复数域中Gamma函数与不完整Gamma函数导数的研究
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"这篇研究论文探讨了非正整数下Gamma函数Γ(α)和不完整Gamma函数Γ(α,z)的导数的一些性质。作者 Zhongfeng Sun 和 Huizeng Qin 提出了一些递归关系，特别是关于Γ(α)和Γ(α,z)的高阶导数的表达式。他们发现Γ(α)的n阶导数在α = -m时可以表示为低阶导数的线性组合，而Γ(α,z)的n阶偏导数可以用Γ(α,z)的低阶偏导数和基本函数的组合来表示。这些结果有助于确定与Γ(α)和Γ(α,z)相关的特殊积分的封闭形式，这些积分可以用Riemann zeta函数和特殊常数来表达。
II. GAMMA FUNCTION AND INCOMPLETE GAMMA FUNCTION
Gamma函数Γ(α)是复数α大于零时的积分定义，它是一个重要的特殊函数，在数学和物理中有广泛应用。不完整Gamma函数Γ(α,z)则是一个扩展，允许在复平面上的α值，并通过一个截止点z将积分限制在有限区间内。这两个函数在概率论、统计学和量子场论等领域中扮演着关键角色。
III. DERIVATIVES OF GAMMA AND INCOMPLETE GAMMA FUNCTIONS
对于非正整数α，论文提出了一种递归方法来计算Γ(α)和Γ(α,z)的导数。对于负整数α = -m，Γ(α)的n阶导数dnΓ/dαn可以通过djΓ/dαj(1)（j = 0,1,...,n+1）的线性组合来表示，其中1是Γ(α)的导数在α = 1处的值。类似地，Γ(α,z)的n阶偏导数∂nΓ/∂αn(−m,z)可以表示为∂jΓ/∂αj(1,z)（j = 0,1,...,n+1）的组合以及一些基本函数。这些关系简化了计算这些导数的过程。
IV. SPECIAL INTEGRALS AND RIEMANN ZETA FUNCTION
利用上述结果，作者能够闭合形式地表示与Γ(α)和Γ(α,z)相关的特殊积分。这些积分可以转化为Riemann zeta函数的形式，这是一个极其重要的无穷级数，与许多深奥的数学问题相关。此外，它们还涉及到一些特殊的常数，如圆周率π、自然对数的底e等。
V. HURWITZ ZETA FUNCTION AND DIGAMMA FUNCTION
论文还提到了Hurwitz zeta函数，这是Riemann zeta函数的一个推广，允许实数或复数的非正参数。同时，Digamma函数（即Γ(α)的导数）在解决与Γ(α)相关的问题时也起到了关键作用。通过这些工具，作者能够深入理解非正整数处Gamma函数的性质。
VI. CONCLUSION
该研究为理解和计算非正整数α下的Gamma函数和不完整Gamma函数的导数提供了新的视角和工具。这些成果不仅在纯数学中具有理论价值，而且在处理涉及这些函数的实际问题时也可能具有实用意义，例如在物理建模和工程计算中。
关键词：不完整Gamma函数，Gamma函数，Neutrix极限，Hurwitz zeta函数，Digamma函数。"
Some Results on the Derivatives of the Gamma
and Incomplete Gamma Function for Non-positive
Integers
Zhongfeng Sun and Huizeng Qin
Abstract—This paper is concerned with some recursive re-
lations of the derivatives of the Gamma function Γ(α) and
incomplete Gamma function
Γ(
α, z
)
for the complex value
of α. In particular,
d
n
Γ
dα
n
(−m)(n, m = 0, 1, 2, . . .) can be
expressed as linear forms in
d
j
Γ
dα
j
(1)(j = 0, 1, . . . , n + 1)
while
∂
n
Γ
∂α
n
(−m, z) can be represented as the combination of
∂
j
Γ
∂α
j
(1, z)(j = 0, 1, . . . , n + 1) and the elementary functions.
With the aid of these results, we can establish the closed forms
of some special integrals associated with Γ(α) and Γ(α, z),
which can be expressed by the Riemann zeta functions and
some special constants.
Index Terms—Incomplete Gamma function, Gamma func-
tion, Neutrix limit, Hurwitz zeta function, Digamma function.
I. INTRODUCTION
T
HE incomplete Gamma function Γ(α, z) was defined
by the following integral [1]
Γ(α, z) =
∞
z
t
α−1
e
−t
dt, (1)
where α ∈ C, |arg(z)| < π, z = 0 and the Gamma function
Γ(α) was defined by
Γ(α) =
∞
0
t
α−1
e
−t
dt, Re(α) > 0, (2)
where α can be extended to all complex numbers except non-
positive integers after an appropriate analytic continuation. In
this paper, we assume that |arg(z)| < π, z = 0 and α ∈ C.
For convenience, we introduce the following notations
N := {1, 2, 3, . . .}, N
0
:= N∪{0}, N
−
0
:= {0, −1, −2, . . .}.
Denote
Γ
(n)
(α) =
d
n
dα
n
Γ(α) =
∞
0
t
α−1
e
−t
ln
n
tdt,
Γ
(n)
(α, z) =
∂
n
∂α
n
Γ(α, z) =
∞
z
t
α−1
e
−t
ln
n
tdt,
(3)
for n ∈ N
0
.
In recent years the issue of the neutrix limit for dealing
with special functions such as the Gamma and incomplete
Gamma function [2]∼[7], the Beta and incomplete Beta
Manuscript received December 28, 2016; revised May 9, 2017. This work
is supported by National Natural Science Foundation of China under Grant
No. 61379009.
Zhongfeng Sun is with School of Science, Shandong University of
Technology, Zibo, Shandong, 255000, P. R. China.
Huizeng Qin is with School of Science, Shandong University of Technol-
ogy, Zibo, Shandong, 255000, P. R. China. Huizeng Qin is the corresponding
author. (e-mail: qin hz@163.com(H.Z.Qin))
function [8]∼[10] and the Hurwitz zeta function [11] for
non-positive integers have attracted much attention. Using
the neutrix limit, Γ
(n)
(−m) (n, m ∈ N
0
) can be defined by
the following neutrix calculus [2]∼[5]
Γ
(n)
(−m) = N − lim
ε→0+0
∞
ε
t
−m−1
e
−t
ln
n
tdt, (4)
where N is the neutrix [12]. Fisher and Kılıc¸man [2]
discussed some recursive relations of the derivatives of the
Gamma function for non-positive integers. However, there
are some mistakes expressed in Theorem 4, 5 in [2] and
the corresponding corrections will be shown in Remark 2.4
and 2.5 in this paper. Fisher et al. [6], [7] used the neutrix
limit to establish the definition of the lower incomplete
Gamma function γ(α, x) and the locally summable function
γ(α, x
+
) = H(x)γ(α, x) for the non-positive integer α,
where H(x) denotes the Heaviside’s function. Subsequently,
¨
Ozc¸a
˘
g et al. [3] studied the partial derivatives of γ(α, x) and
γ(α, x
+
) for non-positive integers. Lin et al. [4] proved that
∂
n
γ
∂α
n
(0, x) can be expressed by
∂
n+1
∂α
n+1
γ(1, x)(n ∈ N
0
) and
the elementary functions.
The paper is structured as follows. Section II establishes
some recursive relations of the derivatives of the Gamma
function. Specially, Γ
(n)
(−m)(n, m ∈ N
0
) can be repre-
sented as linear forms in Γ
(j)
(1)(j = 0, 1, . . . , n + 1).
Section III describes the recurrence formula for the par-
tial derivatives of the incomplete Gamma function. In par-
ticular, Γ
(n)
(−m, z) can be expressed by Γ
(j)
(1, z)(j =
0, 1, . . . , n + 1) and the elementary functions. In Section IV,
we present some examples to investigate the closed forms for
some special integrals and the arbitrary precision calculation
of Γ
(n)
(α) and Γ
(n)
(α, z). A final conclusion is drawn in
Section V.
II. THE RECURSIVE FORMULAS OF THE DERIVATIVES OF
THE GAMMA FUNCTION
Theorem 2.1 Let n ≥ 1 be an integer. Then the recurrence
relation of Γ
(n)
(α)(α ∈ C\N
−
0
) can be expressed as follows,
Γ
(n)
(α) =
ψ(α)Γ(α), n = 1,
(n − 1)!
n−2
k=0
(−1)
n−k
k!
×ζ(n − k, α)Γ
(k)
(α), n > 1,
+ψ(α)Γ
(n−1)
(α)
(5)
where ζ(s, α) is the Hurwitz zeta function defined by
ζ(s, α) =
∞
l=0
1
(l + α)
s
(Re(s) > 1), (6)
IAENG International Journal of Applied Mathematics, 47:3, IJAM_47_3_04
______________________________________________________________________________________
(Advance online publication: 23 August 2017)
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