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Balancing Polynomials, Fibonacci Numbers and Some New Series for \(\pi \)

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Abstract

We evaluate some types of infinite series with balancing and Lucas-balancing polynomials in closed form. These evaluations will lead to some new curious series for \(\pi \) involving Fibonacci and Lucas numbers. Our findings complement those of Castellanos from 1986 to 1989.

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Data sharing not applicable to this article as no special data is used during the current study.

References

  1. Castellanos, D.: Rapidly converging expansions with Fibonacci coefficients. Fibonacci Quart. 24, 70–82 (1986)

    MathSciNet  MATH  Google Scholar 

  2. Castellanos, D.: A generalization of binet’s formula and some of its consequences. Fibonacci Quart. 27(5), 424–438 (1989)

    MathSciNet  MATH  Google Scholar 

  3. Frontczak, R.: On balancing polynomials. Appl. Math. Sci. 13(2), 57–66 (2019)

    MATH  Google Scholar 

  4. Frontczak, R.: Powers of balancing polynomials and some consequences for Fibonacci sums. Int. J. Math. Anal. 13(3), 109–115 (2019)

    Article  Google Scholar 

  5. Frontczak, R.: Relating Fibonacci numbers to bernoulli numbers via balancing polynomials. J. Integer Seq. 22, Article 19.5.3

  6. Frontczak, R., Goy, T.: Combinatorial sums associated with balancing and Lucas-balancing polynomials. Ann. Math. Inform. 52, 97–105 (2020)

    MathSciNet  MATH  Google Scholar 

  7. Frontczak, R., Goy, T.: Additional close links between balancing and Lucas-balancing polynomials. Adv. Stud. Cont. Math. 31, 287–300 (2021)

    MATH  Google Scholar 

  8. Kim, D.S., Kim, T.: On sums of finite products of balancing polynomials. J. Comput. Appl. Math. 377, 112913 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kim, T., Kim, D.S., Dolgy, D.V., Kwon, J.: A note on sums of finite products of Lucas-balancing polynomials. Proc. Jangjeon Math. Soc. 23(1), 1–22 (2020)

    MathSciNet  MATH  Google Scholar 

  10. Kim, T., Kim, D.S., Jang, L.-C., Dolgy, D.V.: Representing by several orthogonal polynomials for sums of finite products of Chebyshev polynomials of the first kind and Lucas polynomials. Adv. Difference Equ. 2019(162), 1–15 (2019)

    MathSciNet  MATH  Google Scholar 

  11. Kim, T., Kim, D.S., Dolgy, D.V., Kwon, J.: Sums of finite products of Chebyshev polynomials of two different types. AIMS Math. 6(11), 12528–12542 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kim, T., Kim, D. S.: Degenerate r-Whitney numbers and degenerate r-Dowling polynomials via boson operators. Adv. in Appl. Math. 140, Paper No. 102394, (2022)

  13. Sloane, N. J., Editor: The on-line encyclopedia of integer sequences. Published electronically at https://oeis.org/A001109

  14. Mező, I.: Several generating functions for second-order recurrence sequences. J. Integer Seq. 12 Article 09.3.7

  15. Chen, H.: On the summation of subseries in closed form. Int. J. Math. Edu. Sci. Tech. 41(4), 538–547 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referees for carefully reading the paper and giving helpful comments and suggestions. The second author would like to thank the University Grant Commission (UGC), India for financial support in the form of a senior research fellowship.

Funding

No funding was received for conducting this study.

Author information

Authors and Affiliations

  1. Landesbank Baden-Württemberg, Am Hauptbahnhof 2, 70173, Stuttgart, Germany

    Robert Frontczak

  2. Department of Mathematics, Central University of Jharkhand, Ranchi, 835205, India

    Kalika Prasad

Authors
  1. Robert Frontczak
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  2. Kalika Prasad
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Contributions

RF and KP wrote the main manuscript text and both authors contributed equally to this work including calculations, verifications, and table preparation. The first draft of the manuscript was written by RF. All authors have read, reviewed and verified the manuscript.

Corresponding author

Correspondence to Kalika Prasad.

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The authors declare that they have no conflict of interest.

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This article does not contain any studies with human participants or animals performed by any of the authors.

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Appendix A: Numerical experiments

Appendix A: Numerical experiments

All series that have been derived in this paper have been verified numerically for different values of the parameters m and n using MATLAB. Here, we present some of the numerical results we obtained, i.e., our results concerning the behavior of the partial sums, paying particular attention to Sections 2 and 3. The reference value for \(\pi \) is

3.141592653589793238462643383279502884197169399375105820974944

59230781...

We begin with a comparison of the series (4) and (14), which is summarized in the first table (Table 1).

Table 1 Comparison of series (4) and series (14)
Full size table

Series (4) converges much faster to \(\pi \) than (14). For instance, for \(n=25\) the partial sum equals 3.1415926535897931755199500028705 and gives 15 correct decimal places of \(\pi \), whereas series (14) only gives one correct decimal place in this case. Even for the big input \(n=100\) (14) is able to get only 6 correct decimal places for \(\pi \).

The series (21)–(24) and their Fibonacci counterparts from (29)-(31) exhibit much better convergence properties as is seen from the next tables. Finally, we state the numerical results for the series containing squared arguments in the numerator (Tables 2, 3, 4, 5).

Table 2 Series with even indexed Lucas coefficients
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Table 3 Series with odd indexed Fibonacci coefficients including Castellaons series (1)
Full size table
Table 4 Series with odd squared Lucas-balancing coefficients
Full size table
Table 5 Comparison of series (46) and series (47)
Full size table

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Frontczak, R., Prasad, K. Balancing Polynomials, Fibonacci Numbers and Some New Series for \(\pi \). Mediterr. J. Math. 20, 207 (2023). https://doi.org/10.1007/s00009-023-02413-2

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