An Asymptotic Expansion for the Bernoulli Numbers of the Second Kind

Gergő Nemes Loránd Eötvös University Pázmány Péter sétány 1/C H-1117 Budapest Hungary

                      nemesgery@gmail.com

Abstract In this paper we derive a complete asymptotic series for the Bernoulli numbers of

the second kind, and provide a recurrence relation for the coefficients.

1 Introduction

The Bernoulli numbers of the second kind b_{n} (also known as the Cauchy numbers, Gregory coefficients or logarithmic numbers) are defined by the generating function

\frac{x}{log (1 + x)} = ∑_{n≥0} b_{n}x^{n}.

The first few are b_{0} = 1, b_{1} = 1/2, b_{2} = −1/12, b_{3} = 1/24, b_{4} = −19/720. The numerators and denominators are given by Sloane’s sequences \underline{A002206} and \underline{A002207}. They are related to the generalized Bernoulli numbers [3, p. 596] by

b_{n} = −\frac{B_{n}^{(n−1)}}{(n − 1) n!}.

The B_{n}^{(t)}’s are the coefficients of the exponential generating function x^{t}/ (e^{x} − 1)^{t}. It is known [1] that

b_{n} = ∫_{0}^{1} \frac{t (t − 1) ⋯ (t − n + 1)}{n!}dt,

which can be shown as follows:

\frac{x}{log (1 + x)} = ∫_{0}^{1} exp (t log (1 + x)) dt = ∫_{0}^{1} (1 + x)^{t} dt = ∫_{0}^{1} ∑_{n≥0} \binom{t}{n}x^{n}dt

= ∑_{n≥0} (∫_{0}^{1} \binom{t}{n}dt) x^{n} = ∑_{n≥0} (∫_{0}^{1} \frac{t (t − 1) ⋯ (t − n + 1)}{n!}dt) x^{n}.

Using the s (n, k) Stirling numbers of the first kind (Sloane’s \underline{A048994}) defined by

t (t − 1) ⋯ (t − n + 1) = ∑_{k=0}^{n} s (n, k) t^{k},

we immediately obtain the representation

b_{n} = \frac{1}{n!} ∑_{k=0}^{n} \frac{s (n, k)}{k + 1}.

The investigation of the asymptotic behavior of these numbers was begun by Steffensen [4], who proved that

b_{n} ∼ \frac{(−1)^{n+1}}{n log^{2} n} =: a_{n}

as n → +∞. However, the ratio b_{n}/a_{n} converges very slowly toward 1, as was pointed out by Davis [1], who derived better approximations including the following:

b_{n} ≈ \frac{(−1)^{n+1} Γ (ξ_{n} + 1)}{n (log^{2} n + π^{2})},

where 0 < ξ_{n} < 1 and Γ is the gamma function. The aim of this paper is to extend Steffensen’s asymptotic approximation into a complete asymptotic expansion in terms of 1/ log n.

2 The asymptotic expansion

Theorem 1. The Bernoulli numbers of the second kind b_{n} have an asymptotic expansion of the form

b_{n} ∼ \frac{(−1)^{n+1}}{n log^{2} n} ∑_{k≥0} \frac{β_{k}}{log^{k} n}  (1)

as n → +∞, where

β_{k} = (−1)^{k} [\frac{d^{k+1}}{ds^{k+1}} (\frac{1}{Γ (s)})]_{s=0} .  (2)

Note that the main term of this asymptotic series is just Steffensen’s approximation. Computing the first few coefficients β_{k}, our expansion takes the form

b_{n} ∼ \frac{(−1)^{n+1}}{n log^{2} n} (1 − \frac{2γ}{log n} − \frac{π^{2} − 6γ^{2}}{2 log^{2} n} + \frac{2π^{2}γ − 4γ^{3} − 8ζ (3)}{log^{3} n} + ⋯) ,

where γ is the Euler-Mascheroni constant and ζ is the Riemann zeta function. In our proof we will use the following special case of Watson’s lemma.

Lemma 2. Let g (s) be a function of the positive real variable s, such that

g (s) = ∑_{k≥1} g_{k}s^{k}

as s → 0+. Then for each nonnegative integer N

∫_{0}^{+∞} g (s) e^{−ms}ds = ∑_{k=0}^{N−1} \frac{(k + 1)!g_{k+1}}{m^{k+2}} + O (\frac{1}{m^{N+2}} )

as m → +∞, provided that this integral converges throughout its range for all sufficiently large m.

For a more general version and proof see, e.g., Olver [2, p. 71] or Wong [5, p. 20]. We will also need sharp bounds for the ratio of two gamma functions.

Lemma 3. For n > 2 and 0 ≤ s ≤ 1 we have

\frac{1}{n} \frac{1}{n^{s}} ≤ \frac{Γ (n − s)}{Γ (n + 1)} ≤ \frac{1}{n} \frac{1}{n^{s}} + \frac{2}{n^{2}} \frac{s}{n^{s}} .

Proof of Lemma 3. Fix n > 1 and let

f_{1} (s) = − (s + 1) log n,
f_{2} (s) = log Γ (n − s) − log Γ (n + 1)

for 0 ≤ s ≤ 1. The function f_{1} is affine while the function f_{2} is convex (since log Γ is convex). Furthermore, f′_{1} (0) = − log n, f′_{2} (0) = −ψ (n), where ψ := Γ′/Γ is the Digamma function. From the simple inequality ψ (n) < log n we see that f′_{1} (0) < f′_{2} (0), hence

\frac{1}{n} \frac{1}{n^{s}} ≤ \frac{Γ (n − s)}{Γ (n + 1)}

holds for n > 1 and 0 ≤ s ≤ 1. To prove the upper bound, we first show that

\frac{Γ (n + a)}{Γ (n + 1)} ≤ \frac{1}{n^{1−a}}  (3)

for n ≥ 1 and 0 ≤ a ≤ 1. Fix n ≥ 1 and let

g_{1} (a) = log Γ (n + a) − log Γ (n + 1) ,
g_{2} (a) = (a − 1) log n

for 0 ≤ a ≤ 1. The function g_{1} is convex while the function g_{2} is affine. Since g_{1} (0) = g_{2} (0) = − log n and g_{1} (1) = g_{2} (1) = 0, the inequality (3) holds. From this it follows that for n > 2 and 0 ≤ s ≤ 1

\frac{Γ (n − s)}{Γ (n + 1)} = \frac{Γ (n + (1 − s))}{(n − s) Γ (n + 1)} ≤ \frac{1}{n} \frac{1}{n^{s} (1 − \frac{s}{n})} ≤ \frac{1}{n} \frac{1}{n^{s}} + \frac{2}{n^{2}} \frac{s}{n^{s}} .

Proof of Theorem 1. As shown by Steffensen,

b_{n} = \frac{(−1)^{n+1}}{π} ∫_{0}^{1} Γ (s + 1) sin (πs) \frac{Γ (n − s)}{Γ (n + 1)}ds.

By Lemma 3 we find that

0 ≤ ∫_{0}^{1} Γ (s + 1) sin (πs) \frac{Γ (n − s)}{Γ (n + 1)}ds − \frac{1}{n} ∫_{0}^{1} Γ (s + 1) sin (πs) e^{−ms}ds

≤ \frac{2}{n^{2}} ∫_{0}^{1} sΓ (s + 1) sin (πs) e^{−ms}ds

where m := log n. Hence, we conclude that

|b_{n} − \frac{(−1)^{n+1}}{πn} ∫_{0}^{1} Γ (s + 1) sin (πs) e^{−ms}ds| ≤ \frac{2}{πn^{2}} ∫_{0}^{1} sΓ (s + 1) sin (πs) e^{−ms}ds

< \frac{2}{πn^{2}} ∫_{0}^{+∞} se^{−ms}ds = \frac{2}{πn^{2} log^{2} n}.

Thus, we derived the asymptotic formula

b_{n} = \frac{(−1)^{n+1}}{πn} ∫_{0}^{1} Γ (s + 1) sin (πs) e^{−ms}ds + O (\frac{1}{n^{2} log^{2} n})

= \frac{(−1)^{n+1}}{n} ∫_{0}^{1} \frac{s}{Γ (1 − s)}e^{−ms}ds + O (\frac{1}{n^{2} log^{2} n})

as n → +∞. Here we used the reflection formula Γ (s + 1) sin (πs) = πs/Γ (1 − s). The function s/Γ (1 − s) is analytic in the range 0 < s < 1 (in fact, it is an entire function), let

\frac{s}{Γ (1 − s)} = ∑_{k≥1} γ_{k}s^{k}.

Define the function Δ (s) in the positive real variable s by

Δ (s) := \begin{cases}s/Γ (1 − s) , & if 0 < s < 1; \\ 0, & if s ≥ 1. \end{cases}

Then our asymptotic formula becomes

b_{n} = \frac{(−1)^{n+1}}{n} ∫_{0}^{+∞} Δ (s) e^{−ms}ds + O (\frac{1}{n^{2} log^{2} n}) .

The integral satisfies the conditions of Watson’s Lemma and we obtain that for each nonnegative integer N

b_{n} = \frac{(−1)^{n+1}}{n} (∑_{k=0}^{N−1} \frac{(k + 1)!γ_{k+1}}{log^{k+2} n} + O (\frac{1}{log^{N+2} n})) + O (\frac{1}{n^{2} log^{2} n})

= \frac{(−1)^{n+1}}{n log^{2} n} (∑_{k=0}^{N−1} \frac{β_{k}}{log^{k} n} + O (\frac{1}{log^{N} n}) + O (\frac{1}{n}))

as n → +∞, where

β_{k} := (k + 1)!γ_{k+1} = [\frac{d^{k+1}}{ds^{k+1}} (\frac{s}{Γ (1 − s)})]_{s=0}

= [\frac{d^{k+1}}{ds^{k+1}} (−\frac{1}{Γ (−s)})]_{s=0}

= (−1)^{k} [\frac{d^{k+1}}{ds^{k+1}} (\frac{1}{Γ (s)})]_{s=0} .

Since for every N ≥ 0

\frac{1}{n} = o (\frac{1}{log^{N} n})

as n → +∞, we have proved the theorem.

3 Recurrence for the coefficients β_{k}

Here we derive a recurrence formula for the coefficients β_{k} in the asymptotic expansion (1). Since the reciprocal of the Gamma function is an entire function, we can write it as a power series around 0, say

\frac{1}{Γ (s)} = ∑_{k≥1} α_{k}s^{k}.

According to the formula for the Taylor coefficients and equation (2), we have

α_{k} = \frac{1}{k!} [\frac{d^{k}}{ds^{k}} (\frac{1}{Γ (s)})]_{s=0} = \frac{(−1)^{k−1} β_{k−1}}{k!} .  (4)

It is known that α_{1} = 1, α_{2} = γ and

kα_{k+1} = γα_{k} − ∑_{j=2}^{k} (−1)^{j} ζ (j) α_{k−j+1}

for k ≥ 2 (cf. [3, p. 139]). This can be seen as follows. The Digamma function has the power series

ψ (s + 1) = −γ + ∑_{k≥2} (−1)^{k} ζ (k) s^{k−1}

(see, e.g., [3, p. 139]) and differentiating

\frac{1}{Γ (s + 1)} = \frac{1}{sΓ (s)} = ∑_{k≥1} α_{k}s^{k−1}

we find the power series for

−\frac{Γ′ (s + 1)}{Γ^{2} (s + 1)} = −\frac{ψ (s + 1)}{Γ (s + 1)},

but this power series can be obtained by Cauchy multiplication of the two previous ones. In this way we get the recursion formula for the α_{k}’s. From this recursion formula and (4) it follows that β_{0} = 1, β_{1} = −2γ and

kβ_{k} = −γ (k + 1) β_{k−1} − ∑_{j=2}^{k} \binom{k + 1}{j}j!ζ (j) β_{k−j}

for k ≥ 2.

4 Acknowledgement

I would like to thank the anonymous referee for his/her thorough, constructive and helpful comments and suggestions on the manuscript.

References

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

[2] F. W. J. Olver, Asymptotics and Special Functions. A. K. Peters. Reprint, with corrections, of original Academic Press edition, 1974.

[3] F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark (eds.), NIST Handbook of Mathematical Functions. Cambridge University Press, New York, 2010.

[4] J. F. Steffensen, On Laplace’s and Gauss’ summation-formulas, Skandinavisk Aktuariettidskrift (1924), 2–4.

[5] R. Wong, Asymptotic Approximations of Integrals. Academic Press. Reprinted with corrections by SIAM, Philadelphia, PA, 2001.

2010 Mathematics Subject Classification: Primary 11B83; Secondary 41A60. Keywords: Bernoulli numbers of the second kind, asymptotic expansions, gamma function.

(Concerned with sequences \underline{A002206}, \underline{A002207}, and \underline{A048994}.)

Received January 7 2011; revised version received March 27 2011. Published in Journal of Integer Sequences, April 15 2011.

Return to Journal of Integer Sequences home page.