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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