关于递推关系αa_(i-1,j-1)+βa_(i-1,j)=a_(i,j)

On the recurrence relation αα_(i-1,j-1)+βα_(i-1,j)=α_(i,j)

研究了满足ααi-1,j-1+βαi-1,j=αi,j的序列{αi,j}.用发生函数法得到了n+1阶矩阵A=(αi,j)(n+1)-(n+1)的精确表达式.用数学归纳法证明(1-βx-axy)中一般项xiyi(i≥j)的系数为αjβi-j i+n-1 n-1 ij.导出了一些有关二项式系数(nk)的新的组合恒等式.

Recurrence sequence {αi,j} with ααi-1,j-1+βαi-1,j=αi,j is studied. The explicit expression of n+1 order matrix A = (αi,j)(n+1)×(n+1) is obtained by using generating function. The coefficient of x1y1 in (1 - βx - axy)-n is proved to be αjβi-j i+n-1 n-1 ij by mathematical induction. In addition, some new combinatorial identities related to binomial coefficient are derived.