See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/345751124
An Improper Integral with a Square Root
Preprint · October 2016
DOI: 10.20944/preprints201610.0089.v1
CITATIONS
READS
5
132
1 author:
Feng Qi
Texas
975 PUBLICATIONS 20,358 CITATIONS
SEE PROFILE
Some of the authors of this publication are also working on these related projects:
Determinantal Expressions, Integral Representations, and Analytic Properties of Sequences and Polynomials in Combinatorics and Number Theory View project
The Bell Polynomials of the Second Kind and Their Applications View project
All content following this page was uploaded by Feng Qi on 16 November 2020.
The user has requested enhancement of the downloaded file.
DOI: 10.15393/j3.art.20**.****
Probl. Anal. Issues Anal. Vol. *, No. *, 20**
Feng Qi
AN IMPROPER INTEGRAL, THE BETA FUNCTION, THE
WALLIS RATIO, AND THE CATALAN NUMBERS
Abstract. In the paper, the author presents closed and unified
expressions for a sequence of improper integrals in terms of the
beta function and the Wallis ratio. Hereafter, the author derives
integral representations for the Catalan numbers originating from
combinatorics.
Please cite this article as “Feng Qi, An improper integral, the
beta function, the Wallis ratio, and the Catalan numbers, Problemy Analiza–Issues of Analysis 7 (25) (2018), no. 1, 104–115;
available online at https://doi.org/10.15393/j3.art.2018.4370.”
Key words: improper integral; closed expression; unified expression; beta function; Wallis ratio; integral representation; Catalan
number.
2010 Mathematical Subject Classification: Primary 11B65;
Secondary 11B75, 11B83, 26A39, 26A42, 33B15.
§ 1. Introduction
In mathematics, a closed form is a mathematical expression that can
be evaluated in a finite number of operations, it may contain constants,
variables, four arithmetic operations, and elementary functions, but usually no limit.
Let a be a positive number. For n ≥ 0, define
r
Z a
a+x
In =
xn
d x.
(1)
a−x
−a
In [1, Section 3], Dana-Picard and Zeitoun computed I0 = aπ and found
a closed form of In for n ∈ N in three steps:
© Petrozavodsk State University, 2020
Improper Integral, . . . , and Catalan Numbers
105
1) establishing a formula of recurrence between In and In+1 in terms
of
Z π/2
Sn =
sinn θ d θ;
(2)
−π/2
2) establishing an equation for In in terms of Sn ;
3) and establishing different expressions for odd values and even values
of n.
Consequently, they deduced an integral representation of the Catalan
numbers which originate from combinatorics and number theory.
The aim of this note is to discuss again the sequence In , to present
closed and unified expressions for the sequence In in terms of the beta
function and the Wallis ratio, to derive integral representations for the
Catalan numbers, and to correct some errors and typos appeared in [1,
Section 3].
§ 2. Closed and unified expressions for In
The sequence In can be computed by several methods below.
Theorem 1. For n ∈ N, the sequence In can be computed by
"
#
n+1
n
1
1
1
+
(−1)
1
+
(−1)
+
,
In = an+1 π
n
n+1
B 12 , n2
B 21 , n+1
2
(3)
where
Z
B(p, q) =
1
t
p−1
(1 − t)
q−1
Z
dt =
0
0
and
Z
Γ(z) =
∞
tp−1
Γ(p)Γ(q)
dt =
p+q
(1 + t)
Γ(p + q)
(4)
∞
tz−1 e−t d t
0
for Re(p), Re(q) > 0 and Re(z) > 0 denote the Euler integrals of the
second kinds (or say, the classical beta and gamma functions) respectively.
Proof. By some properties of definite integral and straightforward computation, we can write
r
Z 0
Z a r
a
+
x
a+x
dx +
xn
dx
In =
xn
a−x
a−x
0
−a
106
Feng Qi
s
Z a r
a + (−y)
a+x
=
(−y)
d(−y) +
xn
dx
a − (−y)
a−x
a
0
r
Z a
Z a r
a
−
y
a+x
=
(−1)n y n
dy +
xn
dx
a+y
a−x
0
0
r
r
Z a a−x
a+x
n
n
x (−1)
=
+
dx
a+x
a−x
0
Z a
(a + x) + (−1)n (a − x)
√
dx
=
xn
a2 − x2
0
Z a
a[1 + (−1)n ] + x[1 − (−1)n ]
√
dx
=
xn
a2 − x2
0
Z a
Z a
xn
xn+1
n
n
√
√
= a[1 + (−1) ]
d x + [1 − (−1) ]
d x.
a 2 − x2
a2 − x2
0
0
Z
0
n
In [6, Theorem 3.1], it was obtained that
Z
0
a
√ n Γ n2 + 21
xn
√
dx = πa
nΓ n2
a2 − x2
for a > 0 and n ≥ 0. Accordingly, considering
√
1
= π,
Γ
2
(5)
we acquire
1
Γ n+1
2 + 2 + [1 − (−1) ] π a
In = a[1 + (−1) ] π a
nΓ n2
(n + 1)Γ n+1
! 2
√
Γ n+1
Γ n2 + 1
n
2 = π an+1 [1 + (−1)n ]
+
[1
−
(−1)
]
nΓ n2
(n + 1)Γ n+1
2
"
#
n Γ n+1
n Γ n +1
1
+
(−1)
1
−
(−1)
2 +
2
= an+1 π
n
n + 1 Γ 12 Γ n+1
Γ 12 Γ n2
2
"
#
n
n+1
1
+
(−1)
1
1
+
(−1)
1
+
.
= an+1 π
n
n+1
B 21 , n2
B 21 , n+1
2
n
√
n
nΓ 2
+
1
2
The proof of Theorem 1 is complete. n
√
n+1
Improper Integral, . . . , and Catalan Numbers
107
Theorem 2. For n ≥ 0, the sequence In can be computed by
1 n+1
1 n+1
n
In = a
[1 + (−1) ]B
,
2
2
2
1 n+2
n+1
+ 1 + (−1)
B
,
. (6)
2
2
Proof. Changing the variable of integration by x = at in (1) gives
r
Z 1 r
Z 1
a + at
1+t
n
n+1
tn
(at)
adt = a
dt
In =
a − at
1−t
−1
−1
Z 0 r
Z 1 r
1+t
1+t
n+1
n
n
=a
t
dt +
t
dt
1−t
1−t
−1
0
r
Z 1
Z 1 r
1−s
1+t
n+1
n
n
=a
(−s)
ds +
dt
t
1+s
1−t
0
0
r
r
Z 1 1
−
t
1+t
n+1
n
n
+
=a
t (−1)
dt
1+t
1−t
0
Z 1 1+t
n+1
n
n 1−t
=a
t (−1) √
+√
dt
1 − t2
1 − t2
0
Z 1
Z 1
tn
tn+1
n+1
n
n
√
√
=a
[1 + (−1) ]
d t + [1 − (−1) ]
dt
1 − t2
1 − t2
0
0
Z π/2
sinn s
n+1
n
p
cos s d s
=a
[1 + (−1) ]
0
1 − sin2 s
Z π/2
sinn+1 s
n
p
+ [1 − (−1) ]
cos s d s
0
1 − sin2 s
Z π/2
Z π/2
n
n+1
n+1
n
n
=a
[1 + (−1) ]
sin s d s + [1 − (−1) ]
sin
sds .
0
Further making use of the formula
Z π/2
1
t+1 1
t
sin x d x = B
,
,
2
2 2
0
0
t > −1
in [6, Remark 6.4] yields
1 n+1
1 n+2
n+1
n 1
n 1
In = a
[1 + (−1) ] B
,
+ [1 − (−1) ] B
,
2
2
2
2
2
2
108
Feng Qi
1 n+1
1 n+1
1 n+2
n
n+1
= a
[1 + (−1) ]B
,
+ 1 + (−1)
B
,
.
2
2
2
2
2
The proof of Theorem 2 is complete. Corollary. For m ≥ 0, the sequences I2m and I2m+1 can be closedly
computed by
(2m − 1)!!
I2m = πa2m+1
(2m)!!
and
I2m+1 = πa2(m+1)
(2m + 1)!!
,
(2m + 2)!!
where the double factorial of negative odd integers −2n − 1 is defined by
(−2n − 1)!! =
(−1)n
2n n!
= (−1)n
,
(2n − 1)!!
(2n)!
n ≥ 0.
Proof. From the recurrence relation
Γ(x + 1) = xΓ(x),
x>0
and the identity (5), we obtain
1
1
(2m − 1)!!
(2m − 1)!! √
=
Γ
=
π.
Γ m+
2
2m
2
2m
By this equality and the last expression in (4), we derive
 1
Γ 2 Γ(m)


n = 2m

1
n
1 ,
Γ 2 Γ 2
1 n
Γ
m
+
2
=
B
,
=

Γ 12 Γ m + 12
2 2
Γ n+1

2

, n = 2m + 1
Γ(m + 1)
√

π (m − 1)!
(2m − 2)!!




n = 2m
, n = 2m;
 (2m−1)!! √ ,
2
(2m
−
1)!!
π
m
2
=
= √ (2m−1)!! √
(2m − 1)!!


π


π
, n = 2m + 1.
 π 2m

, n = 2m + 1
(2m)!!
m!
Substituting this into (6) reveals
1 2m + 1
(2m − 1)!!
1 2m+1
I2m = a
2B
,
= a2m+1 π
2
2
2
(2m)!!
(7)
Improper Integral, . . . , and Catalan Numbers
109
and
I2m+1
1 2(m+1)
1 2m + 3
(2m + 1)!!
= a
2B
,
= a2(m+1) π
.
2
2
2
(2m + 2)!!
The proof of Corollary 2 is complete. § 3. Integral representations for the Catalan numbers
The Catalan numbers Cn for n ≥ 0 form a sequence of natural numbers
that occur in various counting problems in combinatorial mathematics.
The nth Catalan
number can be expressed in terms of the central binomial
2n
coefficients n by
1
2n
Cn =
.
(8)
n+1 n
Theorem 3. For n ≥ 0 and a > 0, the Catalan numbers Cn can be
represented by
r
Z a
1
1 4n
a+x
x2n
dx
Cn =
2n+1
πn+1a
a−x
−a
Z a
1 22n+1 1
x2n
(9)
√
=
dx
π n + 1 a2n 0
a 2 − x2
Z
1 22n+1 π/2 2n
=
sin x d x
π n+1 0
and
1 22n+1 1
Cn =
π 2n + 1 a2n+2
2n+2
Z
a
r
x
2n+1
−a
a
a+x
dx
a−x
1 2
1
x2n+2
√
dx
π 2n + 1 a2n+2 0
a2 − x2
Z π/2
1 22n+2
=
sin2n+2 x d x.
π 2n + 1 0
Z
=
(10)
Proof. From the recurrence relation (7) and the identity (5), it is not
difficult to show that the Catalan numbers Cn can be expressed in terms
of the gamma function Γ by
4n Γ(n + 1/2)
Cn = √
,
π Γ(n + 2)
n ≥ 0.
110
Feng Qi
This implies that
1 4n
1
1
Cn =
B
,n +
.
πn+1
2
2
(11)
Taking n = 2p in (6) and utilizing (11) lead to
1 2p + 1
p+1
2p+1
,
= a2p+1 π p Cp
I2p = a
B
2
2
4
which is equivalent to
4n
1
1 4n
1
Cn =
I
=
2n
n + 1 a2n+1 π
π n + 1 a2n+1
a
Z
r
x
2n
−a
a+x
d x.
a−x
The first formula in (9) thus follows.
By similar argument to the deduction of (11), we can discover
Cn =
4n+1
(2n + 1)(2n + 2) B
1
1
2, n
+1
,
n ≥ 0.
Employing this identity and setting n = 2p + 1 in (3) figures out
I2p+1 = a2p+2
2π
2p + 2 B
1
1
2, p
+1
= a2p+2
2π (2p + 1)(2p + 2)
Cp
2p + 2
4p+1
which can be rearranged as
1 22p+1
1 1 22p+1
Cp = 2p+2
I2p+1 =
a
π 2p + 1
π a2p+2 2p + 1
1
Z
a
r
x
2p+1
−a
a+x
d x.
a−x
The first formula in (10) is thus proved.
The rest integral representations follow from techniques used in the
proofs of Theorems (1) and (2) and from changing variable of integration. § 4. Remarks
Finally, we state several remarks on our main results.
Remark 1. The expressions in Corollary 2 and the integral representation (9) correct [1, Proposition 3.1 and Corollary 3.2] respectively.
Improper Integral, . . . , and Catalan Numbers
111
Remark 2. Since
1 t+1
1 t
2π
B
,
B
,
=
2 2
2 2
t
for t > 0, the formulas (3) and (6) can be transferred to each other.
However, the formula (6) looks simpler.
Remark 3. Considering (8), we can rewrite the integral representations
in (9) and (10) as
r
Z a
a+x
2n
1 4n
2n
x
dx
=
2n+1
πa
a−x
n
−a
Z
1 22n+1 a
x2n
√
=
dx
π a2n 0
a2 − x2
Z
1 2n+1 π/2 2n
= 2
sin x d x
π
0
and
2n
n
1 22n+1 (n + 1) 1
=
π
2n + 1
a2n+2
Z
a
r
x2n+1
−a
a+x
dx
a−x
Z a
1 22n+2 (n + 1) 1
x2n+2
√
dx
=
π
2n + 1
a2n+2 0
a2 − x2
Z
1 22n+2 (n + 1) π/2 2n+2
=
sin
x d x.
π
2n + 1
0
for n ≥ 0.
Remark 4. It is well known that the Wallis ratio is defined by
(2n − 1)!!
(2n)!
1 Γ n + 1/2
Wn =
= 2n
=√
, n ≥ 0.
(2n)!!
2 (n!)2
π Γ(n + 1)
As a result, we have
I2m = πa2m+1 Wm ,
m≥0
and
I2m+1 = πa2m+2 Wm+1 ,
m ≥ 0.
112
Feng Qi
The Wallis ratio has been studied and applied by many mathematicians. For more information, please refer to the survey article [5] and the
paper [11], for example, and plenty of literature therein.
Remark 5. In [2], the formula
√
Z π/2
π Γ
t
sin x d x =
2 Γ
0
t+1
2 t+2 ,
2
t > −1
(12)
was stated. See also [5, p. 16, Eq. (2.18)]. In [3, p. 142, Eq. 5.12.2], the
formula
Z π/2
1
sin2a−1 θ cos2b−1 θ d θ = B(a, b), Re(a), Re(b) > 0
(13)
2
0
was listed. By (12) or (13), we can find that the quantity Sn defined
in (2) is
Z 0
Z π/2
n
Sn =
sin x d x +
sinn x d x
−π/2
Z
=
0
π/2
n
n
Z
π/2
(−1) sin x d x +
0
0
n
n+1 1
1 + (−1)
B
,
.
=
2
2
2
sinn x d x
Remark 6. In [9, Theorem 2.3], among other things, the integral formulas
λ
Z b
b−t
λπ
d t = (b − a)
,
t−a
sin(λπ)
a
λ
λ
Z b
1
π
b
b−t
dt =
−1 ,
t−a
t
sin(λπ)
a
a
λ
λ X
`
Z b
k
b−t
π
hλi` k − 1
1
b
1
a
dt =
1−
t−a
tk+1
sin(λπ) a
ak
`! ` − 1
b
a
`=0
for b > a > 0, k ∈ N, and λ ∈ (−1, 1) \ {0} were derived, where
n−1
 Q (x − k), n ≥ 1
hxin = k=0

1,
n=0
Improper Integral, . . . , and Catalan Numbers
113
is called the falling factorial. In [9, Remark 4.4], the integral formula
Z b
a
b−t
t−a
λ
1 b−t
ln
dt
t t−a

λ
λ
b
b
b
π


ln − π cot(λπ)
− 1 , λ 6= 0

sin(λπ)
a
a
a
=
2

1
b

 ln
,
λ=0
2
a
was concluded from [9, Theorem 2.3]. By comparing the forms of these integrals and In , we naturally propose a problem: can one closedly compute
the integrals
Z b
a
b−t
t−a
λ
Z
α
t d t and
(
α∈
R,
N,
α
t
a
for
b
b−t
t−a
λ
ln
b−t
dt
t−a
b>a>0
b>0>a
and λ ∈ (−1, 1)?
Remark 7. An anonymous reviewer points out that the Catalan numbers Cn emerge frequently in probability, for example, in the closed distribution of the first return to zero of the symmetric coin tossing random
walk, where
(
k
X
1,
p = 12
T0 = inf{k > 0 : Sk = 0}, Sk =
Xj , and Xj =
−1, p = 21
j=1
has distribution
P (T0 = 2n + 2) =
2n
1
1
,
n+1
n n+12
n = 0, 1, 2, . . .
Remark 8. In recent years, the Catalan numbers Cn has been analytically generalized and studied in the papers [10, 12]. For more information, please refer to the survey articles [7, 8] and closely-related references
therein.
114
Feng Qi
Remark 9. This paper is a slightly revised version of the preprint [4].
Acknowledgements
The author thanks Mr. Qing Zou at University of Iowa and other
three anonymous reviewers for their careful corrections to and valuable
comments on the original version of this paper.
References
[1] T. Dana-Picard and D. G. Zeitoun, Parametric improper integrals, Wallis
formula and Catalan numbers, Internat. J. Math. Ed. Sci. Tech. 43 (2012),
no. 4, 515–520; Available online at https://doi.org/10.1080/0020739X.
2011.599877.
[2] D. K. Kazarinoff, On Wallis’ formula, Edinburgh Math. Notes 1956
(1956), no. 40, 19–21.
[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; Available online at http://dlmf.nist.gov/.
[4] F. Qi, An improper integral with a square root, Preprints 2016,
2016100089, 8 pages; Available online at https://doi.org/10.20944/
preprints201610.0089.v1.
[5] F. Qi, Bounds for the ratio of two gamma functions, J. Inequal. Appl. 2010
(2010), Article ID 493058, 84 pages; Available online at https://doi.org/
10.1155/2010/493058.
[6] F. Qi, Parametric integrals, the Catalan numbers, and the beta function,
Elem. Math. 72 (2017), no. 3, 103–110; Available online at https://doi.
org/10.4171/EM/332.
[7] F. Qi and B.-N. Guo, Integral representations of Catalan numbers and their
applications, Mathematics 5 (2017), no. 3, Article 40, 31 pages; Available
online at https://doi.org/10.3390/math5030040.
[8] F. Qi and B.-N. Guo, Some properties and generalizations of the Catalan, Fuss, and Fuss–Catalan numbers, Chapter 5 in Mathematical Analysis and Applications: Selected Topics, First Edition, 101–133; Edited by
Michael Ruzhansky, Hemen Dutta, and Ravi P. Agarwal; Published by
John Wiley & Sons, Inc. 2018; Available online at https://doi.org/10.
1002/9781119414421.ch5.
[9] F. Qi and B.-N. Guo, The reciprocal of the weighted geometric mean is
a Stieltjes function, Bol. Soc. Mat. Mex. (3) 24 (2018), no. 1, 181–202;
Available online at https://doi.org/10.1007/s40590-016-0151-5.
Improper Integral, . . . , and Catalan Numbers
115
[10] F. Qi, M. Mahmoud, X.-T. Shi, and F.-F. Liu, Some properties of
the Catalan–Qi function related to the Catalan numbers, SpringerPlus
(2016), 5:1126, 20 pages; Available online at https://doi.org/10.1186/
s40064-016-2793-1.
[11] F. Qi and C. Mortici, Some best approximation formulas and inequalities
for the Wallis ratio, Appl. Math. Comput. 253 (2015), 363–368; Available
online at https://doi.org/10.1016/j.amc.2014.12.039.
[12] F. Qi, X.-T. Shi, M. Mahmoud, and F.-F. Liu, The Catalan numbers:
a generalization, an exponential representation, and some properties, J.
Comput. Anal. Appl. 23 (2017), no. 5, 937–944.
Institute of Mathematics, Henan Polytechnic University
Jiaozuo, Henan, 454010, China
College of Mathematics, Inner Mongolia University for Nationalities
Tongliao, Inner Mongolia, 028043, China
Department of Mathematics, College of Science
Tianjin Polytechnic University
Tianjin, 300387, China
E-mail: qifeng618@gmail.com, qifeng618@hotmail.com, qifeng618@qq.com
URL: https://qifeng618.wordpress.com
View publication stats