A Note on Some Recent Results for the Bernoulli Numbers of the Second Kind
aa r X i v : . [ m a t h . N T ] J a n A Note on Some Recent Results for the Bernoulli Numbersof the Second Kind
Iaroslav V. Blagouchine
University of Toulon, France&Steklov Institute of Mathematics at St.-Petersburg, Russia.
Abstract
In a recent issue of the
Bulletin of the Korean Mathematical Society , Qi and Zhang discovered an inter-esting integral representation for the Bernoulli numbers of the second kind (also known as
Gregory’scoefficients , Cauchy numbers of the first kind , and the reciprocal logarithmic numbers ). The same represen-tation also appears in many other sources, either with no references to its author, or with references tovarious modern researchers. In this short note, we show that this representation is a rediscovery of anold result obtained in the 19th century by Ernst Schr ¨oder. We also demonstrate that the same integralrepresentation may be readily derived by means of complex integration. Moreover, we discoveredthat the asymptotics of these numbers were also the subject of several rediscoveries, including veryrecent ones. In particular, the first-order asymptotics, which are usually (and erroneously) creditedto Johan F. Steffensen, actually date back to the mid-19th century, and probably were known evenearlier.
Keywords:
Bernoulli number of the second kind, Gregory coefficient, Cauchy number, logarithmicnumber, Schr ¨oder, rediscovery, state of art, complex analysis, theory of complex variable, contourintegration, residue theorem.
I. Rediscovery of Schr ¨oder’s integral formula
In a recent article in the
Bulletin of the Korean Mathematical Society [10], several results concerningthe Bernoulli numbers of the second kind were presented.We recall that these numbers (OEIS A002206 and A002207), which we denote below by G n , are Email address: [email protected], [email protected],[email protected]. (Iaroslav V. Blagouchine)
Note to the readers of the 3rd arXiv version: this version is a copy of the journal version of thearticle, which has been published in the Journal of Integer Sequences, vol. 20, no. 3, Article 17.3.8,pp. 1-7, 2017. URL: https://cs.uwaterloo.ca/journals/JIS/VOL20/Blagouchine/blag5.htmlArtcile history: submitted 20 December 2016, accepted 26 January 2017, published 27 January 2017.The layout of the present version and its page numbering differ from the journal version, but thecontent, the numbering of equations and the numbering of references are the same. For any furtherreference to the material published here, please, use the journal version of the paper, which you canalways get for free from the journal site (Journal of Integer Sequences is an open access journal). igure 1: A fragment of p. 112 from Schr ¨oder’s paper [11]. Schr ¨oder’s C ( − ) n are exactly our G n . rational G = +
12 , G = −
112 , G = +
124 , G = − G = + G = − G = + G = − Gregory’s coefficients , (reciprocal) logarithmic numbers , Bernoulli numbers of the secondkind , normalized generalized Bernoulli numbers B ( n − ) n and normalized Cauchy numbers of the first kindC n . Usually, these numbers are defined either via their generating function u ln ( + u ) = + ∞ ∑ n = G n u n , | u | < G n = C n n ! = lim s → n − B ( s − ) s ( s − ) s ! = n ! ˆ x ( x − ) ( x − ) · · · ( x − n + ) dx , n ∈ N .It is well known that G n are alternating G n = ( − ) n − | G n | and decreasing in absolute value; theybehave as (cid:0) n ln n (cid:1) − at n → ∞ and may be bounded from below and from above accordingly toformulas (55)–(56) from [3]. For more information about these important numbers, see [3, pp. 410–415], [2, p. 379], and the literature given therein (nearly 50 references).Now, the first main result of [10, p. 987] is Theorem 1. It states: the Bernoulli numbers of the second Our G n are exactly b n from [10] and c ( ) n ,1 n ! from [4, Sect. 5]. Despite a venerable history, these numbers still lack a standardnotation and various authors may use different notation for them. ind may be represented as followsG n = ( − ) n + ∞ ˆ dt (cid:0) ln ( t − ) + π (cid:1) t n , n ∈ N . (2)The same representation appears in a slightly different form G n = ( − ) n + ∞ ˆ du (cid:0) ln u + π (cid:1) ( u + ) n , n ∈ N , (3)in [5, pp. 473–474] and [4, Sect. 5], and is called Knessl’s representation and the Qi integral representation respectively. Furthermore, various internet sources provide the same (or equivalent) formula, eitherwith no references to its author or with references to different modern writers and/or their papers.However, the integral representation in question is not novel and is not due to Knessl nor to Qi andZhang; in fact, this representation is a rediscovery of an old result. In a little-known paper of theGerman mathematician Ernst Schr ¨oder [11], written in 1879, one may easily find exactly the sameintegral representation on p. 112; see Fig. 1. Moreover, since this result is not difficult to obtain, it ispossible that the same integral representation was obtained even earlier.
II. Simple derivation of Schr ¨oder’s integral formula by means of the complex integration
Schr ¨oder’s integral formula [11, p. 112] may, of course, be derived in various ways. Below, wepropose a simple derivation of this formula based on the method of contour integration.If we set u = − z −
1, then equality (1) may be written as z + z − π i = − + ∞ ∑ n = (cid:12)(cid:12) G n (cid:12)(cid:12) ( z + ) n , | z + | < C (see Fig. 2), where n ∈ N , and thenletting R → ∞ , r →
0, we have by the residue theorem ‰ C dz ( + z ) n ( ln z − π i ) = R ˆ r . . . dz + ˆ C R . . . dz + r ˆ R . . . dz + ˆ C r . . . dz R → ∞ r → == ∞ ˆ (cid:26) x − π i − x + π i (cid:27) · dx ( + x ) n = π i ∞ ˆ ( + x ) n · dx ln x + π == π i res z = − ( + z ) n ( ln z − π i ) = π in ! · d n dz n z + z − π i (cid:12)(cid:12)(cid:12)(cid:12) z = − = π i (cid:12)(cid:12) G n (cid:12)(cid:12) , Put t = + u . igure 2: Integration contour C ( r and R are radii of the small and big circles respectively, where r ≪ R ≫ since (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ C R dz ( + z ) n ( ln z − π i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O (cid:18) R n − ln R (cid:19) = o ( ) , R → ∞ , n ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ C r dz ( + z ) n ( ln z − π i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O (cid:16) r ln r (cid:17) = o ( ) , r → z = − ( n + ) th order. Thiscompletes the proof. Note that above derivations are valid only for n ≥
1, and so is Schr ¨oder’s integralformula, which may also be regarded as one of the generalizations of G n to the continuous values of n . III. Several remarks on the asymptotics for the Bernoulli numbers of the second kind
The first-order asymptotics | G n | ∼ (cid:0) n ln n (cid:1) − at n → ∞ are usually credited to Johan F. Stef-fensen [12, pp. 2–4], [13, pp. 106–107], [9, p. 29], [7, p. 14, Eq. (14)], [8], who found it in 1924. How-ever, in our recent work [3, p. 415] we noted that exactly the same result appears in Schr ¨oder’s workwritten 45 years earlier, see Fig. 3, and the order of the magnitude of the closely related numbers iscontained in a work of Jacques Binet dating back to 1839 [1]. In 1957 Davis [7, p. 14, Eq. (14)] im-proved this first-order approximation slightly by showing that | G n | ∼ Γ ( + ξ ) (cid:0) n ln n + n π (cid:1) − at n → ∞ for some ξ ∈ [
0, 1 ] , without noticing that 7 years earlier S. C. Van Veen had already obtainedthe complete asymptotics for them [14, p. 336], [9, p. 29]. Equivalent complete asymptotics were re- The same first-order asymptotics also appear in [6, p. 294], but without the source of the formula. By the “closely related numbers” we mean the so-called
Cauchy numbers of the second kind (OEIS A002657 and A002790),and numbers I ′ ( k ) , see [3, pp. 410–415, 428–429]. The comment going just after Eq. (93) [3, p. 429] is based on the statementsfrom [1, pp. 231, 339]. The Cauchy numbers of the second kind C n and Gregory’s coefficients G n are related to each other viathe relationship nC n − − C n = n ! | G n | , see [3, p. 430]. igure 3: A fragment of p. 115 from Schr ¨oder’s paper [11]. cently rediscovered in slightly different forms by Charles Knessl [5, p. 473], and later by Gerg˝o Nemes[8]. An alternative demonstration of the same result was also presented by the author [3, p. 414]. IV. Acknowledgments
The author is grateful to Jacqueline Lorfanfant from the University of Strasbourg for sending ascanned version of [11].
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