Research Article
BibTex RIS Cite

Year 2022, Volume: 5 Issue: 2, 185 - 190, 30.06.2022

Muhammet Cihat Dağlı , Feng Qi

https://doi.org/10.53006/rna.1002272
Cited By: 1

Abstract

References

  • [1] G.-S. Cheon and M. El-Mikkawy, Generalized harmonic numbers with Riordan arrays, J. Number Theory 128 (2008), no. 2, 413?425; available online at https://doi.org/10.1016/j.jnt.2007.08.011.
  • [2] J. Choi and H. M. Srivastava, Some summation formulas involving harmonic numbers and generalized harmonic numbers, Math. Comput. Modelling 54 (2011), 2220-2234; available online at https://doi.org/10.1016/j.mcm.2011.05.032.
  • [3] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co., 1974; available online at https://doi.org/10.1007/978-94-010-2196-8.
  • [4] G. Dattoli, S. Licciardi, E. Sabia, and H. M. Srivastava, Some properties and generating functions of generalized harmonic numbers, Mathematics 7 (2019), no. 4, Article No. 577; available online at https://doi.org/10.3390/math7070577.
  • [5] Ö. Duran, N. Ömür, and S. Koparal, On sums with generalized harmonic, hyperharmonic and special numbers, Miskolc Math. Notes 21 (2020), no. 2, 791-803; available online at https://doi.org/10.18514/MMN.2020.3458.
  • [6] C.-J. Feng and F.-Z. Zhao, Some results for generalized harmonic numbers, Integers 9 (2009), no. 5, 605?619; available online at https://doi.org/10.1515/INTEG.2009.048.
  • [7] A. Gertsch, Nombres harmoniques généralisés, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 1, 7-10; available online at https://doi.org/10.1016/S0764-4442(97)80094-8. (French)
  • [8] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison- Wesley Publishing Company, Reading, MA, 1994.
  • [9] B.-N. Guo and F. Qi, Sharp inequalities for the psi function and harmonic numbers, Analysis (Berlin) 34 (2014), no. 2, 201-208; available online at https://doi.org/10.1515/anly-2014-0001.
  • [10] L. Kargin and M. Can, Harmonic number identities via polynomials with r-Lah coefficients, C. R. Math. Acad. Sci. Paris 358 (2020), no. 5, 535-550; available online at https://doi.org/10.5802/crmath.53.
  • [11] L. Kargin, M. Cenkci, A. Dil, and M. Can, Generalized harmonic numbers via poly-Bernoulli polynomials, arXiv (2008), available online at https://arxiv.org/abs/2008.00284v2.
  • [12] T. Kim and D. S. Kim, Some identities on derangement and degenerate derangement polynomials, Advances in Mathematical Inequalities and Applications, 265-277, Trends Math., Birkhäuser/Springer, Singapore, 2018; available online at https: //doi.org/10.1007/978-981-13-3013-1_13.
  • [13] T. Kim, D. S. Kim, D. V. Dolgy, and J. Kwon, Some identities of derangement numbers, Proc. Jangjeon Math. Soc. 21 (2018), no. 1, 125-141.
  • [14] T. Kim, D. S. Kim, G.-W. Jang, and J. Kwon, A note on some identities of derangement polynomials, J. Inequal. Appl. 2018, Paper No. 40, 17 pages; available online at https://doi.org/10.1186/s13660-018-1636-8.
  • [15] E. Munarini, Callan-like identities, Online J. Anal. Comb. No. 14 (2019), 20 pp.
  • [16] D.-W. Niu, Y.-J. Zhang, and F. Qi, A double inequality for the harmonic number in terms of the hyperbolic cosine, Turkish J. Anal. Number Theory 2 (2014), no. 6, 223-225; available online at http://dx.doi.org/10.12691/tjant-2-6-6.
  • [17] N. Ömür and G. Bilgin, Some applications of the generalized hyperharmonic numbers of order r, H r n (α), Adv. Appl. Math.Sci. 17 (2018), no. 9, 617-627.
  • [18] F. Qi, A determinantal representation for derangement numbers, Glob. J. Math. Anal. 4 (2016), no. 3, 17-17; available online at https://doi.org/10.14419/gjma.v4i3.6574.
  • [19] F. Qi, Diagonal recurrence relations for the Stirling numbers of the first kind, Contrib. Discrete Math. 11 (2016), 22-30; available online at https://doi.org/10.11575/cdm.v11i1.62389.
  • [20] F. Qi, Three closed forms for convolved Fibonacci numbers, Results Nonlinear Anal. 3 (2020), no. 4, 185-195.
  • [21] F. Qi, M. C. Dagh, and W.-S. Du, Determinantal forms and recursive relations of the Delannoy two-functional sequence, Adv. Theory Nonlinear Anal. Appl. 4 (2020), no. 3, 184-193; available online at https://doi.org/10.31197/atnaa.772734.
  • [22] F. Qi and B.-N. Guo, A diagonal recurrence relation for the Stirling numbers of the first kind, Appl. Anal. Discrete Math. 12 (2018), no. 1, 153?165; available online at https://doi.org/10.2298/AADM170405004Q.
  • [23] F. Qi and B.-N. Guo, Explicit formulas for derangement numbers and their generating function, J. Nonlinear Funct. Anal. 2016, Article ID 45, 10 pages.
  • [24] F. Qi, J.-L. Wang, and B.-N. Guo, A recovery of two determinantal representations for derangement numbers, Cogent Math. (2016), 3:1232878, 7 pages; available online at https://doi.org/10.1080/23311835.2016.1232878.
  • [25] F. Qi, J.-L. Wang, and B.-N. Guo, A representation for derangement numbers in terms of a tridiagonal determinant, Kragujevac J. Math. 42 (2018), no. 1, 7-14; available online at https://doi.org/10.5937/KgJMath1801007F.
  • [26] F. Qi, J.-L. Zhao, and B.-N. Guo, Closed forms for derangement numbers in terms of the Hessenberg determinants, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 112 (2018), no. 4, 933-944; available online at https: //doi.org/10.1007/s13398-017-0401-z.
  • [27] J. M. Santmyer, A Stirling like sequence of rational numbers, Discret. Math. 171 (1997), no. 3, 229-235; available online at https://doi.org/10.1016/S0012-365X(96)00082-9.
  • [28] A. Sofo and H. M. Srivastava, A family of shifted harmonic sums, Ramanujan J. 37 (2015), no. 1, 89-108; available online at https://doi.org/10.1007/s11139-014-9600-9.
  • [29] N. M. Temme, Special Functions: An Introduction to Classical Functions of Mathematical Physics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996; available online at http://dx.doi.org/10.1002/9781118032572.
  • [30] M. L. Wachs, On q-derangement numbers, Proc. Amer. Math. Soc. 106 (1989), no. 1, 273-278; available online at https: //doi.org/10.2307/2047402.
  • [31] C. Y. Wang, P. Miska, and I. Mez®, The r-derangement numbers, Discret. Math. 340 (2017), 1681-1692; available online at https://doi.org/10.1016/j.disc.2016.10.012.
  • [32] F.-Z. Zhao, The log-balancedness of generalized derangement numbers, Ramanujan J. (2021), in press; available online at https://doi.org/10.1007/s11139-021-00402-1.

Several recurrence relations and identities on generalized derangement numbers

Year 2022, Volume: 5 Issue: 2, 185 - 190, 30.06.2022

Muhammet Cihat Dağlı , Feng Qi

https://doi.org/10.53006/rna.1002272
Cited By: 1

Abstract

In the paper, with aid of generating functions, the authors present several recurrence relations and identities for generalized derangement numbers involving generalized harmonic numbers and the Stirling numbers of the first kind.

Keywords

recurrence relation identity generalized derangement number generalized harmonic numbers Stirling numbers of the first kind generating function

References

  • [1] G.-S. Cheon and M. El-Mikkawy, Generalized harmonic numbers with Riordan arrays, J. Number Theory 128 (2008), no. 2, 413?425; available online at https://doi.org/10.1016/j.jnt.2007.08.011.
  • [2] J. Choi and H. M. Srivastava, Some summation formulas involving harmonic numbers and generalized harmonic numbers, Math. Comput. Modelling 54 (2011), 2220-2234; available online at https://doi.org/10.1016/j.mcm.2011.05.032.
  • [3] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co., 1974; available online at https://doi.org/10.1007/978-94-010-2196-8.
  • [4] G. Dattoli, S. Licciardi, E. Sabia, and H. M. Srivastava, Some properties and generating functions of generalized harmonic numbers, Mathematics 7 (2019), no. 4, Article No. 577; available online at https://doi.org/10.3390/math7070577.
  • [5] Ö. Duran, N. Ömür, and S. Koparal, On sums with generalized harmonic, hyperharmonic and special numbers, Miskolc Math. Notes 21 (2020), no. 2, 791-803; available online at https://doi.org/10.18514/MMN.2020.3458.
  • [6] C.-J. Feng and F.-Z. Zhao, Some results for generalized harmonic numbers, Integers 9 (2009), no. 5, 605?619; available online at https://doi.org/10.1515/INTEG.2009.048.
  • [7] A. Gertsch, Nombres harmoniques généralisés, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 1, 7-10; available online at https://doi.org/10.1016/S0764-4442(97)80094-8. (French)
  • [8] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison- Wesley Publishing Company, Reading, MA, 1994.
  • [9] B.-N. Guo and F. Qi, Sharp inequalities for the psi function and harmonic numbers, Analysis (Berlin) 34 (2014), no. 2, 201-208; available online at https://doi.org/10.1515/anly-2014-0001.
  • [10] L. Kargin and M. Can, Harmonic number identities via polynomials with r-Lah coefficients, C. R. Math. Acad. Sci. Paris 358 (2020), no. 5, 535-550; available online at https://doi.org/10.5802/crmath.53.
  • [11] L. Kargin, M. Cenkci, A. Dil, and M. Can, Generalized harmonic numbers via poly-Bernoulli polynomials, arXiv (2008), available online at https://arxiv.org/abs/2008.00284v2.
  • [12] T. Kim and D. S. Kim, Some identities on derangement and degenerate derangement polynomials, Advances in Mathematical Inequalities and Applications, 265-277, Trends Math., Birkhäuser/Springer, Singapore, 2018; available online at https: //doi.org/10.1007/978-981-13-3013-1_13.
  • [13] T. Kim, D. S. Kim, D. V. Dolgy, and J. Kwon, Some identities of derangement numbers, Proc. Jangjeon Math. Soc. 21 (2018), no. 1, 125-141.
  • [14] T. Kim, D. S. Kim, G.-W. Jang, and J. Kwon, A note on some identities of derangement polynomials, J. Inequal. Appl. 2018, Paper No. 40, 17 pages; available online at https://doi.org/10.1186/s13660-018-1636-8.
  • [15] E. Munarini, Callan-like identities, Online J. Anal. Comb. No. 14 (2019), 20 pp.
  • [16] D.-W. Niu, Y.-J. Zhang, and F. Qi, A double inequality for the harmonic number in terms of the hyperbolic cosine, Turkish J. Anal. Number Theory 2 (2014), no. 6, 223-225; available online at http://dx.doi.org/10.12691/tjant-2-6-6.
  • [17] N. Ömür and G. Bilgin, Some applications of the generalized hyperharmonic numbers of order r, H r n (α), Adv. Appl. Math.Sci. 17 (2018), no. 9, 617-627.
  • [18] F. Qi, A determinantal representation for derangement numbers, Glob. J. Math. Anal. 4 (2016), no. 3, 17-17; available online at https://doi.org/10.14419/gjma.v4i3.6574.
  • [19] F. Qi, Diagonal recurrence relations for the Stirling numbers of the first kind, Contrib. Discrete Math. 11 (2016), 22-30; available online at https://doi.org/10.11575/cdm.v11i1.62389.
  • [20] F. Qi, Three closed forms for convolved Fibonacci numbers, Results Nonlinear Anal. 3 (2020), no. 4, 185-195.
  • [21] F. Qi, M. C. Dagh, and W.-S. Du, Determinantal forms and recursive relations of the Delannoy two-functional sequence, Adv. Theory Nonlinear Anal. Appl. 4 (2020), no. 3, 184-193; available online at https://doi.org/10.31197/atnaa.772734.
  • [22] F. Qi and B.-N. Guo, A diagonal recurrence relation for the Stirling numbers of the first kind, Appl. Anal. Discrete Math. 12 (2018), no. 1, 153?165; available online at https://doi.org/10.2298/AADM170405004Q.
  • [23] F. Qi and B.-N. Guo, Explicit formulas for derangement numbers and their generating function, J. Nonlinear Funct. Anal. 2016, Article ID 45, 10 pages.
  • [24] F. Qi, J.-L. Wang, and B.-N. Guo, A recovery of two determinantal representations for derangement numbers, Cogent Math. (2016), 3:1232878, 7 pages; available online at https://doi.org/10.1080/23311835.2016.1232878.
  • [25] F. Qi, J.-L. Wang, and B.-N. Guo, A representation for derangement numbers in terms of a tridiagonal determinant, Kragujevac J. Math. 42 (2018), no. 1, 7-14; available online at https://doi.org/10.5937/KgJMath1801007F.
  • [26] F. Qi, J.-L. Zhao, and B.-N. Guo, Closed forms for derangement numbers in terms of the Hessenberg determinants, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 112 (2018), no. 4, 933-944; available online at https: //doi.org/10.1007/s13398-017-0401-z.
  • [27] J. M. Santmyer, A Stirling like sequence of rational numbers, Discret. Math. 171 (1997), no. 3, 229-235; available online at https://doi.org/10.1016/S0012-365X(96)00082-9.
  • [28] A. Sofo and H. M. Srivastava, A family of shifted harmonic sums, Ramanujan J. 37 (2015), no. 1, 89-108; available online at https://doi.org/10.1007/s11139-014-9600-9.
  • [29] N. M. Temme, Special Functions: An Introduction to Classical Functions of Mathematical Physics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996; available online at http://dx.doi.org/10.1002/9781118032572.
  • [30] M. L. Wachs, On q-derangement numbers, Proc. Amer. Math. Soc. 106 (1989), no. 1, 273-278; available online at https: //doi.org/10.2307/2047402.
  • [31] C. Y. Wang, P. Miska, and I. Mez®, The r-derangement numbers, Discret. Math. 340 (2017), 1681-1692; available online at https://doi.org/10.1016/j.disc.2016.10.012.
  • [32] F.-Z. Zhao, The log-balancedness of generalized derangement numbers, Ramanujan J. (2021), in press; available online at https://doi.org/10.1007/s11139-021-00402-1.
There are 32 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Muhammet Cihat Dağlı 0000-0003-2859-902X

Feng Qi

Publication Date June 30, 2022
Published in Issue Year 2022 Volume: 5 Issue: 2

Cite

APA Dağlı, M. C., & Qi, F. (2022). Several recurrence relations and identities on generalized derangement numbers. Results in Nonlinear Analysis, 5(2), 185-190. https://doi.org/10.53006/rna.1002272
AMA Dağlı MC, Qi F. Several recurrence relations and identities on generalized derangement numbers. RNA. June 2022;5(2):185-190. doi:10.53006/rna.1002272
Chicago Dağlı, Muhammet Cihat, and Feng Qi. “Several Recurrence Relations and Identities on Generalized Derangement Numbers”. Results in Nonlinear Analysis 5, no. 2 (June 2022): 185-90. https://doi.org/10.53006/rna.1002272.
EndNote Dağlı MC, Qi F (June 1, 2022) Several recurrence relations and identities on generalized derangement numbers. Results in Nonlinear Analysis 5 2 185–190.
IEEE M. C. Dağlı and F. Qi, “Several recurrence relations and identities on generalized derangement numbers”, RNA, vol. 5, no. 2, pp. 185–190, 2022, doi: 10.53006/rna.1002272.
ISNAD Dağlı, Muhammet Cihat - Qi, Feng. “Several Recurrence Relations and Identities on Generalized Derangement Numbers”. Results in Nonlinear Analysis 5/2 (June 2022), 185-190. https://doi.org/10.53006/rna.1002272.
JAMA Dağlı MC, Qi F. Several recurrence relations and identities on generalized derangement numbers. RNA. 2022;5:185–190.
MLA Dağlı, Muhammet Cihat and Feng Qi. “Several Recurrence Relations and Identities on Generalized Derangement Numbers”. Results in Nonlinear Analysis, vol. 5, no. 2, 2022, pp. 185-90, doi:10.53006/rna.1002272.
Vancouver Dağlı MC, Qi F. Several recurrence relations and identities on generalized derangement numbers. RNA. 2022;5(2):185-90.

Cited By

ON SUMS WITH THE r-DERANGEMENT NUMBERS
Journal of Science and Arts
https://doi.org/10.46939/J.Sci.Arts-23.2-a14