An asymptotic series in Ramanujan's second notebook (Entry 10, Chapter 3) is
concerned with the behavior of the expected value of $\phi(X)$ for large
$\lambda$ where $X$ is a Poisson random variable with mean $\lambda$ and $\phi$
is a function satisfying certain growth conditions. We generalize this by
studying the asymptotics of the expected value of $\phi(X)$ when the
distribution of $X$ belongs to a suitable family indexed by a convolution
parameter. Examples include the problem of inverse moments for distribution
families such as the binomial or the negative binomial.Comment: To appear, Ramanujan 

A. Gut

B.C. Berndt

E. Marciniak

E.L. Grab

F.F. Stephan

F.N. David

G.A. Rempala

M. Abramowitz

M. Žnidarič

N.L. Johnson

P. Flajolet

P. Jacquet

R. Willink

R.E. Krichevskiy

R.J. Evans

T. Wuyungaowa and Wang

Yaming Yu

English

arXiv

Springer - Publisher Connector

On an asymptotic series of Ramanujan

An asymptotic series in Ramanujan’s second notebook (Entry&nbsp;10, Chap.&nbsp;3) is concerned with the behavior of the expected value of φ(X) for large λ where X is a Poisson random variable with mean λ and φ is a function satisfying certain growth conditions. We generalize this by studying the asymptotics of the expected value of φ(X) when the distribution of X belongs to a suitable family indexed by a convolution parameter. Examples include the binomial, negative binomial, and gamma families. Some formulas associated with the negative binomial appear new

Yu, Yaming

eScholarship - University of California

Crossref

Ramanujan J (2009) 20: 179–188
DOI 10.1007/s11139-009-9169-x
On an asymptotic series of Ramanujan
Yaming Yu
Received: 22 May 2008 / Accepted: 1 April 2009 / Published online: 12 August 2009
© The Author(s) 2009. This article is published with open access at Springerlink.com
Abstract An asymptotic series in Ramanujan’s second notebook (Entry 10, Chap. 3)
is concerned with the behavior of the expected value of φ(X) for large λ where X is
a Poisson random variable with mean λ and φ is a function satisfying certain growth
conditions. We generalize this by studying the asymptotics of the expected value
of φ(X) when the distribution of X belongs to a suitable family indexed by a con-
volution parameter. Examples include the binomial, negative binomial, and gamma
families. Some formulas associated with the negative binomial appear new.
Keywords Asymptotic expansion · Binomial distribution · Central moments ·
Cumulants · Gamma distribution · Negative binomial distribution · Poisson
distribution · Ramanujan’s notebooks
Mathematics Subject Classification (2000) Primary 34E05 · Secondary 60E05
1 An asymptotic series of Ramanujan
A version (modified from [2]) of Entry 10 in Chap. 3 of Ramanujan’s second note-
book reads
Theorem 1 Let φ(x), x ∈ [0,∞), denote a function of at most polynomial growth as
x tends to ∞. Suppose there exist constants x0 > 0 and A ≥ 1, and a function G(x)
of at most polynomial growth as x → ∞ such that for each nonnegative integer m
and all x > x0, the derivatives φ(m)(x) exist and satisfy
∣
∣
∣
∣
φ(m)(x)
m!
∣
∣
∣
∣
≤ G(x)
(
A
x
)m
. (1)
Y. Yu ()
Department of Statistics, University of California, Irvine 92697, USA
e-mail: yamingy@uci.edu
180 Y. Yu
Assume that there exists a positive constant c such that
G(x)  e−c
√
x (2)
as x → ∞. Put
φ∞(x) = e−x
∞
∑
k=0
xkφ(k)
k! .
Then for any fixed positive integer M ,
φ∞(x) = φ(x) +
2M−2
∑
n=2
n
∑
k=	(n+1)/2
+1
bknx
n−k+1 φ(n)(x)
n! + O
(
G(x)x−M
)
, (3)
as x tends to ∞, where 	(n + 1)/2
 denotes the integer part of (n + 1)/2 and the
numbers bkn are defined recursively by
bkk = 1, k ≥ 2;
bkn = 0, n < k or n > 2k − 2;
bk+1,n+1 = nbk,n−1 + (n − k + 1)bkn, k ≤ n ≤ 2k − 1.
This result may seem hard to penetrate at first glance. Its relevance, however, is
easily appreciated through interesting examples such as [2, 3]
e−x
∞
∑
k=0
√
kxk
k! =
√
x
(
1 − 1
8x
− 7
128x2
+ O
(
1
x3
))
(4)
and
e−x
∞
∑
k=0
xk log(k + 1)
k! = log(x) +
1
2x
+ 1
12x2
+ O
(
1
x3
)
,
both valid as x → ∞; by choosing φ(x) = log(x +1) in (3), an asymptotic formula
for the Shannon entropy of the Poisson distribution can also be obtained (see [6]).
The first goal of this note is a formal, probabilistic derivation of Theorem 1. The
starting point is the observation that φ∞(x) = Eφ(U), where U is a Poisson random
variable with mean x and E denotes expectation. Based on this we present in Sect. 2
a more general version (Theorem 2) of Theorem 1 by considering U distributed as
some distribution other than the Poisson, e.g., a gamma distribution or a binomial
distribution. As illustrations, we derive asymptotic expansions for digamma functions
and for inverse moments of certain positive random variables. We prove Theorem 2
in Sect 3.
Noting that φ∞(x) = Eφ(U) where U has the Po(x) distribution, we expand
φ(U) as a Taylor series
φ(U) = φ(x) +
2M−2
∑
n=1
(U − x)nφ(n)(x)
n! + · · · ,
On an asymptotic series of Ramanujan 181
and formally take the expectation term by term:
Eφ(U) = φ(x) +
2M−2
∑
n=1
E(U − x)nφ(n)(x)
n! + · · · (5)
The quantity μn = E(U − x)n is the nth central moment of the Po(x) distribution.
The first few μn’s are
(μ1,μ2,μ3,μ4) = (0, x, x,3x2 + x),
and they obey the well-known recursion (see [19], Lemma 3, for example)
μn+1 = x
(
dμn
dx
+ nμn−1
)
, n ≥ 2,
from which we obtain, by comparing the coefficients of xn−k+1 and using the defini-
tion of bkn,
μn =
n
∑
k=0
bknx
n−k+1, n ≥ 2. (6)
The double sum in (3) is the result of substituting (6) in (5) and noting that bkn = 0
if k ≤ 	(n + 1)/2
. Based on this it is also clear that μn is a polynomial in x of
degree 	n/2
, which implies that, given the condition (1), the “leading term of the
remainder” in (5),
E(U − x)2M−1φ(2M−1)(x)
(2M − 1)! =
μ2M−1φ(2M−1)(x)
(2M − 1)! ,
is O(G(x)x−M).
The above derivation is, of course, strictly formal. However, it can be made rigor-
ous under the stated conditions; see Berndt [2] and Evans [5]. Berndt actually proved
a modification of (3) where the order of summation over k and n on the right hand
side is inverted and certain higher order terms of the resulting sum are absorbed in
the O(G(x)x−M) term.
2 A general version
The formal derivation in Sect. 1 suggests that it is possible to generalize Theorem 1
if we let U have a suitable distribution other than the Poisson. Noting the key role
played by the central moments of U , we give a version of Theorem 1 by imposing
conditions on the moment generating function (mgf) of U . An introduction to mo-
ment generating functions can be found in probability texts such as Gut [9]. A useful
property is that, if an mgf exists in a neighborhood of zero, then for all m ≥ 1, the
mth moment exists and can be obtained by differentiating the mgf m times.
182 Y. Yu
Theorem 2 Let φ(x), x ∈ [0,∞), denote a Borel measurable function that can be
bounded in absolute value by a polynomial in x. Let M be a fixed positive integer.
Suppose there exist a constant A ≥ 1 and a function G(x) of at most polynomial
growth such that for 1 ≤ m ≤ 2M and all sufficiently large x, the derivatives φ(m)(x)
exist and satisfy
∣
∣
∣
∣
φ(m)(x)
m!
∣
∣
∣
∣
≤ G(x)
(
A
x
)m
. (7)
Assume there exist η ∈ (0,1) and a constant B such that for all sufficiently large x,
G(y) ≤ BG(x) whenever |y − x| ≤ ηx. (8)
Let  be an unbounded subset of [0,∞) and let Ux,x ∈ , be a family of nonnegative
random variables. Assume there exist a constant δ > 0 and a function g(s) such that
for all x ∈ , the mgf of Ux exists in the interval (−δ, δ) and satisfies
EesUx = exg(s), s ∈ (−δ, δ). (9)
Assume g′(0) = 1 in addition. Then
Eφ(Ux) = φ(x) +
2M−2
∑
n=2
n
∑
k=	(n+1)/2
+1
cknx
n−k+1 φ(n)(x)
n! + O
(
G(x)x−M
)
, (10)
as x tends to ∞, where ckn are constants that depend only on the function g(s), and
are determined by
E(Ux − x)n =
n
∑
k=0
cknx
n−k+1, n ≥ 2.
Evidently, Theorem 1 is the special case g(s) = es − 1, except for the assumption
(8) on G(x) which replaces (2). This new assumption does not appear very restrictive
as we shall see from the examples later in this section; however it does make the proof
of Theorem 2 more straightforward. We also relax the assumption on φ(x) slightly
by requiring only 2M derivatives.
It should be emphasized that the function g(s) in (9) does not depend on x. Also
note that g(s) is analytic in s ∈ (−δ, δ) given the existence of the mgf. Aside from the
Poisson, examples of distribution families that satisfy (9) include the binomial, neg-
ative binomial, and gamma families. In general, suppose Y1, Y2, . . . is a sequence of
independent and identically distributed (i.i.d.), nonnegative, nondegenerate random
variables whose mgf exists in a neighborhood of zero. Then the family of random
variables {∑nk=1 Yk, n = 1,2, . . .} (n is known as a convolution parameter) has mgf
E exp
(
s
n
∑
k=1
Yk
)
= exp
(
n log
(
EesY1
))
, n = 1,2, . . . ,
which is of the form (9) with g(s) = (EY1)−1 log(EesY1), if we index the family by
its mean x = nEY1. This shows that Theorem 2 is potentially applicable to a wide
range of problems.
On an asymptotic series of Ramanujan 183
Example 1 For a fixed p ∈ (0,1), consider the family Ux,x = p,2p,3p, . . . , where
Ux has the binomial distribution Bi(n,p), n = x/p. The first few central moments of
Ux are given by (q = 1 − p)
(μ2,μ3,μ4) = (qx, (q − p)qx,3q2x2 + q(1 − 6pq)x).
Since Bi(n,p) is a sum of n i.i.d. Bernoulli(p) random variables, (9) is satisfied.
• Given r (real) and a > 0, let φ(x) = G(x) = (x + a)−r . It is easy to verify (7)
and (8); thus we have
n
∑
k=0
(
n
k
)
pkqn−k(k + a)−r = (np + a)−r + qr(r + 1)
2
(np)(np + a)−r−2
− (q − p)qr(r + 1)(r + 2)
6
(np)(np + a)−r−3
+ q
2r(r + 1)(r + 2)(r + 3)
8
(np)2(np + a)−r−4
+ O(n−r−3), (11)
as n → ∞.
• If we let φ(x) = x−r , x ≥ 1 and φ(x) = 0, x < 1, then we obtain an asymptotic
expansion for
n
∑
k=1
(
n
k
)
pkqn−kk−r (12)
by simply substituting a = 0 in the right hand side of (11). When r is a posi-
tive integer, (12) is sometimes known as the r th inverse moment of the binomial.
The problem of inverse moments has a long history in statistics (see, for example,
Stephan [16], Grab and Savage [8], and David and Johnson [4]). More recently,
expansions for (12) have been considered by Marciniak and Wesolowski [14] (see
also Rempala [15]) for r = 1, and by Žnidaricˇ [19] for general r . Žnidaricˇ [19] also
gives a brief historical account with many references.
A special case corresponding to r = −1/2 is
n
∑
k=0
(
n
k
)
pkqn−k
√
k = √np
[
1 − q/8
np
+ (q − p)q/16 − 15q
2/128
(np)2
+ O
(
1
n3
)]
,
which is the binomial analog of (4) considered by Ramanujan [2].
• Let φ(x) = log(x + β) for a fixed β > 0 and let G(x) ≡ 1. We have (as n → ∞)
n
∑
k=0
(
n
k
)
pkqn−k log(k + β)
= log(np + β) − npq
2
(np + β)−2
184 Y. Yu
+ (q − p)q
3
(np)(np + β)−3 − 3q
2
4
(np)2(np + β)−4
+ O(n−3). (13)
The problem of approximating the left hand side of (13) appears in Krichevskiy
[13] in an information theoretic context; see also Jacquet and Szpankowski [10, 11],
who give an alternative derivation of (13) using the method of analytic poissonization
and depoissonization. Flajolet [7] also considers similar problems using singularity
analysis.
Example 2 For a fixed p ∈ (0,1), consider the negative binomial family NB(n,p)
whose probability mass function is f (k;n,p) = ( n+k−1
k
)
pnqk , k = 0,1, . . . , where
q = 1 − p. The mean is nq/p and the first few central moments are
(μ2,μ3,μ4) =
(
nq
p2
,
nq(1 + q)
p3
,
[
3 + 6
n
+ p
2
nq
]
(nq)2
p4
)
.
Similar to the binomial case, as NB(n,p) is a sum of n i.i.d. geometric(p) random
variables, (9) is satisfied and Theorem 2 is applicable for an appropriate φ(x).
• Take φ(x) = (x + a)−r , a > 0. We have, as n → ∞,
∞
∑
k=0
(
n + k − 1
k
)
pnqk(k + a)−r
=
(
nq
p
+ a
)−r
+ r(r + 1)
2p
(
nq
p
)(
nq
p
+ a
)−r−2
− (1 + q)r(r + 1)(r + 2)
6p2
(
nq
p
)(
nq
p
+ a
)−r−3
+ r(r + 1)(r + 2)(r + 3)
8p2
(
nq
p
)2 (
nq
p
+ a
)−r−4
+ O(n−r−3). (14)
• As in the binomial case, we obtain an asymptotic expansion for
∞
∑
k=1
(
n + k − 1
k
)
pnqkk−r (15)
for real r by substituting a = 0 in the right hand side of (14). Expansions for (15)
have been considered by Marciniak and Wesolowski [14] and Rempala [15] for the
special case r = 1, and by Wuyungaowa and Wang [18] for integer r ≥ 0.
• Let φ(x) = log(x + β) for a fixed β > 0. We have
∞
∑
k=0
(
n + k − 1
k
)
pnqk log(k + β) = log
(
nq
p
+ β
)
− 1
2p
(
nq
p
)(
nq
p
+ β
)−2
On an asymptotic series of Ramanujan 185
+ (1 + q)
3p2
(
nq
p
)(
nq
p
+ β
)−3
− 3
4p2
(
nq
p
)2 (
nq
p
+ β
)−4
+ O(n−3).
Example 3 Consider the gamma family Gam(x,1), x > 0, whose density function is
f (u;x) = ux−1e−u/(x), u > 0. The mean is x and the first few central moments
are
(μ2,μ3,μ4) = (x,2x,3x2 + 6x).
The moment generating function is (1 − s)−x , s < 1, which is of the form (9) with
g(s) = − log(1 − s).
Take φ(x) = G(x) = x log(x). We have
1
(x)
∫ ∞
0
u log(u)ux−1e−udu = x log(x) + 1
2
− 1
12x
+ O
(
log(x)
x2
)
,
as x → ∞. Noting (x + 1) = x(x), we may write
′(x + 1)
(x + 1) = log(x) +
1
2x
− 1
12x2
+ O
(
log(x)
x3
)
, (16)
which is a familiar asymptotic formula for the digamma function ([1], p. 259). By
expanding for one more term we can replace O(x−3 log(x)) by O(x−3) in (16). A full
asymptotic expansion can be recovered by applying (10) and using the following
recursion between the central moments of Gam(x,1) (see [17]):
μk = (k − 1)(μk−1 + xμk−2), k ≥ 2.
3 Proof of Theorem 2
Our proof follows Berndt [2]. In the setting of Theorem 2 we have
Lemma 1 Let h(u), u ∈ [0,∞), be a Borel measurable function that can be bounded
in absolute value by a polynomial. Then for a fixed t ∈ (0,1), both EI (Ux <
tx)h(Ux) and EI (Ux > x/t)h(Ux) tend to 0 exponentially fast as x tends to ∞,
where I (·) is the indicator function.
Proof Observe that xg(s), the cumulant generating function of Ux , is an analytic
function of s (real) in a neighborhood of zero. Because g(0) = 0, g′(0) = 1 and
t ∈ (0,1), we may choose r, 	 > 0 small enough such that both g(r) < r/t and g(r +
	) < r/t . Since |h(u)| is bounded by a polynomial, there exists a constant D such
that |h(u)| < e	u + D for all u ∈ [0,∞). We have
|EI (Ux > x/t)h(Ux)| ≤ E(e	Ux + D)er(Ux−x/t)
= ex[g(r+	)−r/t] + Dex[g(r)−r/t],
186 Y. Yu
which tends to zero exponentially as x → ∞. The proof for EI (Ux < tx)h(Ux) is
similar and hence omitted. 
Proof of Theorem 2 Throughout we assume that x is sufficiently large. Define inter-
vals I1 = [0, (1 − η)x), I2 = [(1 − η)x, (1 + η)x) and I3 = [(1 + η)x,∞), where η
is as specified in (8).
By Lemma 1, both
EI (Ux ∈ I1)φ(Ux) (17)
and
EI (Ux ∈ I3)φ(Ux) (18)
tend to zero exponentially as x → ∞.
Consider the Taylor polynomial
ψ(y) =
2M−1
∑
k=0
φ(k)(x)
k! (y − x)
k.
Since for any y ∈ I1,
|ψ(y)| ≤
2M−1
∑
k=0
∣
∣
∣
∣
φ(k)(x)
k!
∣
∣
∣
∣
xk ≡ q(x),
we have
|EI (Ux ∈ I1)ψ(Ux)| ≤ q(x)EI (Ux ∈ I1).
From (7) it follows that q(x) has at most polynomial growth as x → ∞; by Lemma 1
we know that
EI (Ux ∈ I1)ψ(Ux) (19)
tends to zero exponentially as x → ∞.
Similarly, for any y ∈ I3, we have
|ψ(y)| ≤
2M−1
∑
k=0
∣
∣
∣
∣
φ(k)(x)
k!
∣
∣
∣
∣
yk,
and hence
|EI (Ux ∈ I3)ψ(Ux)| ≤
2M−1
∑
k=0
∣
∣
∣
∣
φ(k)(x)
k!
∣
∣
∣
∣
EI (Ux ∈ I3)Ukx .
By Lemma 1, each of EI (Ux ∈ I3)Ukx , k ≤ 2M − 1, tends to zero exponentially as
x → ∞. By (7), each of φ(k)(x) has at most polynomial growth as x → ∞. Overall
EI (Ux ∈ I3)ψ(Ux) (20)
On an asymptotic series of Ramanujan 187
tends to zero exponentially as x → ∞.
For any y ∈ I2, there exists some point ζ between x and y such that
|φ(y) − ψ(y)| =
∣
∣
∣
∣
φ(2M)(ζ )
(2M)!
∣
∣
∣
∣
|y − x|2M
≤ G(ζ)
(
A
(1 − η)x
)2M
|y − x|2M
≤ BG(x)
(
A
(1 − η)x
)2M
|y − x|2M,
where (7) and (8) are used in the inequalities. Letting C = B[A/(1 − η)]2M , we have
|EI (Ux ∈ I2)[φ(Ux) − ψ(Ux)]| ≤ CG(x)
x2M
E|Ux − x|2M.
We now consider the nth central moment of Ux , μn = E(Ux − x)n, as a function
of x. (Note that the mean of Ux is x as EUx = xg′(0) = x.) Expand xg(s) around
s = 0 to get
xg(s) =
∞
∑
j=1
xg(j)(0)sj
j ! .
Note that the coefficient xg(j)(0) is the j th cumulant of Ux , and, according to the
well-known relation between central moments and cumulants (see [12] or [17], for
example)
μn =
n−2
∑
j=0
(
n − 1
j
)
μjxg
(n−j)(0), n ≥ 2, (21)
with μ0 = 1 and μ1 = 0. Based on (21), it is easy to show by induction that μn is
a polynomial in x of degree at most 	n/2
, its coefficients depending only on the
function g(s). Hence, for large x we have E(Ux − x)2M = O(xM) and
EI (Ux ∈ I2)[φ(Ux) − ψ(Ux)] = O(G(x)x−M).
Combined with the exponentially small items (17), (18), (19) and (20), this gives
E[φ(Ux) − ψ(Ux)] = O(G(x)x−M).
It remains to calculate Eψ(Ux). We have, by the definition of ckn,
Eψ(Ux) =
2M−1
∑
n=0
E(Ux − x)n φ
(n)(x)
n!
= φ(x) +
2M−1
∑
n=2
n
∑
k=0
cknx
n−k+1 φ(n)(x)
n!
188 Y. Yu
= φ(x) +
2M−1
∑
n=2
n
∑
k=	(n+1)/2
+1
cknx
n−k+1 φ(n)(x)
n!
= φ(x) +
2M−2
∑
n=2
n
∑
k=	(n+1)/2
+1
cknx
n−k+1 φ(n)(x)
n! + O(G(x)x
−M).
Note that the inner sum over k is curtailed because the degree of μn is at most 	n/2
,
i.e., ckn = 0 if k ≤ 	(n + 1)/2
. As a consequence of (7), the term corresponding to
n = 2M − 1 in the outer sum is written as O(G(x)x−M) in the last equality. The
proof of (10) is now complete. 
Acknowledgements The author would like to thank the reviewer for his/her valuable comments.
Open Access This article is distributed under the terms of the Creative Commons Attribution Noncom-
mercial License which permits any noncommercial use, distribution, and reproduction in any medium,
provided the original author(s) and source are credited.
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On an Asymptotic Series of Ramanujan

On an Asymptotic Series of Ramanujan

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