二项式系数与Lucas数四次幂的一个关系  

A relation between binomial coefficients and four-powers of Lucas numbers

作　　者：陈小芳 CHEN Xiaofang(School of Mathematics and Statistics,Weinan Normal University,Weinan Shaanxi 714099)

机构地区：[1]渭南师范学院数学与统计学院,陕西渭南714099

出　　处：《首都师范大学学报（自然科学版）》2023年第2期8-10,共3页Journal of Capital Normal University：Natural Science Edition

基　　金：陕西省科技厅科学研究计划资助项目(2021JM-520)。

摘　　要：若(^(n)_(i))=n!/i!(n-i)!(n,i∈N^(*),且n≥1)表示二项式系数,第l个Lucas数是L_(l),其中l是非负整数;对任意正整数n和非负整数k,数列{(^(n)_(i))}_(i=0)^(n)和{L_(k+i)^(p)}_(i=0)^(n)的卷积为l(k,p,n)=(^(n)_(0))L_(k+i)^(p)+…+(^(n)_(n))L_(k+n)^(p)=∑_(i=0)^(n)(^(n)_(i))L_(k+i)^(p)。本文利用初等数论方法证明了p=4时,等式l(k,4,n)=3^(n)L_(4k+2n)+4(-1)^(k+n)L_(2k+n)+3×2^(n+1)成立。Let(^(n)_(i))=n!/i!(n-i)!(n,i∈N^(*),and n≥1)be a binomial coefficient.Supposing that l is a nonnegative integer and Ll is the l-th Lucas number,furthermore,when n is a fixed positive integer and k is a fixed nonnegative integer,the convolution of the sequence{(^(n)_(i))}_(i=0)^(n)and{L_(k+i)^(p)}_(i=0)^(n)is l(k,p,n)=(^(n)_(0))L_(k+i)^(p)+…+(^(n)_(n))L_(k+n)^(p)=∑_(i=0)^(n)(^(n)_(i))L_(k+i)^(p).When p=4,the identity l(k,4,n)=3^(n)L_(4k+2n)+4(-1)^(k+n)L_(2k+n)+3×2^(n+1)is proved by elementary number theory method.

关 键 词：二项式系数 LUCAS数 四次幂 卷积 

分 类 号：O156.2[理学—数学]