| In recent years,many scholars have studied the reciprocal sums of recursive sequences and the floor of the tail term of Riemann zeta function,and obtained the floor formulas including the reciprocal sums of Fibonacci sequences,Pell sequences,higher-order linear recursive sequences and their subsequences,and the reciprocal sums of higher power.Based on this,we consider the extension of the floor problem of the tail term of Fibonacci zeta function,and take the exponent of the sequence in the floor problem of reciprocal sums as a fraction.In addition,we will investigate the related properties of generalized Fibonacci sequence(Dn=a·Dn-1+b·Dn-2(n≥2),D0=c,D1=d).At present,there are studies on the reciprocal sums and reciprocal product of generalized Fibonacci sequence,and there are restrictions on the selection of coefficients in the recursive formula,that is,the coefficient a of Dn-1 is required to be greater than the coefficient b of Dn-2.Therefore,we hope to study a more general case,that is,the number sequence with the coefficient of Dn-1 smaller than the coefficient of Dn-2,and consider the floor problems of its reciprocal sums and reciprocal product.And we study its weighted sum formulas.Firstly,this paper studies the reciprocal sums of fractional powers of Fibonacci sequence and product of adjacent terms,and obtains the exact expression of[(∑k=n∞1/Fk1/s)-1].By using the elementary method,we study the reciprocal sums of the generalized form of term and odd term in a class of generalized Fibonacci sequence {Gn},and obtain the floor formula in the form of[(∑k=n∞1/Gpk)-1].Then,we extend it to the more general generalized sequence｛Hn},study the reciprocal sums and reciprocal product of the sequence {Hn｝,give a more accurate estimation,and give the related identities.Finally,several kinds of weighted sums of a class of generalized Fibonacci sequence {Fn} and generalized Tribonacci sequence {Tn} are studied. |
