Reciprocal series involving Horadam numbers

K.  Adegoke; R.  Frontczak; T.  Goy

doi:10.37863/umzh.v75i3.6874

K. Adegoke Obafemi Awolowo University, Nigeria https://orcid.org/0000-0002-3933-0459
R. Frontczak Landesbank Baden–Württemberg, Stuttgart, Germany https://orcid.org/0000-0002-9373-5297
T. Goy Vasyl Stefanyk Precarpathian National University, Ivano-Frankivsk, Ukraine

DOI: https://doi.org/10.37863/umzh.v75i3.6874

Keywords: Horadam sequence, Fibonacci sequence, Lucas sequence, reciprocal sum

Abstract

UDC 511.176

We evaluate some new three-parameter families of finite reciprocal sums involving Horadam numbers. We are also able to state the results for the associated infinite sums. Some Fibonacci and Lucas sums are presented as examples.

References

K. Adegoke, Generalizations of reciprocal Fibonacci–Lucas sums of Brousseau, J. Integer Seq., 21, Article 18.1.6 (2018).

R. André-Jeannin, Lambert series and the summation of reciprocals in certain Fibonacci–Lucas-type sequences, Fibonacci Quart., 28, № 3, 223–226 (1990).

R. André-Jeannin, Summation of reciprocals in certain second-order recurring sequences, Fibonacci Quart., 35, № 1, 68–74 (1997).

P. S. Bruckman, I. J. Good, A generalization of a series of de Morgan, with applications of Fibonacci type, Fibonacci Quart., 14, № 3, 193–196 (1976).

D. Duverney, I. Shiokawa, On series involving Fibonacci and Lucas numbers. I, Proc. DARF Conf. 2007, AIP Conf. Proc., 976, 62–76 (2008). DOI: https://doi.org/10.1063/1.2841912

B. Farhi, Summation of certain infinite Lucas-related series, J. Integer Seq., 22, Article 19.1.6 (2019).

R. Frontczak, Closed-form evaluations of Fibonacci–Lucas reciprocal sums with three factors, Notes Number Theory Discrete Math., 23, № 2, 104–116 (2017).

R. Frontczak, Further results on arctangent sums with applications to generalized Fibonacci numbers, Notes Number Theory Discrete Math., 23, № 1, 39–53 (2017).

R. Frontczak, On two new classes of Fibonacci and Lucas reciprocal sums with subscripts in arithmetic progression, Appl. Math. Sci., 11, № 25, 1211–1219 (2017). DOI: https://doi.org/10.12988/ams.2017.7392

I. J. Good, A reciprocal series of Fibonacci numbers, Fibonacci Quart., 12, № 4, 346 (1974).

V. E. Hoggatt (Jr.), M. Bicknell, A primer for the Fibonacci numbers, Part XV. Variations on summing a series of reciprocals of Fibonacci numbers, Fibonacci Quart., 14, № 3, 272–276 (1976).

V. E. Hoggatt (Jr.), M. Bicknell, A reciprocal series of Fibonacci numbers with subscripts $2^nk$, Fibonacci Quart., 14, № 5, 453–455 (1976).

A. F. Horadam, Basic properties of a certain generalized sequence of numbers, Fibonacci Quart., 3, № 3, 161–176 (1965).

A. F. Horadam, Elliptic functions and Lambert series in the summation of reciprocals in certain recurrence-generated sequences, Fibonacci Quart., 26, № 2, 98–114 (1988).

A. F. Horadam, Generalization of a result of Morgado, Port. Math., 44, № 2, 131–136 (1987).

H. Hu, Z.-W. Sun, J.-X. Liu, Reciprocal sums of second-order recurrent sequences, Fibonacci Quart., 39, № 3, 214–220 (2001).

E. Kılıç, D. Ersanli, New reciprocal sums involving finite products of second order recursions, Miskolc Math. Notes, 20, № 2, 1039–1050 (2019). DOI: https://doi.org/10.18514/MMN.2019.2810

E. Kılıç, H. Prodinger, Closed form evaluation of Melham's reciprocal sums, Miskolc Math. Notes, 18, № 1, 251–264 (2017). DOI: https://doi.org/10.18514/MMN.2017.1284

V. Laohakosol, K. Kuhapatanakul, Reciprocal sums of generalized second order recurrence sequences, Fibonacci Quart., 46/47, № 4, 316–325 (2008/2009).

R. S. Melham, Finite reciprocal sums involving summands that are balanced products of generalized Fibonacci numbers, J. Integer Seq., 17, Article 14.6.5 (2014).

R. S. Melham, More on finite sums that involve reciprocals of products of generalized Fibonacci numbers, Integers, 14, #A4 (2014). DOI: https://doi.org/10.1515/9783110298161.560

R. S. Melham, On certain families of finite reciprocal sums that involve generalized Fibonacci numbers, Fibonacci Quart., 53, № 4, 323–334 (2015).

R. S. Melham, Reciprocal series of squares of Fibonacci related sequences with subscripts in arithmetic progression, J. Integer Seq., 18, Article 15.8.7 (2015).

R. S. Melham, A. G. Shannon, On reciprocal sums of Chebyshev related sequences, Fibonacci Quart., 33, № 3, 194–202 (1995).

D. A. Miller, Problem H-237, Fibonacci Quart., 12, № 3, 309 (1974).

S. Rabinowitz, Algorithmic simplification of reciprocals sums, F. T. Howard (ed.), Applications of Fibonacci Numbers, Springer, Dordrecht (1999), p. 277–292. DOI: https://doi.org/10.1007/978-94-011-4271-7_27

S. Rabinowitz, Algorithmic summation of reciprocals of products of Fibonacci numbers, Fibonacci Quart., 37, № 2, 122–127 (1999).

N. J. A. Sloane, ed., The On-Line Encyclopedia of Integer Sequences; https://oeis.org.

R. E. Whitney, ed., Advanced problem and solutions, Fibonacci Quart., 14, № 1, 88–92 (1976).

F.-Z. Zhao, Notes on reciprocal series related to Fibonacci and Lucas numbers, Fibonacci Quart., 37, № 3, 254–257 (1999).