关于Lucas数立方与二项式数的卷积公式

对于非负整数l，L_l表示第l个Lucas数；$\left( {array}{l}n\\i{array} \right) = \frac{{n!}}{{i!\left( {n - i} \right)!}}$为二项式系数；对于非负整数l和k以及正整数n，设l(k, 3, n)是数列$\left\{ {\left( {array}{l}n\\i{array} \right)} \right\}_{i = 0}^n$和$\left\{ {L_{k + i}^3} \right\}_{i = 0}^n$的卷积，即l(k, 3, n)=$\left( {array}{l}n\\0{array} \right)L_k^3 + \left( {array}{l}n\\1{array} \right)L_{k + 1}^3 + \cdots + \left( {array}{l}n\\n{array} \right)L_{k + n}^3 = \sum\limits_{i = 0}^n {\left( {array}{l}n\\i{array} \right)L_{k + i}^3} $。文章证明了k≥n时，l(k, 3, n)=2nL_3k+2n+3(－1)k+nL_k－n; 当k < n时，l(k, 3, n)=2nL_3k+2n+3L_n－k成立。

A Relationship Between Binomial Coefficients and Cubes of Lucas Numbers

For nonnegative integer l, L_l is the lth Lucas Number and $\left( {array}{l}n\\i{array} \right) = \frac{{n!}}{{i!\left( {n - i} \right)!}}$ is the binomial coeffient. For any nonnegative integer l, k and positive integer n, l(k, 3, n) denotes the convolution of sequence $\left\{ {\left( {array}{l}n\\i{array} \right)} \right\}_{i = 0}^n$ and $\left\{ {L_{k + i}^3} \right\}_{i = 0}^n$, namely, l(k, 3, n)=$\left( {array}{l}n\\0{array} \right)L_k^3 + \left( {array}{l}n\\1{array} \right)L_{k + 1}^3 + \cdots + \left( {array}{l}n\\n{array} \right)L_{k + n}^3 = \sum\limits_{i = 0}^n {\left( {array}{l}n\\i{array} \right)L_{k + i}^3} $. According to the definition of the Fibonacci sequence and by using the knowledge of elementary number theory, it is proved that l(k, 3, n) is equal to 2nL_3k+2n+3(-1)k+nL_k-n or 2nL_3k+2n+3L_n-k when k≥n or not.