关于一类非常系数线性递推式的解的显式表示

THE SOLUTION OF A CLASS OF LINEAR RECURRENCE WITH VARIABLE COEFFICIENTS

本文给出了两个指标的非常系数的线性递推式的显式解 .有关方法 ,避免了由于解高阶线性代数方程所带来的困难 .其结果 ,为求解组合计数中相应定解问题 ,提供了一个明确的计算公式 .

In this paper we consider the following class of linear recurrence with variable coefficients with two indicesu i,j =f(i,j)u i-1,j-1 +g(i,j)u i-q,j-q +h(i,j), u i,0 =c i,0 ,u 0,j =c 0,j (i,j=0,1,…),u i,j =0(i<0 or j<0),where i,j=1,2,…,q≥2,f(i,j),g(i,j) and h(i,j) (i,j≥1) are variable numbers,c i,0  and c 0,j (i,j=0,1,…) are vrbitrary constants.Its general solution is given by the following formulau i,j =F(i,j;j,j-qm)c i-j,0 +∑j-1n=0｛F(i,j,j-n-1,j-n-1-qm)｝ ×h(i-j+n+1,n+1)(i>j), F(i,j;i,i-qm)c 0,j-i +∑i-1n=0｛F(i,j;i-n-1,i-n-1-qm)｝ ×h(n+1,j-i+n+1)(j≥i) (i,j=1,2,…)