一类以广义Fibonacci数为系数的线性不定方程

A class of linear diophantine equations with generalized Fibonacci number coefficients

设a,b为整数,b≠0。广义Fibonacci序列{un}定义为u0=0,u1=1,un+2=aun+1+bun(n≥0)。设a,b,c,n,k,m,r为整数,求解关于t1,…,tm-r的不定方程(+1-)1m ri i k m ii?t e u c=∑=(k>0,m-1>r≥0,c∈Z,ei=±1,i=1,…,m-r)给出了求解例子,并较详细说明了在构造F-L恒等式方面的应用。

Let a, b be integers, b ≠0, Defining the generalized Fibonacci sequence {un} as： u0 = 0, u1 = 1, un＋2=aun＋1 ＋ bun（n≥0）, we solved the linear Diophantine equation with unknowns t1 …… tm-r.,r∑i=1tieiuk（m＋1-i）=c（k〉0,m-1〉r≥0,c∈Z,ei=±1,i=1,……,m-r）Examples tbr solving equations are given. Applications to constructing F-L identities are stated in detail.