Recurrences and congruences for higher order geometric polynomials and related numbers

  • L. Kargın Akdeniz Univ., Antalya, Turkey
  • M. Cenkci Akdeniz Univ., Antalya, Turkey
Keywords: Higher order geometric polynomials, p-Bernoulli numbers, congruences

Abstract

UDC 517.5

We obtain new recurrence relations, an explicit formula, and convolution identities for higher order geometric polynomials. These relations generalize known results for geometric polynomials, and lead to congruences for higher order geometric polynomials, particularly for $p$-Bernoulli numbers.

References

C. Ahlbach, J. Usatine, N. Pippenger, Barred preferential arrangement, Electron. J. Combin., 20, № 2, 1 – 18 (2013); https://doi.org/10.37236/2482 DOI: https://doi.org/10.37236/2482

A. A. Asgari, M. Jahangiri, On the periodicity problem of residual $r$-Fubini sequences, J. Integer Seq. 21, Article 18.4.5 (2018).

A. Z. Broder, The $r$-Stirling numbers, Discrete Math., 49, 241 – 259 (1984); https://doi.org/10.1016/0012-365X(84)90161-4 DOI: https://doi.org/10.1016/0012-365X(84)90161-4

K. N. Boyadzhiev, A series transformation formula and related polynomials, Int. J. Math. and Math. Sci., 2005, № 23, 3849 – 3866 (2005); https://doi.org/10.1155/IJMMS.2005.3849 DOI: https://doi.org/10.1155/IJMMS.2005.3849

K. N. Boyadzhiev, Apostol – Bernoulli functions, derivative polynomials and Eulerian polynomials, Adv. Appl. Discrete Math., 1, № 2, 109 – 122 (2008).

K. N. Boyadzhiev, Exponential polynomials, Stirling numbers, and evaluation of some gamma integrals, Abstr. and Appl. Anal., Article ID 168672 (2009); https://doi.org/10.1155/2009/168672 DOI: https://doi.org/10.1155/2009/168672

K. N. Boyadzhiev, Close encounters with the Stirling numbers of the second kind, Math. Mag., 85, № 4, 252 – 266 (2012); https://doi.org/10.4169/math.mag.85.4.252 DOI: https://doi.org/10.4169/math.mag.85.4.252

K. N. Boyadzhiev, A. Dil, Geometric polynomials: properties and applications to series with zeta values, Anal. Math., 42, № 3, 203 – 224 (2016); https://doi.org/10.1007/s10476-016-0302-y DOI: https://doi.org/10.1007/s10476-016-0302-y

L. Comtet, Advanced combinatorics, Riedel, Dordrech, Boston 1974. DOI: https://doi.org/10.1007/978-94-010-2196-8

M. E. Dasef, S. M. Kautz, Some sums of some importance, College Math. J., 28, 52 – 55 (1997). DOI: https://doi.org/10.1080/07468342.1997.11973831

T. Diagana, H. Ma¨ıga, Some new identities and congruences for Fubini numbers, J. Number Theory, 173, 547 – 569 (2017); https://doi.org/10.1016/j.jnt.2016.09.032 DOI: https://doi.org/10.1016/j.jnt.2016.09.032

A. Dil, V. Kurt, Investigating geometric and exponential polynomials with Euler-Seidel matrices, J. Integer Seq., 14, № 4, Article 11.4.6, 12 pp. (2011).

A. Dil, V. Kurt, Polynomials related to harmonic numbers and evaluation of harmonic number series II, Appl. Anal. and Discrete Math., 5, № 2, 212 – 229 (2011); https://doi.org/10.2298/AADM110615015D DOI: https://doi.org/10.2298/AADM110615015D

A. Dil, V. Kurt, Polynomials related to harmonic numbers and evaluation of harmonic number series I, Integers, 12, Article A38 (2012).

R. L. Graham, D. E. Knuth, O. Patashnik, Concrete mathematics, Addison-Wesley Publ. Co., New York (1994).

O. A. Gross, Preferential arrangements, Amer. Math. Monthly, 69, 4 – 8 (1962), https://doi.org/10.2307/2312725 DOI: https://doi.org/10.1080/00029890.1962.11989826

F. T. Howard, Congruences for the Stirling numbers and associated Stirling numbers, Acta Arith., 55, № 1, 29 – 41 (1990); https://doi.org/10.4064/aa-55-1-29-41 DOI: https://doi.org/10.4064/aa-55-1-29-41

L. C. Hsu, P. J.-S. Shiue, A unified approach to generalized Stirling numbers, Adv. Appl. Math., 20, № 3, 366 – 384 (1998). DOI: https://doi.org/10.1006/aama.1998.0586

D. H. Kauffman, H. Dolores, Note on preferential arrangements, Amer. Math. Monthly., 70, Article 62 (1963); https://doi.org/10.2307/2312790 DOI: https://doi.org/10.2307/2312790

L. Kargın, Some formulae for products of geometric polynomials with applications, J. Integer Seq., 20, № 4, Article 17.4.4 (2017).

L. Kargın, $p$-Bernoulli and geometric polynomials, Int. J. Number Theory, 14, № 2, 595 – 613 (2018). DOI: https://doi.org/10.1142/S1793042118500665

L. Kargın, R. B. Corcino, Generalization of Mellin derivative and its applications, Integral Transforms and Spec. Funct., 27, № 8, 620 – 631 (2016); https://doi.org/10.1080/10652469.2016.1174701 DOI: https://doi.org/10.1080/10652469.2016.1174701

L. Kargın, B. Çekim, Higher order generalized geometric polynomials, Turkish J. Math., 42, № 3, 887 – 903 (2018), https://doi.org/10.3906/mat-1705-95 DOI: https://doi.org/10.3906/mat-1705-95

B. C. Kellner, Identities between polynomials related to Stirling and harmonic numbers, Integers, 14, Paper No. A54, 22 pp. (2014).

D. S. Kim, T. Kim, H.-I. Kwon, J.-W. Park, Two variable higher-order Fubini polynomials, J. Korean Math. Soc., 55, № 4, 975 – 986 (2018); https://doi.org/10.4134/JKMS.j170573

I. Mező, The $r$-Bell numbers, J. Integer Seq., 14, № 1, Article 11.1.1 (2011).

I. Mező, Periodicity of the last digits of some combinatorial sequences, J. Integer Seq., 17, № 1, Article 14.1.1, 18 pp. (2014).

M. Mihoubi, H. Belbachir, Linear recurrences for $r$-Bell Polynomials, J. Integer Seq., 17, № 10, Article 14.10.6, 10 pp. (2014).

M. Mihoubi, S. Taharbouchet, Identities and congruences involving the geometric polynomials, Miskolc Math. Notes., 20, № 1, 395 – 408 (2019). DOI: https://doi.org/10.18514/MMN.2019.2498

M. Rahmani, On $p$-Bernoulli numbers and polynomials, J. Number Theory., 157, 350 – 366 (2015); https://doi.org/10.1016/j.jnt.2015.05.019 DOI: https://doi.org/10.1016/j.jnt.2015.05.019

S. M. Tanny, On some numbers related to the Bell numbers, Canad. Math. Bull., 17, № 5,

– 738 (1974), https://doi.org/10.4153/CMB-1974-132-8 DOI: https://doi.org/10.4153/CMB-1974-132-8

Published
17.12.2021
How to Cite
Kargın, L., and M. Cenkci. “Recurrences and Congruences for Higher Order Geometric Polynomials and Related Numbers”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 12, Dec. 2021, pp. 1619 -37, doi:10.37863/umzh.v73i12.1080.
Section
Research articles