Uniform Asymptotic Expansion for the Incomplete Beta Function
In [Temme N.M., Special functions. An introduction to the classical functions of mathematical physics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996, Section 11.3.3.1] a uniform asymptotic expansion for the incomplete beta function was derived. It was not obvious from...
Saved in:
| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Date: | 2016 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2016
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/148002 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Uniform Asymptotic Expansion for the Incomplete Beta Function / G. Nemes, A.B. Olde Daalhuis // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 4 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-148002 |
|---|---|
| record_format |
dspace |
| spelling |
Nemes, G. Olde Daalhuis, A.B. 2019-02-16T16:26:10Z 2019-02-16T16:26:10Z 2016 Uniform Asymptotic Expansion for the Incomplete Beta Function / G. Nemes, A.B. Olde Daalhuis // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 4 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 41A60; 33B20 DOI:10.3842/SIGMA.2016.101 https://nasplib.isofts.kiev.ua/handle/123456789/148002 In [Temme N.M., Special functions. An introduction to the classical functions of mathematical physics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996, Section 11.3.3.1] a uniform asymptotic expansion for the incomplete beta function was derived. It was not obvious from those results that the expansion is actually an asymptotic expansion. We derive a remainder estimate that clearly shows that the result indeed has an asymptotic property, and we also give a recurrence relation for the coefficients. This research was supported by a research grant (GRANT11863412/70NANB15H221) from the National Institute of Standards and Technology. The authors thank the anonymous referees for their helpful comments and suggestions on the manuscript. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Uniform Asymptotic Expansion for the Incomplete Beta Function Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Uniform Asymptotic Expansion for the Incomplete Beta Function |
| spellingShingle |
Uniform Asymptotic Expansion for the Incomplete Beta Function Nemes, G. Olde Daalhuis, A.B. |
| title_short |
Uniform Asymptotic Expansion for the Incomplete Beta Function |
| title_full |
Uniform Asymptotic Expansion for the Incomplete Beta Function |
| title_fullStr |
Uniform Asymptotic Expansion for the Incomplete Beta Function |
| title_full_unstemmed |
Uniform Asymptotic Expansion for the Incomplete Beta Function |
| title_sort |
uniform asymptotic expansion for the incomplete beta function |
| author |
Nemes, G. Olde Daalhuis, A.B. |
| author_facet |
Nemes, G. Olde Daalhuis, A.B. |
| publishDate |
2016 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
In [Temme N.M., Special functions. An introduction to the classical functions of mathematical physics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996, Section 11.3.3.1] a uniform asymptotic expansion for the incomplete beta function was derived. It was not obvious from those results that the expansion is actually an asymptotic expansion. We derive a remainder estimate that clearly shows that the result indeed has an asymptotic property, and we also give a recurrence relation for the coefficients.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/148002 |
| citation_txt |
Uniform Asymptotic Expansion for the Incomplete Beta Function / G. Nemes, A.B. Olde Daalhuis // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 4 назв. — англ. |
| work_keys_str_mv |
AT nemesg uniformasymptoticexpansionfortheincompletebetafunction AT oldedaalhuisab uniformasymptoticexpansionfortheincompletebetafunction |
| first_indexed |
2025-11-25T20:55:48Z |
| last_indexed |
2025-11-25T20:55:48Z |
| _version_ |
1850543546812071936 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 101, 5 pages
Uniform Asymptotic Expansion
for the Incomplete Beta Function
Gergő NEMES and Adri B. OLDE DAALHUIS
Maxwell Institute and School of Mathematics, The University of Edinburgh,
Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK
E-mail: Gergo.Nemes@ed.ac.uk, A.B.Olde.Daalhuis@ed.ac.uk
URL: http://www.maths.ed.ac.uk/~gnemes/, http://www.maths.ed.ac.uk/~adri/
Received September 12, 2016, in final form October 21, 2016; Published online October 25, 2016
http://dx.doi.org/10.3842/SIGMA.2016.101
Abstract. In [Temme N.M., Special functions. An introduction to the classical functions
of mathematical physics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New
York, 1996, Section 11.3.3.1] a uniform asymptotic expansion for the incomplete beta func-
tion was derived. It was not obvious from those results that the expansion is actually an
asymptotic expansion. We derive a remainder estimate that clearly shows that the result
indeed has an asymptotic property, and we also give a recurrence relation for the coefficients.
Key words: incomplete beta function; uniform asymptotic expansion
2010 Mathematics Subject Classification: 41A60; 33B20
1 Introduction
For positive real numbers a, b and x ∈ [0, 1], the (normalised) incomplete beta function Ix(a, b)
is defined by
Ix(a, b) =
1
B(a, b)
∫ x
0
ta−1(1− t)b−1 dt,
where B(a, b) denotes the ordinary beta function:
B(a, b) =
∫ 1
0
ta−1(1− t)b−1 dt =
Γ(a)Γ(b)
Γ(a+ b)
(see, e.g., [2, Section 8.17(i)]). In this paper, we will use the notation of [2, Section 8.18(ii)].
The incomplete beta function plays an important role in statistics in connection with the beta
distribution (see, for instance, [1, pp. 210–275]). Large parameter asymptotic approximations
are useful in these applications. For fixed x and b, one could use the asymptotic expansion
Ix(a, b) =
xa(1− x)b−1
aB(a, b)
2F1
(
1, 1− b
a+ 1
;
x
x− 1
)
∼ xa(1− x)b−1
aB(a, b)
∞∑
n=0
(1− b)n
(a+ 1)n
(
x
x− 1
)n
, (1)
as a→ +∞. The right-hand side of (1) converges only for x ∈ [0, 12), but for any fixed x ∈ [0, 1)
it is still useful when used as an asymptotic expansion as a → +∞. For more details, see [3,
Section 11.3.3]. However, it is readily seen that (1) breaks down as x → 1. Since this limit
has significant importance in applications, Temme derived in [3, Section 11.3.3.1] an asymptotic
expansion as a→ +∞ that holds uniformly for x ∈ (0, 1]. His result can be stated as follows.
mailto:Gergo.Nemes@ed.ac.uk
mailto:A.B.Olde.Daalhuis@ed.ac.uk
http://www.maths.ed.ac.uk/~gnemes/
http://www.maths.ed.ac.uk/~adri/
http://dx.doi.org/10.3842/SIGMA.2016.101
http://dlmf.nist.gov/8.17.i
http://dlmf.nist.gov/8.18.ii
http://dx.doi.org/10.1002/9781118032572.ch11
http://dx.doi.org/10.1002/9781118032572.ch11
2 G. Nemes and A.B. Olde Daalhuis
Theorem 1. Let ξ = − lnx. Then for any f ixed positive integer N and fixed positive real b,
Ix(a, b) =
Γ(a+ b)
Γ(a)
(
N−1∑
n=0
dnFn +O
(
a−N
)
F0
)
, (2)
as a→ +∞, uniformly for x ∈ (0, 1]. The functions Fn = Fn(ξ, a, b) are defined by the recurrence
relation
aFn+1 = (n+ b− aξ)Fn + nξFn−1, (3)
with
F0 = a−bQ(b, aξ), F1 =
b− aξ
a
F0 +
ξbe−aξ
aΓ(b)
,
and Q(a, z) = Γ(a, z)/Γ(a) is the normalised incomplete gamma function (see [2, Section 8.2(i)]).
The coefficients dn = dn(ξ, b) are defined by the generating function(
1− e−t
t
)b−1
=
∞∑
n=0
dn(t− ξ)n. (4)
In particular,
d0 =
(
1− x
ξ
)b−1
, d1 =
xξ + x− 1
(1− x)ξ
(b− 1)d0.
They satisfy the recurrence relation
ξ(n+ 1)(n+ 2)d0dn+2 = ξ
n∑
m=0
(m+ 1)
(
n− 2m+ 1 +
m− n− 1
b− 1
)
dm+1dn−m+1
+
n∑
m=0
(m+ 1)
(
n− 2m− 2− ξ +
m− n
b− 1
)
dm+1dn−m
+
n∑
m=0
(1−m− b)dmdn−m. (5)
In the case that b = 1, we have d0 = 1 and dn = 0 for n ≥ 1.
Our contribution is the remainder estimate in (2) and the recurrence relation (5). In fact, it is
not at all obvious from (3) that the sequence {Fn}∞n=0 has an asymptotic property as a→ +∞.
We will show that for any non-negative integer n,
0 < Fn+1 ≤
n+ β
a
Fn, (6)
where β = max(1, b).
In [4, Section 38.2.8] the function Fn is identified as a Kummer U -function:
Fn =
ξn+be−aξn!
Γ(b)
U(n+ 1, n+ b+ 1, aξ).
http://dlmf.nist.gov/8.2.i
http://dx.doi.org/10.1142/9789814612166_0038
Uniform Asymptotic Expansion for the Incomplete Beta Function 3
2 Proof of the main results
We proceed similarly as in [3, Section 11.3.3.1] and start with the integral representation
Ix(a, b) =
1
B(a, b)
∫ +∞
ξ
tb−1e−at
(
1− e−t
t
)b−1
dt. (7)
We substitute the truncated Taylor series expansion(
1− e−t
t
)b−1
=
N−1∑
n=0
dn(t− ξ)n + rN (t)
into (7) and obtain
Ix(a, b) =
Γ(a+ b)
Γ(a)
(
N−1∑
n=0
dnFn +RN (a, b, x)
)
,
where Fn is given by the integral representation
Fn =
1
Γ(b)
∫ +∞
ξ
tb−1e−at(t− ξ)n dt =
e−aξ
Γ(b)
∫ +∞
0
(τ + ξ)b−1τne−aτ dτ, (8)
and the remainder term RN (a, b, x) is defined by
RN (a, b, x) =
1
Γ(b)
∫ +∞
ξ
tb−1e−atrN (t) dt. (9)
The recurrence relation (3) can be obtained from (8) via a simple integration by parts.
Let, for a moment,
cn(a, b) =
∫ +∞
0
(τ + ξ)b−1τne−aτ dτ.
Then via integration by parts we find
acn+1(a, b) = (n+ b)cn(a, b) + ξ(1− b)cn(a, b− 1). (10)
We make the observation that
0 ≤ ξcn(a, b− 1) = ξ
∫ +∞
0
(τ + ξ)b−2τne−aτ dτ ≤ cn(a, b). (11)
It follows from (10) and (11) that
acn+1(a, b) ≤
{
(n+ 1)cn(a, b) if 0 < b ≤ 1,
(n+ b)cn(a, b) if b ≥ 1.
Since Fn = e−aξcn(a, b)/Γ(b), this inequality implies (6).
To obtain the remainder estimate in (2), we use the Cauchy integral representation
rN (t) =
(t− ξ)N
2πi
∮
{ξ,t}
(
1−e−τ
τ
)b−1
(τ − t)(τ − ξ)N
dτ, (12)
http://dx.doi.org/10.1002/9781118032572.ch11
4 G. Nemes and A.B. Olde Daalhuis
where the contour encircles the points ξ and t once in the positive sense. From the integral
representation (9), we have that 0 ≤ ξ ≤ t. Thus, in the case that N ≥ 1, we can deform the
contour in (12) to the path
[1 +∞i, 1 + πi] ∪ [1 + πi,−1 + πi] ∪ [−1 + πi,−1− πi]
∪ [−1− πi, 1− πi] ∪ [1− πi, 1−∞i].
For the integrals along the final three portions of the path, we have the estimates∣∣∣∣∣∣∣
1
2πi
∫ −1−πi
−1+πi
(
1−e−τ
τ
)b−1
(τ − t)(τ − ξ)N
dτ
∣∣∣∣∣∣∣ ≤
max
(
(e− 1)b−1,
(
e+1√
π2+1
)b−1)
(1 + ξ)N+1
,
∣∣∣∣∣∣∣
1
2πi
∫ 1−πi
−1−πi
(
1−e−τ
τ
)b−1
(τ − t)(τ − ξ)N
dτ
∣∣∣∣∣∣∣ ≤
max
((
e±1+1√
π2+1
)b−1)
πN+2
, (13)
and ∣∣∣∣∣∣∣
1
2πi
∫ 1−∞i
1−πi
(
1−e−τ
τ
)b−1
(τ − t)(τ − ξ)N
dτ
∣∣∣∣∣∣∣ ≤
1
2π
∫ +∞
π
max
((
1± e−1
)b−1)(
s2 + 1
)(1−b)/2√
s2 + (1− t)2
(
s2 + (1− ξ)2
)N/2 ds
≤
max
((
1± e−1
)b−1)
2π
∫ +∞
π
(
s2 + 1
)(1−b)/2
sN+1
ds, (14)
respectively. The integrals along the first two portions can be estimated similarly to (13)
and (14). Hence, for 0 ≤ ξ ≤ t and N ≥ 1, we have
|rN (t)| ≤ CN (b)(t− ξ)N ,
where the constant CN (b) does not depend on ξ. Using this result in the integral representa-
tion (9), we can infer that
|RN (a, b, x)| ≤ CN (b)FN .
Finally, combining this result with the inequalities (6), we obtain the required remainder estimate
in (2).
The reader can check that the function f(t) =
(
1−e−t
t
)b−1
is a solution of the nonlinear
differential equation
tf(t)f ′′(t)− b− 2
b− 1
tf ′2(t) + (t+ 2)f(t)f ′(t) + (b− 1)f2(t) = 0.
If we substitute the Taylor series (4) into this differential equation and rearrange the result, we
obtain the recurrence relation (5).
Acknowledgements
This research was supported by a research grant (GRANT11863412/70NANB15H221) from the
National Institute of Standards and Technology. The authors thank the anonymous referees for
their helpful comments and suggestions on the manuscript.
Uniform Asymptotic Expansion for the Incomplete Beta Function 5
References
[1] Johnson N.L., Kotz S., Balakrishnan N., Continuous univariate distributions, Vol. 2, 2nd ed., Wiley Series
in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Inc., New
York, 1995.
[2] Olver F.W.J., Lozier D.W., Boisvert R.F., Clark C.W. (Editors), NIST handbook of mathematical functions,
U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC, Cam-
bridge University Press, Cambridge, 2010, Release 1.0.13 of 2016-09-16, available at http://dlmf.nist.
gov/.
[3] Temme N.M., Special functions. An introduction to the classical functions of mathematical physics, A Wiley-
Interscience Publication, John Wiley & Sons, Inc., New York, 1996.
[4] Temme N.M., Asymptotic methods for integrals, Series in Analysis, Vol. 6, World Scientific Publishing Co.
Pte. Ltd., Hackensack, NJ, 2015.
http://dlmf.nist.gov/
http://dlmf.nist.gov/
http://dx.doi.org/10.1002/9781118032572
http://dx.doi.org/10.1002/9781118032572
http://dx.doi.org/10.1142/9195
1 Introduction
2 Proof of the main results
References
|