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New Series Involving Harmonic Numbers and Squared Central Binomial Coefficients John Campbell

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New Series Involving Harmonic Numbers and Squared Central Binomial Coefficients John Campbell New series involving harmonic numbers and squared central binomial coefficients John Campbell To cite this version: John Campbell. New series involving harmonic numbers and squared central binomial coefficients. 2018. hal-01774708 HAL Id: hal-01774708 https://hal.archives-ouvertes.fr/hal-01774708 Preprint submitted on 23 Apr 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. NEW SERIES INVOLVING HARMONIC NUMBERS AND SQUARED CENTRAL BINOMIAL COEFFICIENTS JOHN MAXWELL CAMPBELL ABSTRACT. Recently, there have been a variety of in- triguing discoveries regarding the symbolic computation of series containing central binomial coefficients and harmonic- type numbers. In this article, we present a vast generaliza- tion of the recently-discovered harmonic summation formula 1 2 2 1 X 2n Hn Γ 4 ln(2) = p4 1 − n 32n 4 π π n=1 through creative applications of an integration method that we had previously introduced and applied to prove new 1 Ramanujan-like formulas for π . We provide explicit closed- form expressions for natural variants of the above series that cannot be evaluated by state-of-the-art computer algebra systems, such as the elegant symbolic evaluation 1 2n2 1 2 p X Hn 2Γ 4π3=2 + 16 π ln(2) n = 8 − 4 − 32n(n + 1) π3=2 1 2 n=1 Γ 4 introduced in our present paper. We also discuss some re- lated problems concerning binomial series containing alter- nating harmonic numbers. We also introduce a new class of 1 harmonic summations for Catalan's constant G and π such as the series 1 2n2 X Hn 32G − 64 ln(2) n = 16 + − 16 ln(2) 16n(n + 1)2 π n=1 which we prove through a variation of our previous integra- 1 tion method for constructing π series. 1991 AMS Mathematics subject classification. 33E20, 33B15. Keywords and phrases. Harmonic number, Central binomial coefficient, Infinite series, Symbolic computation, Gamma function. Received by the editors April 23, 2018. 1 2 JOHN MAXWELL CAMPBELL 1. Introduction. In [6], it was noted that the elegant Ramanujan- like formula 1 2n2 X Hn 12 − 16 ln(2) (1.1) n = 16n(2n − 1)2 π n=1 may be proven through an application of a differential operator with respect to a parameter involved in a known hypergeometric identity; this technique is also used in [7] to prove (1.1). A similar strategy was also applied to prove an equivalent formulation of the known equation 1 2n2 X Hn 16 ln(2) (1.2) n = 4 − (n + 1)16n π n=1 in 2017 in [11]. Both (1.1) and (1.2) are special cases of a very useful result introduced in [4] on the evaluation of series containing factors 2n2 of the form n Hn for indices n 2 N, which was proven through the use of a beta-like integral transform. While applying parameter derivatives to classical hypergeometric identities only produces specific results on harmonic summations containing squared central binomial coefficients as a factor in the summand, the integration technique presented in [4] may be applied much more generally. The power of the integration strategy considered in [4] motivates the exploration of further applications of the integral transform given in [4], as well as investigations on applications of analogues and variants of this integral transform. On this latter note, a new integral transform Tln;arcsin given in [5] in 1 2017 was used to evaluate new series for π containing factors of the form 2n2 0 0 1 1 n H2n in [5], letting H2n = 1 − 2 + · · · − 2n denote the alternating harmonic number of order 2n for n 2 N. The operator Tln;arcsin, which is analogous to the integral transform introduced in [4], is used in [5] to prove the elegant formula 1 2n2 4G − 12 ln 2 + 6 X H2n (1.3) = n π 16n(2n − 1)2 n=1 for Catalan's constant G, which is also proven through the use of the 0 generating function for (CnH2n : n 2 N0) that was evaluated in [3], th letting Cn denote the n entry in the Catalan sequence. As discussed 2n2 NEW SERIES INVOLVING n Hn 3 in [4], it is not in general feasible to use known generating functions for 2n sequences such as n Hn : n 2 N0 to evaluate series with summands 2n2 containing expressions of the form n Hn. In general, the evaluation of infinite series involving products of entries of the harmonic sequence 2n2 and expressions such as n is difficult, and often leads to surprising and elegant results in the field of classical analysis. The problem of determining explicit symbolic evaluations for sum- mations containing entries of harmonic-type sequences and central bi- nomial coefficients is a deep and interesting subject that has been ex- plored through the use of many different kinds of classical analysis- based techniques. In [9], a variety of infinite summations involving gen- eralized harmonic numbers and central binomial coefficients are evalu- ated through the use of beta-like integrals. In [8], a more abstract way of \depicting" harmonic-like numbers is used, writing 1 1 1 H = σ 1; ; ;:::; n 1 2 3 n and H2 − H(2) 1 1 1 n n = σ 1; ; ;:::; ; 2 2 2 3 n letting X σm (x1; x2; : : : ; xn) = xk1 xk2 ··· xkm 1≤k1<k2<···<km≤n denote the elementary symmetric function of order m. The authors in [8] mainly explore the symbolic evaluation of infinite series in- volving central binomial coefficients as well as expressions such as 1 1 1 1 σm 1; 32 ;:::; (2n−1)2 and σm 1; 22 ;:::; (n−1)2 . The results put forth in [8] are nicely representative of how mathematical problems concerning the symbolic computation of series involving harmonic-like numbers can be closely connected with seemingly unrelated subjects in the theory of symmetric functions and in number theory. The series expansions for powers of the inverse sine mapping proven in [2] involve central binomial coefficients and \nested" harmonic-type 4 JOHN MAXWELL CAMPBELL multisums, including k−1 n −1 nN−1−1 X 1 X1 1 X 1 2 2 ··· 2 : (2n1) (2n2) (2nN ) n1=1 n2=1 nN =1 Motivated in part by the main results from [2], and in particular the classical infinite series identity 1 (2) 2 z 4 X H arcsin = n−1 z2n 3 2 22n n=1 n n known to Ramanujan [1], the authors in [12] determine congruences for p−1 X Hn−1(2) p tn (mod p) d2n n=1 n n for prime numbers p and for special values of d, generalizing congruence results given by Zhi-Wei Sun in [15], in which the elegant formula 1 (2) π3 X 2nH = n−1 48 2n n=1 n n is also introduced. This discussion further illustrates how researching new kinds of subjects concerning summations containing harmonic- like numbers together with central binomial coefficients can lead to unexpected results in both applied analysis and number theory, and surprising connections between these disciplines. In [10], new hypergeometric identities related to Ramanujan-like 1 series for π are proven using WZ-pairs, and an equivalent formulation of the beautiful Ramanujan-type [17] formula 1 n 3 9 2 X 1 2n 2Γ 8 4 ln(2) − (4n + 1)Hn = − 64 n 7 2 5 2 π n=1 3Γ 8 Γ 4 is employed in the derivation of one of the main identities given in [10]. The related formula 1 n 5 X 1 2n 4(5 ln(2) − 2π) − (2 − 5(4n + 1)Hn) = 1024 n 3 4 n=0 3Γ 4 2n2 NEW SERIES INVOLVING n Hn 5 was recently proven in [17] through the use of a parameter derivative applied to a classical 6F5(−1) series identity. These striking results strongly inspire us to explore new techniques for computing series with 2n summands with a factor of the form Hn and fixed powers of n in the numerator. The beautiful formula p 1 3 6 1 X 2n H0 Γ (8 3 ln 2 − 3π) (1.4) 2n = 3 n 28n 24 · 22=3π4 n=1 is proven in [18] through the use of special values of the multi- dimensional integral s Z n X 2πxki (1.5) Wn(s) := e dx; n [0;1] k=1 which, as discussed in [18], is used in the analysis of uniform planar random walks in the case whereby every step is a unit step, with the direction being random. In particular, the definite integral in (1.5) is equal to the sth moment of the distance in a given random walk, measured from the origin of the plane after a total of n 2 N steps are taken. The delightful binomial-harmonic series given in (1.4) is proven through an identity for W3(s), which shows how series with binomial powers and entries in harmonic-like sequences can have direct applications in the theory of random walks, further motivating the exploration of new applications of the main techniques introduced in [4]. The conjectural equality 1 2n2 1 2n2 X 2 X H2n (1.6) n H0 = n n16n 2n 3 (2n + 1)16n n=1 n=0 was introduced in 2014 by Sun, with this conjecture being given alongside a number of conjectural p-adic congruences for truncated sums involving squared central binomial coefficients and harmonic numbers, in [14].
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