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Year 2021, Volume: 34 Issue: 2, 493 - 504, 01.06.2021
https://doi.org/10.35378/gujs.705885

Abstract

References

  • [1] Conway, J. H., Guy, R. K., “The book of numbers”, Springer, New York, (1996).
  • [2] Dumont, D., “Matrices d'Euler-Seidel”, Semin. Lothar. Comb., 5 (1981).
  • [3] Dil, A., Mezö, I., “A symmetric algorithm for hyperharmonic and Fibonacci numbers”, Appl. Math. Comput., 206: 942-951, (2008).
  • [4] Bahşi, M., Mezö, I., Solak, S., “A symmetric algorithm for hyper-Fibonacci and hyper-Lucas numbers”, Ann. Math. Inform., 43:19-27, (2014).
  • [5] Mezö, I., Ramirez, J. L., “A q-Symmetric algorithm and its applications to some combinatorial sequences”, Online J. Anal. Comb., 12: 1-13, (2017).
  • [6] Dil, A., Kurt, V., “Investigating geometric and exponential polynomials with Euler-Seidel matrices”, J. Integer Seq., 14: 1-12, (2011).
  • [7] Dil, A., Mezö, I., “Euler-Seidel method for certain combinatorial numbers and a new characterization of Fibonacci sequence”, Cent. Eur. J. Math., 7(2): 310-321, (2009).
  • [8] Ramirez, J. L., Shattuck, M., “A multi-parameter generalization of the symmetric algorithm”, Math. Slovaca, 68(4): 699-712, (2018).
  • [9] Holliday, S., Komatsu, T., “On the sum of reciprocal generalized Fibonacci numbers”, Integers, 11: 441-455, (2011).
  • [10] Ohtsuka, H., Nakamura, S., “On the sum of reciprocal Fibonacci numbers”, Fibonacci Quart., 46 (2):153-159, (2008).
  • [11] Rabinowitz, S., “Algorithmic summation reciprocals of products of Fibonacci numbers”, Fibonacci Quart., 37: 122-127, (1999).
  • [12] Tuglu, N., Kızılateş, C., Kesim, S., “On the harmonic and hyperharmonic Fibonacci numbers”, Adv. Difference Equ., 2015: 297, (2015).
  • [13] Tuglu, N., Kızılateş C., “On the norms of circulant and r-circulant matrices with the hyperharmonic Fibonacci numbers”, J.Inequal. Appl., 2015: 253, (2015).
  • [14] Tuglu, N., Kızılateş, C., “On the norms of some special matrices with the harmonic Fibonacci numbers”, Gazi University Journal of Science, 28 (3): 447-501, (2015).
  • [15] Shapiro, L. Getu, W. S., Woan, W. J., Woodson, L. C., “The Riordan group”, Discrete Appl. Math., 34: 229-239, (1991).
  • [16] Cheon, G. S., Hwang, S. G., Lee, S. G., “Several polynomials associated with the harmonic numbers”, Discrete Appl.Math., 155: 2573-2584, (2007).
  • [17] Cheon, G. S., Mikkawy, M. E. A., “Generalized harmonic numbers with Riordan arrays”, J. Number Theory, 128: 413-425, (2008).
  • [18] Munarini, E., “Riordan matrices and sums of harmonic numbers”, Appl. Anal. Discrete Math., 5: 176-200, (2011).
  • [19] Matala-Aho, T. and Vaananen, K., “On Approximation Measures of q-Logarithms”, Bull. Austral. Math. Soc., 58: 15-31, (1998).
  • [20] Murty, M. R., “The Fibonacci Zeta Function, Automorphic Representations and L-Functions”, Tata Institue of Fundamental Research, Hindustan Book Agency, New Delhi, 1-17, (2013).

Some Identities of Harmonic and Hyperharmonic Fibonacci Numbers

Year 2021, Volume: 34 Issue: 2, 493 - 504, 01.06.2021
https://doi.org/10.35378/gujs.705885

Abstract

This paper is concerned with the combinatorial identities of the harmonic and the hyperharmonic Fibonacci numbers. By using the symmetric algorithm, we get some identities which improve the usual results and generalize known equations. Moreover, with the help of concept of Riordan array, we obtain the generating functions for these numbers and a variety of identities are derived.

References

  • [1] Conway, J. H., Guy, R. K., “The book of numbers”, Springer, New York, (1996).
  • [2] Dumont, D., “Matrices d'Euler-Seidel”, Semin. Lothar. Comb., 5 (1981).
  • [3] Dil, A., Mezö, I., “A symmetric algorithm for hyperharmonic and Fibonacci numbers”, Appl. Math. Comput., 206: 942-951, (2008).
  • [4] Bahşi, M., Mezö, I., Solak, S., “A symmetric algorithm for hyper-Fibonacci and hyper-Lucas numbers”, Ann. Math. Inform., 43:19-27, (2014).
  • [5] Mezö, I., Ramirez, J. L., “A q-Symmetric algorithm and its applications to some combinatorial sequences”, Online J. Anal. Comb., 12: 1-13, (2017).
  • [6] Dil, A., Kurt, V., “Investigating geometric and exponential polynomials with Euler-Seidel matrices”, J. Integer Seq., 14: 1-12, (2011).
  • [7] Dil, A., Mezö, I., “Euler-Seidel method for certain combinatorial numbers and a new characterization of Fibonacci sequence”, Cent. Eur. J. Math., 7(2): 310-321, (2009).
  • [8] Ramirez, J. L., Shattuck, M., “A multi-parameter generalization of the symmetric algorithm”, Math. Slovaca, 68(4): 699-712, (2018).
  • [9] Holliday, S., Komatsu, T., “On the sum of reciprocal generalized Fibonacci numbers”, Integers, 11: 441-455, (2011).
  • [10] Ohtsuka, H., Nakamura, S., “On the sum of reciprocal Fibonacci numbers”, Fibonacci Quart., 46 (2):153-159, (2008).
  • [11] Rabinowitz, S., “Algorithmic summation reciprocals of products of Fibonacci numbers”, Fibonacci Quart., 37: 122-127, (1999).
  • [12] Tuglu, N., Kızılateş, C., Kesim, S., “On the harmonic and hyperharmonic Fibonacci numbers”, Adv. Difference Equ., 2015: 297, (2015).
  • [13] Tuglu, N., Kızılateş C., “On the norms of circulant and r-circulant matrices with the hyperharmonic Fibonacci numbers”, J.Inequal. Appl., 2015: 253, (2015).
  • [14] Tuglu, N., Kızılateş, C., “On the norms of some special matrices with the harmonic Fibonacci numbers”, Gazi University Journal of Science, 28 (3): 447-501, (2015).
  • [15] Shapiro, L. Getu, W. S., Woan, W. J., Woodson, L. C., “The Riordan group”, Discrete Appl. Math., 34: 229-239, (1991).
  • [16] Cheon, G. S., Hwang, S. G., Lee, S. G., “Several polynomials associated with the harmonic numbers”, Discrete Appl.Math., 155: 2573-2584, (2007).
  • [17] Cheon, G. S., Mikkawy, M. E. A., “Generalized harmonic numbers with Riordan arrays”, J. Number Theory, 128: 413-425, (2008).
  • [18] Munarini, E., “Riordan matrices and sums of harmonic numbers”, Appl. Anal. Discrete Math., 5: 176-200, (2011).
  • [19] Matala-Aho, T. and Vaananen, K., “On Approximation Measures of q-Logarithms”, Bull. Austral. Math. Soc., 58: 15-31, (1998).
  • [20] Murty, M. R., “The Fibonacci Zeta Function, Automorphic Representations and L-Functions”, Tata Institue of Fundamental Research, Hindustan Book Agency, New Delhi, 1-17, (2013).
There are 20 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Mathematics
Authors

Miraç Çetin 0000-0002-9588-0295

Can Kızılateş 0000-0002-7958-4226

Fatma Yeşil Baran 0000-0001-8613-2706

Naim Tuglu 0000-0002-7277-0034

Publication Date June 1, 2021
Published in Issue Year 2021 Volume: 34 Issue: 2

Cite

APA Çetin, M., Kızılateş, C., Yeşil Baran, F., Tuglu, N. (2021). Some Identities of Harmonic and Hyperharmonic Fibonacci Numbers. Gazi University Journal of Science, 34(2), 493-504. https://doi.org/10.35378/gujs.705885
AMA Çetin M, Kızılateş C, Yeşil Baran F, Tuglu N. Some Identities of Harmonic and Hyperharmonic Fibonacci Numbers. Gazi University Journal of Science. June 2021;34(2):493-504. doi:10.35378/gujs.705885
Chicago Çetin, Miraç, Can Kızılateş, Fatma Yeşil Baran, and Naim Tuglu. “Some Identities of Harmonic and Hyperharmonic Fibonacci Numbers”. Gazi University Journal of Science 34, no. 2 (June 2021): 493-504. https://doi.org/10.35378/gujs.705885.
EndNote Çetin M, Kızılateş C, Yeşil Baran F, Tuglu N (June 1, 2021) Some Identities of Harmonic and Hyperharmonic Fibonacci Numbers. Gazi University Journal of Science 34 2 493–504.
IEEE M. Çetin, C. Kızılateş, F. Yeşil Baran, and N. Tuglu, “Some Identities of Harmonic and Hyperharmonic Fibonacci Numbers”, Gazi University Journal of Science, vol. 34, no. 2, pp. 493–504, 2021, doi: 10.35378/gujs.705885.
ISNAD Çetin, Miraç et al. “Some Identities of Harmonic and Hyperharmonic Fibonacci Numbers”. Gazi University Journal of Science 34/2 (June 2021), 493-504. https://doi.org/10.35378/gujs.705885.
JAMA Çetin M, Kızılateş C, Yeşil Baran F, Tuglu N. Some Identities of Harmonic and Hyperharmonic Fibonacci Numbers. Gazi University Journal of Science. 2021;34:493–504.
MLA Çetin, Miraç et al. “Some Identities of Harmonic and Hyperharmonic Fibonacci Numbers”. Gazi University Journal of Science, vol. 34, no. 2, 2021, pp. 493-04, doi:10.35378/gujs.705885.
Vancouver Çetin M, Kızılateş C, Yeşil Baran F, Tuglu N. Some Identities of Harmonic and Hyperharmonic Fibonacci Numbers. Gazi University Journal of Science. 2021;34(2):493-504.