摘要
著名的Fibonacci数列{Fn},其中F0=F1=1,Fn+1=Fn+Fn-1,(n=1,2,…),在许多实际问题中都有着极其广泛的应用.Fibonacci数列通项的得出方法多种多样.在文献[2]用生成函数的方法得出了Fibonacci数列通项的基础上,将Fibonacci数列由各项取自然数推广至各项取任意实数,得到广义Fibonacci数列,其中R0=a,R1=b,Rn+1=uRn+vRn-1(n=1,2,…).其中a,b,u,v∈R.并用生成函数的方法得出推广后的广义Fibonacci数列的通项.希望这种方法可应用在求有关递推数列的通项中.
The famous Fibonacci sequence {F_n},F_0=F_1=1,F_(n+1)=F_n+F_(n-1)(n=1,2,…),are widely used in many practical situations. Many methods can be used to get the general term of Fibonacci sequence, for example, by way of generating function as in reference [2]. This paper explains that the general term of generalized Fibonacci sequence can be got by the way of each number's having its natural number extended to arbitrary real number, in which R_0=a,R_1=b,R_(n+1)=uR_n+vR_(n-1) (n=1,2,……),and a、b、u、v∈R. The generating function can also be used to get the extended general term of the generalized Fibonacci sequence. This method can also be used to get the general term of the relevant recursive sequence.
