prove the property of beta function
Answers
Answer:
A Note on the Beta Function And Some Properties
of Its Partial Derivatives
Nina Shang, Aijuan Li, Zhongfeng Sun, and Huizeng Qin
Abstract—In this paper, the partial derivatives Bp,q(x, y) =
∂
q+p
∂xp∂yq B(x, y) of the Beta function B(x, y) are expressed in
terms of a finite number of the Polygamma function, where p
and q are non-negative integers, x and y are complex numbers.
In particular, Bp,q(x, y) can be expressed by the Riemann zeta
function if x is equal to n or n +
1
2
and y is equal to m
or m +
1
2
, where n and m are integers. Furthermore, many
integral functions associated with B(x, y) and Bp,q(x, y) can
be expressed as the closed forms.
Index Terms—Riemann zeta function, Beta function, Gamma
function, Polygamma function, Digamma function, closed form.
I. INTRODUCTION
I
N mathematics, the Beta function was studied by Euler
and Legendre as a special function. It is usually defined
by
B(x, y) = ∫ 1
0
t
x−1
(1 − t)
y−1
dt (1)
for Rex > 0 and Rey > 0. It is often applied in many fields
such as mathematical equations and probability theory. Its
definition was extended to complex numbers values of x and
y by using the neutrix limit in [1]. Furthermore, the partial
derivatives of the Beta function on the complex numbers x
and y exist a close relationship with many special functions
and special integrals. For example, the following relation was
proved in [2]
∫ 1
0
t
x−1
(1 − t)
y−1
lnp
tlnq
(1 − t)dt = Bp,q(x, y) (2)
for integers p, q > 0 and q + Rex, p + Rey > 0, where
Bp,q(x, y) = ∂
p+q
∂xp∂yq B(x, y). Moreover, K. S. KOlbig gave ¨
the closed expressions of (2) for x = 0 and y = 1 in [3].
Putting t = sin2 u in (2), we have
∫ π
2
0
sin2x−1 u cos2y−1 u lnp
sin u lnq
cos udu
= 2−p−q−1Bp,q(x, y).
(3)
K. S. KOlbig also gave the closed expression of (3) for ¨ x =
y =
1
2
in [4]. Note that the most effective way of computing
(1) and (2) is based on power series expansion, even for
integral equations[5]. Moreover, many scholars have studied
different Integro-differential equation by different methods
in [6] and [7].
In this paper, we concern about the recurrence formulas
and the closed forms of Bp,q(x, y) in (1) and (2). Also, we
Manuscript received March 22, 2014. This work is supported by National
Natural Science Foundation of China under Grant No. 61379009.
Nina Shang is with School of Science, Shandong University of Technology, Zibo, Shandong, 255049, P. R. China.
Aijuan Li, Zhongfeng Sun and Huizeng Qin are with School of
Science, Shandong University of Technology, Zibo, Shandong, 255049,
P. R. China. Huizeng Qin is the corresponding author. (e-mail: qinhz [email protected](H.Z.Qin))
consider the existence condition of the closed forms and the
representations of Bp,q(x, y).
From
1
B(x,y)
=
2
x+y−1
(x+y−1)
π
∫ π/2
0
cos(x − y)t cosx+y−2
tdt,
(see [8]) and Γ(z)Γ(1 − z) = π
sin πz
, where Γ(z) is the
Gamma function, we see that the following formulas
∫ π
0
(
2 cos θ
2
)2x
cos yθdθ
=
B(2x + 1, y − x) sin π(y − x),
x − y < 0, Rex > −
1
2
,
−B(2x + 1, −x − y) sin π(x + y),
x + y < 0, Rex > −
1
2
,
(4)
and
∫ π
0
t
q
(
2 cos t
2
)2x
cos (
yt +
qπ
2
)
lnp
(
2 cos t
2
)
dt
=
(−1)p
2
p
∑p
j=0
π
p−jC
j
p
∑q
k=0
π
kC
k
q
∑
j
u=0
(−1)u2
uC
u
j
·
Bu,q+j−u−k(2x + 1, y − x) sin aπ,
a =
2y−2x+p+k−j
2
, Re(x − y) < 0,
−
(−1)q
2
p
∑p
j=0
π
p−jC
j
p
∑q
k=0
(−π)
kC
k
q
∑
j
u=0
(−1)j−u2
uC
u
j
·
Bu,q+j−u−k(2x + 1, −x − y) sin bπ,
b =
2x+2y+p+k−j
2
, Re(x + y) < 0,
(5)
exist for non-negative integers p, q and Rex > −
1
2
.
If we can establish the closed forms of Bp,q(x, y), then
the above integrals can be expressed as the associated closed
forms. It is well known that an expression is said to be the
closed form expression if it can be expressed analytically in
terms of a finite number of the Riemann zeta function and
some special constants γ, π, where γ denotes Euler .