On Infinite Series of Reciprocals of Generalized Bi-Periodic Fibonacci Numbers

We define the generalized bi-periodic Fibonacci sequence {fn} by the recurrence relation fn+1=afn+cfn−1 if n is even and fn+1=bfn+cfn−1 if n is odd, with initial conditions f0=0 and f1=1. This sequence generalizes the well-known Fibonacci sequence {Fn}, which has the same recurrence relation with a=b=c=1. In this paper, we focus on the values of the series of reciprocals of generalized bi-periodic Fibonacci numbers as ∑n=1∞c2n−1f2n, ∑n=0∞cncn+f2n+1 and ∑n=1∞(−c)n−1fnfn+1, generalizing the well-known values of these series for the Fibonacci sequence ∑n=0∞1F2n=7−52, ∑n=0∞11+F2n+1=52 and ∑n=1∞(−1)n−1fnfn+1=5−12.