom ials[J].Acta A rith,2006,125:21-39.

[5]Pan Hao,Sun Zhiwei.New identities involving Bernou lli and Eu ler polynom ials[J].Combin.Theory Ser. A,2006,125:156-175.

[6]劉國棟,羅輝.一些包含Chebyshev多項(xiàng)式和Stirling數(shù)的恒等式[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),2010,26(2): 177-182.

Som e iden tities on the quad ratic sum of Lucas polynom ials

Yang Ruini,Dong Xiaoru
(Department of Mathematics,Northwest University,X i′an 710127,China)

W ith the properties of generalized Lucas polynom ials Ln(x,y)and themethod of analysis,we obtain a list of identities connecting the sum of L2n(x,y)by work ing w ith Tn(x,y;tx2)and generating functions of Bernou lli polynom ials Bi(t)and Eu ler polynom ials Ei(t).Simultaneously,some corollaries of these resu lts can be easily estab lished.Thus some results obtained in some essays are generalized.

Lucas polynom ial,Bernou llipolynom ial,Eu ler polynom ial

O156.4

A

1008-5513(2013)04-0433-08

2013-05-12.

陜西省教育廳科學(xué)研究項(xiàng)目(11JK 0470).

楊瑞妮(1987-),碩士生,研究方向：數(shù)論.

2010 M SC:11B39