Lambert Series in the Summation of Reciprocals in Gaussian Fibonacci Sequences

Abstract

In this paper, we consider infinite sums derived from the reciprocals of the Gaussian Fibonacci numbers. New expressions of these sums are obtained in terms of Lambert series.

Keywords

• Gaussian Fibonacci number,Reciprocal sum,Lambert series

References

1. Berzseny, G., Gaussian Fibonacci numbers, Fibonacci Quarterly, 15(1977), 233--236.
2. Bilgici, G., Two generalizations of Lucas sequence, Applied Mathematics and Computation, 245(2014), 526--538.
3. Bolat, C., Köse H., On the properties of k-Fibonacci numbers, Int. J. Contemp. Math. Sciences, 5(2010), 1097--1105.
4. Edson, M., Yayenie, O., A new generalization of Fibonacci sequence and extended binet's formula}, Integers, 9(2009), 639--654.
5. Elsner, C., Shimomura, S., ShiokawaI. I., Algebraic relations for reciprocal sums of Fibonacci numbers, Acta Arith., 130(2007), 37--60.
6. Elsner, C., Shimomura, S., ShiokawaI. I., Algebraic relations for reciprocal sums of odd terms in Fibonacci numbers, Ramanujan J., 17(2008), 429--446.
7. Falcon, S., Plaza, A., On the Fibonacci k-numbers, Chaos Soliton Fract., 32(2007), 1615--1624.
8. Falcon, S., On the k-Lucas numbers, Int. J. Contemp. Math. Sciences, 6(2011),1039--1050.

9. Horadam, A. F., Complex Fibonacci numbers and Fibonacci quaternions, Amer. Math. Monthly, 70(1963), 289--291.
10. Horadam, A. F., Elliptic functions and Lambert series in the summation of reciprocals in certain recurrence-generated sequences, Fibonacci Quarterly, 26(1988), 98--114.
11. Jordan, J.H., Gaussian Fibonacci and Lucas numbers, Fibonacci Quarterly, 3(1965), 315--318.
12. Koshy, T., Fibonacci and Lucas Numbers with Applications, John Wiley & Sons, New York, 2001.
13. Ohtsuka, H., Nakamura, S., On the sum of reciprocal Fibonacci numbers, Fibonacci Quart., 46/47(2008/2009), 153--159.