关于两个线性递推序列

Note on Two Bindered Linear Recurrence Relations

用发生函数的办法考察了线性递推关系bi,j=αbi-1,j+βbi,j-1和ci,j=αci,j-1+βci,j-1+βαci-1,j-1的特殊情况所确定的矩阵B和C,得到了矩阵B,C的分解B=P[α](b0,0I+ωE)PT[β],C=P[α]DPT[β]和相应行列式的值。发现B,C与Pascal矩阵P有着紧密的联系。

A interesting factorizations of matrix B with entries , ?1,,?1=+ijijijb α bβb and matrix whose entries satisfy , ?1,,?1?1,?1=++ijijijijc αc βcαβc in special case are derived as ( )[][]0,0αωβTB = PbI+EP and [α ][β]TC = PDP respectively. In addition, we solve the corresponding determinants and find that B and C are related to Pascal matrix P.