Harmonic number identities via polynomials with r-Lah coefficients

Levent Kargın; Mümün Can

doi:10.5802/crmath.53

Théorie des nombres

Harmonic number identities via polynomials with r-Lah coefficients
[Identités sur les nombres harmonique via des polynômes à coefficients r-Lah]

Levent Kargın ¹ ; Mümün Can ¹

¹ Department of Mathematics, Akdeniz University, Antalya, Turkey

Comptes Rendus. Mathématique, Volume 358 (2020) no. 5, pp. 535-550.

Résumé
Abstract

Dans cet article, des polynômes à coefficients faisant intervenir les nombres $r$ -Lah sont utilisés pour établir plusieurs formules de sommation en fonction des coefficients binomiaux, des nombres de Stirling et des nombres harmoniques ou hyper-harmoniques. De plus, nous introduisons le nombre asymétrique-hyper-harmonique et nous étudions ses propriétés de base.

In this paper, polynomials whose coefficients involve $r$ -Lah numbers are used to evaluate several summation formulae involving binomial coefficients, Stirling numbers, harmonic or hyperharmonic numbers. Moreover, skew-hyperharmonic number is introduced and its basic properties are investigated.

Reçu le : 2020-02-05
Révisé le : 2020-04-18
Accepté le : 2020-04-19
Publié le : 2020-09-14

DOI : 10.5802/crmath.53

Classification : 11B75, 11B68, 47E05, 11B73, 11B83

Affiliations des auteurs :

Levent Kargın ¹ ; Mümün Can ¹

¹ Department of Mathematics, Akdeniz University, Antalya, Turkey

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Levent Karg{\i}n and M\"um\"un Can},
     title = {Harmonic number identities via polynomials with {r-Lah} coefficients},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {535--550},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {5},
     year = {2020},
     doi = {10.5802/crmath.53},
     language = {en},
}

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Levent Kargın; Mümün Can. Harmonic number identities via polynomials with r-Lah coefficients. Comptes Rendus. Mathématique, Volume 358 (2020) no. 5, pp. 535-550. doi : 10.5802/crmath.53. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.53/

Bibliographie
Cité par

[1] Rachid Ait-Amrane; Hacène Belbachir Non-integerness of class of hyperharmonic numbers, Ann. Math. Inform., Volume 37 (2010), pp. 7-11 | MR | Zbl

[2] Rachid Ait-Amrane; Hacène Belbachir Are the hyperharmonics integral, A partial answer via the small intervals containing primes, C. R. Math. Acad. Sci. Paris, Volume 349 (2011) no. 3-4, pp. 115-117 | DOI | MR | Zbl

[3] Hacène Belbachir; Amine Belkhir Cross recurrence relations for $r$ -Lah numbers, Ars Comb., Volume 110 (2013), pp. 199-203 | MR | Zbl

[4] Arthur T. Benjamin; David Gaebler; Robert Gaebler A combinatorial approach to hyperharmonic numbers, Integers, Volume 3 (2003), pp. 1-9 | MR | Zbl

[5] Arthur T. Benjamin; Gregory O. Preston; Jennifer J. Quinn A Stirling encounter with harmonic numbers, Math. Mag., Volume 75 (2002) no. 2, pp. 95-103 | DOI | MR | Zbl

[6] David H. Borwein; Jonathan M. Bailey; Roland Girgensohn Explicit evaluation of Euler sums, Proc. Edinb. Math. Soc., Volume 38 (1995) no. 2, pp. 277-294 | DOI | MR | Zbl

[7] Khristo N. Boyadzhiev A series transformation formula and related polynomials, Int. J. Math. Math. Sci., Volume 23 (2005), pp. 3849-3866 | DOI | MR | Zbl

[8] Khristo N. Boyadzhiev Harmonic number identities via Euler’s transform, J. Integer Seq., Volume 12 (2009) no. 6, 09.6.1, p. 8 | MR | Zbl

[9] Khristo N. Boyadzhiev Series transformation formulas of Euler type, Hadamard product of series, and harmonic number identities, Indian J. Pure Appl. Math., Volume 42 (2011) no. 5, pp. 371-386 | DOI | MR | Zbl

[10] Khristo N. Boyadzhiev Power series with skew-harmonic numbers, dilogarithms, and double integrals, Tatra Mt. Math. Publ., Volume 56 (2013), pp. 93-108 | MR | Zbl

[11] Khristo N. Boyadzhiev Binomial transform and the backward difference, Adv. Appl. Discrete Math., Volume 13 (2014) no. 1, pp. 43-63 | MR | Zbl

[12] Khristo N. Boyadzhiev Notes on the Binomial Transform. Theory and table with appendix on Stirling transform, World Scientific, 2018 | Zbl

[13] Khristo N. Boyadzhiev; Ayhan Dil Geometric polynomials: properties and applications to series with zeta values, Anal. Math., Volume 42 (2016) no. 3, pp. 203-224 | DOI | MR

[14] Andrei Z. Broder The $r$ -Stirling numbers, Discrete Math., Volume 49 (1984), pp. 241-259 | DOI | Zbl

[15] Mümün Can; Muhammet Cihat Dağli Extended Bernoulli and Stirling matrices and related combinatorial identities, Linear Algebra Appl., Volume 444 (2014), pp. 114-131 | DOI | MR | Zbl

[16] José Luis Cereceda An introduction to hyperharmonic numbers (classroom note), Int. J. Math. Educ. Sci. Technol., Volume 46 (2015) no. 3, pp. 461-469 | DOI | MR | Zbl

[17] Junesang Choi Finite summation formulas involving binomial coefficients, harmonic numbers and generalized harmonic numbers, J. Inequal. Appl., Volume 1 (2013), 49 | MR | Zbl

[18] Wenchang Chu Summation formulae involving harmonic numbers, Filomat, Volume 26 (2012) no. 1, pp. 143-152 | MR | Zbl

[19] Wenchang Chu; De Donno Livia Hypergeometric series and harmonic number identities, Adv. Appl. Math., Volume 34 (2005) no. 1, pp. 123-137 | MR | Zbl

[20] John H. Conway; Richard K. Guy The book of numbers, Springer, 1996 | Zbl

[21] Ayhan Dil; Khristo N. Boyadzhiev Euler sums of hyperharmonic numbers, J. Number Theory, Volume 147 (2015), pp. 490-498 | MR | Zbl

[22] Ayhan Dil; Veli Kurt Polynomials related to harmonic numbers and evaluation of harmonic number series II, Appl. Anal. Discrete Math., Volume 5 (2011) no. 2, pp. 212-229 | MR | Zbl

[23] Ayhan Dil; Veli Kurt Polynomials related to harmonic numbers and evaluation of harmonic number series I, Integers, Volume 12 (2012), a38, pp. 1-18 | MR | Zbl

[24] Ayhan Dil; István Mező A symmetric algorithm for hyperharmonic and Fibonacci numbers, Appl. Math. Comput., Volume 206 (2008) no. 2, pp. 942-951 | MR | Zbl

[25] Ayhan Dil; István Mező; Mehmet Cenkci Evaluation of Euler-like sums via Hurwitz zeta values, Turk. J. Math., Volume 41 (2017) no. 6, pp. 1640-1655 | MR | Zbl

[26] Ayhan Dil; Erkan Muniroğlu Applications of derivative and difference operators on some sequences (2019) (https://arxiv.org/abs/1910.01876)

[27] Philippe Flajolet; Bruno Salvy Euler sums and contour integral representations, Exp. Math., Volume 7 (1998) no. 1, pp. 15-35 | DOI | MR | Zbl

[28] Haydar Göral; Doğa Can Sertbaş Almost all hyperharmonic numbers are not integers, J. Number Theory, Volume 147 (2017), pp. 495-526 | DOI | MR | Zbl

[29] Som Prakash Goyal; R. K. Laddha On the generalized Riemann zeta functions and the generalized Lambert transform, Ganita Sandesh, Volume 11 (1997) no. 2, pp. 99-108 | MR | Zbl

[30] Bai-Ni Guo; Feng Qi Some integral representations and properties of Lah numbers, J. Algebra Number Theory Acad., Volume 4 (2014) no. 3, pp. 77-87

[31] Ken Kamano Dirichlet series associated with hyperharmonic numbers, Mem. Osaka Inst. Tech., Volume 56 (2011) no. 2, pp. 11-15 | MR

[32] Levent Kargin Some formulae for products of geometric polynomials with applications, J. Integer Seq., Volume 20 (2017) no. 4, Article 17.4.4. | MR | Zbl

[33] Levent Kargin; Bayram Çekim Higher order generalized geometric polynomials, Turk. J. Math., Volume 42 (2018) no. 3, pp. 887-903 | MR | Zbl

[34] Levent Kargin; Roberto B. Corcino Generalization of Mellin derivative and its applications, Integral Transforms Spec. Funct., Volume 27 (2016) no. 8, pp. 620-631 | DOI | MR | Zbl

[35] Bernd C. Kellner Identities between polynomials related to Stirling and harmonic numbers, Integers, Volume 14 (2014), A54 | MR | Zbl

[36] Peter M. Knopf The operator ${(x \frac{d}{d x})}^{n}$ and its application to series, Math. Mag., Volume 76 (2003) no. 5, pp. 364-371 | DOI

[37] István Mező About the non-integer property of hyperharmonic numbers, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math., Volume 50 (2007), pp. 13-20 | MR | Zbl

[38] István Mező Analytic extension of hyperharmonic numbers, Online J. Anal. Comb., Volume 4 (2009), 1 | MR | Zbl

[39] István Mező; Ayhan Dil Euler–Seidel method for certain combinatorial numbers and a new characterization of Fibonacci sequence, Cent. Eur. J. Math., Volume 7 (2009) no. 2, pp. 310-321 | MR | Zbl

[40] István Mező; Ayhan Dil Hyperharmonic series involving Hurwitz zeta function, J. Number Theory, Volume 130 (2010) no. 2, pp. 360-369 | DOI | MR | Zbl

[41] Gábor Nyul; Gabriella Rácz The r-Lah numbers, Discrete Math., Volume 338 (2015) no. 10, pp. 1660-1666 | DOI | MR | Zbl

[42] Peter Paule; Carsten Schneider Computer proofs of a new family of harmonic number identities, Adv. Appl. Math., Volume 31 (2003) no. 2, pp. 359-378 | DOI | MR | Zbl

[43] Aleksandar Petojević A note about the Pochhammer symbol, Mathematica Moravica, Volume 12 (2008) no. 1, pp. 37-42 | DOI | MR | Zbl

[44] R. Sita Rama Chandra Rao; A. Siva Rama Sarma Some identities involving the Riemann zeta function, Indian J. Pure Appl. Math., Volume 10 (1979), pp. 602-607 | MR | Zbl

[45] József Sándor; Borislav Crstici Handbook of number theory. Vol II, Kluwer Academic Publishers, 2004 | Zbl

[46] Madjid Sebaoui; Diffalah Laissaoui; G. Guettai; Mourad Rahmani On s-Lah polynomials, Ars Comb., Volume 142 (2019), pp. 111-118 | MR | Zbl

[47] Jürgen Spieß Some identities involving harmonic numbers, Math. Comput., Volume 55 (1990) no. 132, pp. 839-863 | DOI | MR | Zbl

[48] Michael Z. Spivey Combinatorial sums and finite differences, Discrete Math., Volume 307 (2007) no. 24, pp. 3130-3146 | DOI | MR | Zbl

[49] Leopold Theisinger Bemerkung über die harmonische Reihe, Monatsh. Math. Phys., Volume 26 (1915), pp. 132-134 | DOI | Zbl

[50] Ce Xu Euler sums of generalized hyperharmonic numbers, J. Korean Math. Soc., Volume 55 (2018) no. 5, pp. 1207-1220 | MR | Zbl

[51] Qinglun Yan; Yaqing Liu Harmonic number identities involving telescoping method and derivative operator, Integral Transforms Spec. Funct., Volume 28 (2017) no. 10, pp. 703-709 | MR | Zbl

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