A Note on Some Recent Results for the Bernoulli Numbers of the Second Kind

Iaroslav V. Blagouchine SeaTech University of Toulon Avenue de l’Université 83957 La Garde France

           iaroslav.blagouchine@univ-tln.fr
       iaroslav.blagouchine@centrale-marseille.fr

and Steklov Institute of Mathematics at St. Petersburg 27 Fontanka 191023 St. Petersburg Russia

            iaroslav.blagouchine@pdmi.ras.ru

Abstract

In a recent issue of the Bulletin of the Korean Mathematical Society, Qi and Zhang discovered an interesting integral representation for the Bernoulli numbers of the second kind (also known as Gregory’s coefficients, Cauchy numbers of the first kind, and the reciprocal logarithmic numbers). The same representation also appears in many other sources, either with no references to its author, or with references to various modern researchers. In this short note, we show that this representation is a rediscovery of an old result obtained in the 19th century by Ernst Schröder. We also demonstrate that the same integral representation may be readily derived by means of complex integration. Moreover, we discovered that the asymptotics of these numbers were also the subject of several rediscoveries, including very recent ones. In particular, the firstorder asymptotics, which are usually (and erroneously) credited to Johan F. Steffensen, actually date back to the mid-19th century, and probably were known even earlier.

1 Rediscovery of Schröder’s integral formula

In a recent article in the Bulletin of the Korean Mathematical Society [10], several results concerning the Bernoulli numbers of the second kind were presented.

We recall that these numbers (OEIS \underline{A002206} and \underline{A002207}), which we denote below by G_{n}, are rational

\begin{matrix}G_{1} = +\frac{1}{2} , & G_{2} = −\frac{1}{12} , & G_{3} = +\frac{1}{24} , & G_{4} = −\frac{19}{720} , \\ G_{5} = +\frac{3}{160} , & G_{6} = −\frac{863}{60480} , & G_{7} = +\frac{275}{24192} , & G_{8} = −\frac{33953}{3628800} , … \end{matrix}

and were introduced by the Scottish mathematician and astronomer James Gregory in 1670 in the context of area’s interpolation formula. Subsequently, they were rediscovered by many famous mathematicians, including Gregorio Fontana, Lorenzo Mascheroni, Pierre-Simon Laplace, Augustin-Louis Cauchy, Jacques Binet, Ernst Schröder, Oskar Schlömilch, Charles Hermite and many others. Because of numerous rediscoveries these numbers do not have a standard name, and in the literature they are also referred to as Gregory’s coefficients, (reciprocal) logarithmic numbers, Bernoulli numbers of the second kind, normalized generalized Bernoulli numbers B_{n}^{(n−1)} and normalized Cauchy numbers of the first kind C_{1,n}. Usually, these numbers are defined either via their generating function

\frac{u}{ln(1 + u)} = 1 + ∑_{n=1}^{∞} G_{n} u^{n},  |u| < 1 ,  (1)

or explicitly

G_{n} = \frac{C_{1,n}}{n!} = lim_{s→n} \frac{−B_{s}^{(s−1)}}{(s − 1) s!} = \frac{1}{n!}∫_{0}^{1}x (x − 1) (x − 2) ⋯ (x − n + 1) dx , n ∈ ℕ .

It is well known that G_{n} are alternating G_{n} = (−1)^{n−1}|G_{n}| and decreasing in absolute value; they behave as (n ln^{2} n)^{−1} at n → ∞ and may be bounded from below and from above accordingly to formulas (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.^{1} It states: the Bernoulli numbers of the second kind may be represented as follows

G_{n} = (−1)^{n+1}∫_{1}^{∞}\frac{dt}{( ln^{2}(t − 1) + π^{2}) t^{n}} ,  n ∈ ℕ .  (2)

1Our G_{n} are exactly b_{n} from [10] and \frac{c_{n,1}^{(1)}}{n!} from [4, Sect. 5]. Despite a venerable history, these numbers still lack a standard notation and various authors may use different notation for them.

Figure 1: A fragment of p. 112 from Schröder’s paper [11]. Schröder’s C_{n}^{(−1)} are exactly our G_{n}.

The same representation appears in a slightly different form^{2}

G_{n} = (−1)^{n+1}∫_{0}^{∞}\frac{du}{( ln^{2} u + π^{2}) (u + 1)^{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, either with 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 and Zhang; in fact, this representation is a rediscovery of an old result. In a little-known paper of the German mathematician Ernst Schröder [11], written in 1879, one may easily find exactly the same integral representation on p. 112; see Fig. 1. Moreover, since this result is not difficult to obtain, it is possible that the same integral representation was obtained even earlier.

2 Simple derivation of Schröder’s integral formula by means of the complex integration

Schröder’s integral formula [11, p. 112] may, of course, be derived in various ways. Below, we propose 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

\frac{z + 1}{ln z − πi} = −1 + ∑_{n=1}^{∞} |G_{n}| (z + 1)^{n}, |z + 1| < 1 .

Now considering the following line integral along a contour C (see Fig. 2), where n ∈ ℕ, and Figure 2: Integration contour C (r and R are radii of the small and big circles respectively, where r ≪ 1 and R ≫ 1).

2Put t = 1 + u.

then letting R → ∞, r → 0, we have by the residue theorem

∳_{C} \frac{dz}{(1 + z)^{n} (ln z − πi)} = ∫_{r}^{R}… dz + ∫_{C_{R}} … dz + ∫_{R}^{r}… dz + ∫_{C_{r}} … dz =^{\begin{matrix}R→∞ \\ r→0 \end{matrix}}

= ∫_{0}^{∞}\{\frac{1}{ln x − πi} − \frac{1}{ln x + πi}\} · \frac{dx}{(1 + x)^{n}} = 2πi∫_{0}^{∞}\frac{1}{(1 + x)^{n}} · \frac{dx}{ln^{2} x + π^{2}} =

= 2πi res_{z=−1} \frac{1}{(1 + z)^{n} (ln z − πi)} = \frac{2πi}{n!} · \frac{d^{n}}{dz^{n}} \frac{z + 1}{ln z − πi} |_{z=−1}= 2πi |G_{n}| ,

since

| ∫_{C_{R}} \frac{dz}{(1 + z)^{n} (ln z − πi)} | = O (\frac{1}{R^{n−1} ln R}) = o(1) ,  R → ∞ ,  n ≥ 1 ,

| ∫_{C_{r}} \frac{dz}{(1 + z)^{n} (ln z − πi)} | = O (\frac{r}{ln r}) = o(1) ,  r → 0 ,

and because at z = −1 the integrand of the contour integral has a pole of the (n + 1)th order. This completes the proof. Note that above derivations are valid only for n ≥ 1, and so is Schröder’s integral formula, which may also be regarded as one of the generalizations of G_{n} to the continuous values of n.

Figure 3: A fragment of p. 115 from Schröder’s paper [11].

3 Several remarks on the asymptotics for the Bernoulli numbers of the second kind

The first-order asymptotics |G_{n}| ∼ (n ln^{2} n)^{−1} at n → ∞ are usually credited to Johan F. Steffensen [12, pp. 2–4], [13, pp. 106–107], [9, p. 29], [7, p. 14, Eq. (14)], [8], who found it in 1924.^{3} However, in our recent work [3, p. 415] we noted that exactly the same result appears in Schröder’s work written 45 years earlier, see Fig. 3, and the order of the magnitude of the closely related numbers is contained in a work of Jacques Binet dating back to 1839 [1].^{4} In 1957 Davis [7, p. 14, Eq. (14)] improved this first-order approximation slightly by showing that |G_{n}| ∼ Γ(1 + ξ)(n ln^{2} n + nπ^{2})^{−1} at n → ∞ for some ξ ∈ [0, 1] , without noticing that 7 years earlier S. C. Van Veen had already obtained the complete asymptotics for them [14, p. 336], [9, p. 29]. The identical complete asymptotics were recently rediscovered in a slightly different form by Charles Knessl [5, p. 473], and later by Gergő Nemes [8]. An alternative demonstration of the same result was also presented by the author [3, p. 414].

4 Acknowledgments

The author is grateful to Jacqueline Lorfanfant from the University of Strasbourg for sending a scanned version of [11].

References

[1] J. Binet, Mémoire sur les intégrales définies eulériennes et sur leur application à la théorie des suites, ainsi qu’à l’évaluation des fonctions des grands nombres, J. Éc. Roy.

3The same first-order asymptotics also appear in [6, p. 294], but without the source of the formula.

4By the “closely related numbers” we mean the so-called Cauchy numbers of the second kind (OEIS \underline{A002657} and \underline{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 statements from [1, pp. 231, 339]. The Cauchy numbers of the second kind C_{2,n} and Gregory’s coefficients G_{n} are related to each other via the relationship nC_{2,n−1} − C_{2,n} = n! |G_{n}|, see [3, p. 430].

Polytech., 16 (27) (1839), 123–343.

[2] Ia. V. Blagouchine, Expansions of generalized Euler’s constants into the series of polynomials in π^{−2} and into the formal enveloping series with rational coefficients only, J. Number Theory, 158 (2016), 365–396. Corrigendum: 173 (2017), 631–632.

[3] Ia. V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to π^{−1}, J. Math. Anal. Appl., 442 (2016), 404–434.

[4] J. Chikhi, Integral representation for some generalized Poly-Cauchy numbers, preprint,

2016, https://hal.archives-ouvertes.fr/hal-01370757v1.

[5] M. W. Coffey, Series representations for the Stieltjes constants, Rocky Mountain J. Math., 44 (2014), 443–477.

[6] L. Comtet, Advanced Combinatorics. The Art of Finite and Infinite Expansions (revised and enlarged edition), D. Reidel Publishing Company, 1974.

[7] H. T. Davis, The approximation of logarithmic numbers, Amer. Math. Monthly, 64 (1957), 11–18.

[8] G. Nemes, An asymptotic expansion for the Bernoulli numbers of the second kind, J. Integer Seq., 14 (2011), Article 11.4.8.

[9] N. E. Nörlund, Sur les valeurs asymptotiques des nombres et des polynômes de Bernoulli, Rend. Circ. Mat. Palermo, 10 (1) (1961), 27–44.

[10] F. Qi and X.-J. Zhang, An integral representation, some inequalities, and complete monotonicity of Bernoulli numbers of the second kind, Bull. Korean Math. Soc., 52 (3) (2015), 987–998.

[11] E. Schröder, Bestimmung des infinitären Werthes des Integrals ∫_{0}^{1}(u)_{n} du, Z. Math. Phys.,

25 (1880), 106–117.

[12] J. F. Steffensen, On Laplace’s and Gauss’ summation–formulas, Skandinavisk Aktuarietidskrift (Scand. Actuar. J.), 1 (1924), 1–15.

[13] J. F. Steffensen, Interpolation, 2nd edition, Chelsea Publishing, 1950.

[14] S. C. Van Veen, Asymptotic expansion of the generalized Bernoulli numbers B_{n}^{(n−1)} for large values of n (n integer), Indag. Math., 13 (1951), 335–341.

2010 Mathematics Subject Classification: Primary 11B68; Secondary 11B83, 30E20, 11-03, 05A10, 01A55.

Keywords: Bernoulli number of the second kind, Gregory coefficient, Cauchy number, logarithmic number, Schröder, rediscovery, state of art, complex analysis, theory of complex variable, contour integration, residue theorem.

(Concerned with sequences \underline{A002206}, \underline{A002207}, \underline{A002657}, \underline{A002790}, \underline{A195189}, \underline{A075266}, and \underline{A262235}.)

Received December 20 2016; revised versions received January 24 2017; January 25 2017; January 26 2017. Published in Journal of Integer Sequences, January 27 2017.

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