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Harmonic Number Identities Via Polynomials with R-Lah Coefficients

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Harmonic Number Identities Via Polynomials with R-Lah Coefficients Comptes Rendus Mathématique Levent Kargın and Mümün Can Harmonic number identities via polynomials with r-Lah coeYcients Volume 358, issue 5 (2020), p. 535-550. <https://doi.org/10.5802/crmath.53> © Académie des sciences, Paris and the authors, 2020. Some rights reserved. This article is licensed under the Creative Commons Attribution 4.0 International License. http://creativecommons.org/licenses/by/4.0/ Les Comptes Rendus. Mathématique sont membres du Centre Mersenne pour l’édition scientifique ouverte www.centre-mersenne.org Comptes Rendus Mathématique 2020, 358, nO 5, p. 535-550 https://doi.org/10.5802/crmath.53 Number Theory / Théorie des nombres Harmonic number identities via polynomials with r-Lah coeYcients Identités sur les nombres harmonique via des polynômes à coeYcients r-Lah , a a Levent Kargın¤ and Mümün Can a Department of Mathematics, Akdeniz University, Antalya, Turkey. E-mails: [email protected], [email protected]. Abstract. In this paper, polynomials whose coeYcients involve r -Lah numbers are used to evaluate several summation formulae involving binomial coeYcients, Stirling numbers, harmonic or hyperharmonic num- bers. Moreover, skew-hyperharmonic number is introduced and its basic properties are investigated. Résumé. Dans cet article, des polynômes à coeYcients faisant intervenir les nombres r -Lah sont utilisés pour établir plusieurs formules de sommation en fonction des coeYcients 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. 2020 Mathematics Subject Classification. 11B75, 11B68, 47E05, 11B73, 11B83. Manuscript received 5th February 2020, revised 18th April 2020, accepted 19th April 2020. 1. Introduction th The n harmonic number Hn is defined by n X 1 Hn , Æ k 1 k Æ with the assumption H 0. These numbers have a long mathematical history and are seen 0 Æ in various branches of mathematics, especially in number theory. Therefore, there is an enor- mous literature about the identities involving harmonic numbers with binomial coeYcients, ¤ Corresponding author. ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/ 536 Levent Kargın and Mümün Can Stirling numbers and Bernoulli numbers: Benjamin and Quinn [5, Identity 4] used combinato- rial technique to obtain n · ¸ · ¸ X n n 1 k n!Hn Å , k 1 k Æ Æ 2 £ n ¤ Æ where k is the Stirling number of the first kind. Kellner [35] capitalized derivative operator to achieve n ½ ¾ X k 1 n k! n ( 1) Å Hk Bn 1, k 0 ¡ k k 1 Æ 2 ¡ Æ Å © n ª th where k is the Stirling number of the second kind and Bn is the n Bernoulli number. By using finite diVerences, Spivey [48] exhibited many combinatorial sums, for instance, the binomial harmonic identity n à ! k 1 X n ( 1) Å Hn ¡ . Æ k 1 k k Æ This identity was also recorded by Chu [18] and Boyadzhiev [11, 12] with diVerent methods. Utilizing the backward diVerence Boyadzhiev [11,12] reproved the symmetric formula n à ! Hn X k 1 n Hk ( 1) Å . n 1 Æ k 1 ¡ k k 1 Å Æ Å Moreover, numerous evaluation formulas for binomial-harmonic sums (sums involving bino- mial coeYcients and harmonic numbers) are produced by using generating function [47], algo- rithmic methods [42], hypergeometric summation theorems [19], derivative operator [17,18,51], Hadamard multiplication Theorem [9]. (r ) As a generalization of the harmonic numbers, hyperharmonic numbers hn are defined, for r 1, by [20] ¸ n (r ) X (r 1) (0) 1 (r ) hn hk ¡ , with hn , n 1, and h0 0. (1) Æ k 1 Æ n ¸ Æ Æ It is obvious that h(1) H . These numbers have the generating function [4] n Æ n X1 (r ) n ln(1 t) hn t ¡ r (2) n 0 Æ¡ (1 t) Æ ¡ and the explicit representation (cf. [4,20]) à ! (r ) n r 1 hn Å ¡ (Hn r 1 Hr 1). (3) Æ n Å ¡ ¡ ¡ There exist many elegant identities involving hyperharmonic numbers. Some of these identities are exhibited by using combinatorial technique [4], Euler–Siedel matrix [24,39], derivative and diVerence operators [22,23, 26, 38], Pascal type matrix [15]. Whether the properties of harmonic numbers are provided by hyperharmonic numbers are actively studied. For instance, the har- monic number Hn is never an integer except for H1; this is a classical result of Theisinger [49]. (r ) In [37], Mez˝oproved that if r 2 or r 3, the numbers hn are never integers except the trivial Æ Æ r case when n 1. He conjectured that the hyperharmonic numbers h( ) are never integers except Æ n when n 1. This conjecture was handled by Ait–Amrane and Belbachir [1,2], Cereceda [16] and Æ recently, Göral and Sertba¸s[28]. Moreover Euler showed that (see, e.g., [27,44]) m 2 X1 Hn ³ m ´ 1 X¡ N m 1 ³(m 1) ³(m k)³(k 1), m \{1}, n 1 n Æ Å 2 Å ¡ 2 k 1 ¡ Å 2 Æ Æ where ³(s) is the usual Riemann zeta function. The summation on the left-hand side is known as Euler sum. Euler sum is generalized in diVerent means and evaluated in terms of miscellaneous zeta functions. One of the generalizations is Euler sum of hyperharmonic numbers which has also been evaluated in terms of zeta functions [6,21,25,40,50]. C. R. Mathématique, 2020, 358, nO 5, 535-550 Levent Kargın and Mümün Can 537 The aim of this paper is to contribute to the theory of harmonic and hyperharmonic numbers by means of producing identities involving binomial coeYcients, harmonic numbers, Stirling numbers and hyperharmonic numbers. For this purpose we capitalize some families of polyno- mials whose coeYcients involve r -Lah numbers (see Section2 for r -Lah numbers). In fact, these polynomials appear by applying the following Mellin type derivative to appropriate functions: (xD 2r )(xD 2r 1) (xD 2r n 1). Å Å Å ¢¢¢ Å Å ¡ Properties of the arising polynomials give rise to binomial hyperharmonic identity and several summation formulas involving (hyper)harmonic numbers. Moreover, a closed-form evaluation formula for an Euler-type sum is deduced (Theorem8). It should be noted that the studies on the Euler-type sums containing hyperharmonic numbers depend on the lower index [6,21, 25, 31, 40, 50], however, the sum in question is over the upper index. Furthermore, we come across a ¡ generalization of the skew-harmonic numbers Hn defined by n ( 1)k 1 ¡ X Å ¡ Hn ¡ , with H0 0, Æ k 1 k Æ Æ i.e., partial sums of the expansion of log2. We then examine basic properties of these numbers. Additionally, several new formulas for the r -Lah numbers are presented. 2. Preliminaries (n) (0) n (n) Let x x (x 1) (x n 1), x 1, and (x)n ( 1) ( x) denote the rising and falling Æ Å ¢¢¢ Å ¡ Æ Æ ¡ £ n¡¤ © n ª factorial functions. The r -Stirling numbers of the first kind k r and the second kind k r can be defined by [14] n · ¸ n ½ ¾ X n X n (x r )(n) xk ,(x r )n (x) k k k Å Æ k 0 r Å Æ k 0 r £ n ¤ £ n ¤ © n ª © n ª Æ Æ Note that k k 0 and k k 0 are the Stirling numbers of the first and second kind. Æ ¥ n ¦ Æ The r -Lah numbers k r are defined by [3,41] n ¹ º X n (x 2r )(n) (x) , (4) k k Å Æ k 0 r Æ and have the explicit formula à ! ¹nº n! n 2r 1 Å ¡ , (5) l Æ k! k 2r 1 r Å ¡ and the generating function µ ¶k µ ¶2r ¹ º n 1 t 1 X1 n t . (6) k k! 1 t 1 t Æ n k r n! ¡ ¡ Æ In particular ¥ n ¦ ¥ n ¦ is Lah numbers or rarely called Stirling numbers of the third kind [45]. k 0 Æ k The Mellin derivative (xD) x d has been used for many diVerent purposes, such as eval- Æ dx uating some power series, integrals [7, 13,23, 33,36] and also introducing some new families of polynomials [7,22,23,33,34]. When it is applied to a n-times diVerentiable function f we have [7] n ½ ¾ k X n d (xD)n f (x) xk f (x). (7) k Æ k 0 k dx Æ According to the generalizations of the Stirling numbers of the second kind, numerous general- izations of the Mellin derivative have also been studied [7,22,33,34]. Here, as a generalization of the Mellin derivative, we deal with the operator (xD 2r )(n) (xD 2r )(xD 2r 1) (xD 2r n 1). Å Æ Å Å Å ¢¢¢ Å Å ¡ C. R. Mathématique, 2020, 358, nO 5, 535-550 538 Levent Kargın and Mümün Can Then, we have n ¹ º k X n d (xD 2r )(n) f (x) xk f (x), (8) k k Å Æ k 0 r dx Æ which is a companion of (7) and follows from (4) and k k d (xD)k f (x) x f (x). (9) Æ dxk To see (9) we replace x by (xD) in n · ¸ X n k n k (x)n ( 1) ¡ x , Æ k 0 ¡ k Æ and then utilize the identities (7) and n · ¸½ ¾ ( X j k n j 1, n k ( 1) ¡ Æ . j k ¡ j k Æ 0, n k Æ 6Æ We finally want to recall the r -Stirling transform which will be useful in the next sections: n ½ ¾ n · ¸ X n X n a b (n 0) if and only if b ( 1)n k a (n 0).
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