A NOTE ABOUT THE {Ki(z)}
FUNCTIONS
Branko J. Malešević
In the article [10], A. Petojević verified useful properties of the Ki(z) functions
which generalize Kurepa’s [1] left factorial function. In this note, we present
simplified proofs of two of these results and we answer the open question stated
in [10]. Finally, we discuss the differential transcendency of the Ki(z) functions.
A. Petojević [7, p. 3.] considered the family of functions:
vMm(s; a, z) =
(−1)k−1
z +m+ 1− k
L[s; 2F1(a, k − z,m+ 2; 1− t)],(1)
for ℜ(z) > v−m−2, where v∈N is a positive integer; m∈{−1, 0, 1, 2, . . .} is an integer;
s, a, z are complex variables; L[s;F (t)] is Laplace transform and 2F1(a, b, c;x) is
the hypergeometric function (|x| < 1). D- .Kurepa has considered in the articles
[1, p. 151.] and [2, p. 297.] a complex function defined by the integral:
K(z) =
tz − 1
dt,(2)
for ℜ(z)>0. Especially, forKurepa’s functionK(z), it is true thatK(z)=1M0(1; 1, z),
for ℜ(z)>0, according to [10]. For various of values of parameters v,m, s, a, z from (1),
different special functions, as presented in [10], are obtained. A. Petojević has con-
sidered in the article [10, p. 1640.] the following sequence of functions:
Ki(z) =
1M0(1; 1, z + i− 1)− 1M0(1; 1, i− 1)
1M−1(1; 1, i)
for i∈N and ℜ(z)>−i. On the basis of the definition in (3), the following represen-
tation via Kurepa’s function is true:
Ki(z) =
(i−1)!
K(z + i− 1)−K(i− 1)
for i∈N and ℜ(z)>−i+1. Note that K(0)=0 [2, p. 297.] and therefore K1(z)=K(z)
for ℜ(z)>0. Analytical and differential–algebraic properties of Kurepa’s function
K(z) are considered in articles [1− 12] and in many other articles. On the basis
of well-known statements for Kurepa’s function K(z), using representation (4), in
many cases we can get simple proofs for analogous statements for Ki(z) functions.
For example, it is a well-known fact that it is possible to analytically continue Ku-
repa’s function to a meromorphic function with simple poles at integer points z = −1
and z = −m, (m ≥ 3) [2, p. 303.], [3, p. 474.]. Residues of Kurepa’s function at
these poles have the following form [2]:
Research partially supported by the MNTRS, Serbia, Grant No. 144020.
http://arxiv.org/abs/0704.0068v2
2 Branko J. Malešević
z = −1
K(z) = −1 and res
z = −m
K(z) =
(−1)k−1
, (m≥3).(5)
For Kurepa’s function K(z) the infinite point is an essential singularity [3]. Hence,
on the basis of (4), each function Ki(z) is meromorphic with simple poles at integer
points z=−i and z=−(i+m), (m≥2). On the basis of (4) we have:
z = −(i+m)
Ki(z) =
(i−1)!
· res
z = −(i+m)
K(z + i− 1) = 1
(i−1)!
· res
z = −(m+1)
K(z),(6)
where m = 0 or m≥2. Hence:
z = −i
Ki(z)=−
(i−1)!
and res
z = −(i+m)
Ki(z)=
(i−1)!
(−1)k−1
, (m≥2).(7)
For each Ki(z) function the infinite point is an essential singularity. Therefore, we get
Theorem 3.3. from [10]. Next, it is a well-known fact that for Kurepa’s function
the following asymptotic relation K(x) ∼ Γ(x) is true for real x such that x → ∞
and where Γ(x) is the gamma function [2, p. 299.]. Hence, for fixed i ∈N and real
x>−i+1, on the basis of (4), we get:
Ki(x)
Γ(x+ i− 1)
(i−1)!
· K(i+ x− 1)−K(i− 1)
Γ(x+ i− 1)
(i−1)!
Ki(x)
Γ(x+ i)
(i−1)!
· K(i+ x− 1)−K(i− 1)
(x+ i− 1)Γ(x+ i− 1)
0.(9)
Therefore, we get Theorem 3.6. from [10]. Next we give a solution to the open
problem stated in Question 3.7. in [10]. Namely, the following formula in the article
[8, p. 35.] is given:
K(z) =
Ei(1) + iπ
(−1)zΓ(1 + z)Γ(−z,−1)
,(10)
for values z ∈ C\{−1,−2,−3,−4, . . .} and i =
−1. In the previous formula Ei(z)
and Γ(z, a) are exponential integral and incomplete gamma function respectively [8].
Then, for fixed i∈N and values z∈C\{−i,−i− 1,−i− 2,−i− 3, . . .}, on the basis of
(4) and (10), we get:
Ki(z) =
(i−1)!
K(z + i− 1)−K(i− 1)
Ei(1) + iπ
e(i−1)!
(−1)z+i−1Γ(1 + z + i− 1)Γ(−z − i+ 1,−1)
e(i−1)!
−Ei(1) + iπ
e(i−1)!
− (−1)
i−1Γ(i)Γ(−i+ 1,−1)
e(i−1)!
= (−1)ie−1
Γ(1− i,−1)− (−1)z Γ(1− i− z,−1)Γ(i+ z)
(i−1)!
Therefore, the affirmative answer for Question 3.7. from [10] is true for complex
values z∈C\{−i,−i− 1,−i− 2,−i− 3, . . .}.
A note about the {Ki(z)}∞i=1 functions 3
Finally, at the end of this note let us emphasize one differential–algebraic fact for
the sequence of functions Ki(z). On the basis of the formula (17) from the article
[10], we can conclude that each Ki(z) function satisfies the following recurrence re-
lation (i−1)!Ki(z + 1)− (i−1)!Ki(z) = Γ(z + i). The previous relation can be used
to verify the differential transcendency of these functions as discussed in [11, 12].
Therefore, we can conclude that each Ki(z) function is a differential transcendental
function, i.e. it satisfies no algebraic differential equation over the field of complex ra-
tional functions.
REFERENCES
[1] D- . Kurepa: On the left factorial function !n, Mathematica Balkanica 1 (1971), 147−153.
[2] D- . Kurepa: Left factorial function in complex domain, Mathematica Balkanica 3 (1973),
297 − 307.
[3] D. Slavić: On the left factorial function of the complex argument, Mathematica Balkan-
ica 3 (1973), 472− 477.
[4] A. Ivić, Ž. Mijajlović: On Kurepa problems in number theory, Publications de
l’Institut Mathématique, SANU Beograd, 57, (71) (1995), 19 − 28, available at
http://elib.mi.sanu.ac.yu/pages/browse journals.php .
[5] G.V. Milovanović: Expansions of the Kurepa function, Publications de l’Institut
Mathématique, SANU Beograd 57 (71) (1995), 81− 90, available at home page
http://gauss.elfak.ni.ac.yu .
[6] G.V. Milovanović, A. Petojević: Generalized factorial functions, numbers and poly-
nomials, Mathematica Balkanica 16 (2002), 113− 130.
[7] A. Petojević: The function vMm(s; a, z) and some well-known sequences, Journal of
Integer Sequences, Article 02.1.6, Vol. 5 (2002).
[8] B. Malešević: Some considerations in connection with Kurepa’s function, Univerzitet u
Beogradu, Publikacije Elektrotehničkog Fakulteta, Serija Matematika, 14 (2003), 26−36,
available at http://pefmath.etf.bg.ac.yu/ .
[9] B. Malešević: Some inequalities for Kurepa’s function, Journal of Inequalities in
Pure and Applied Mathematics, Vol. 5, Issue 4, Article 84, (2004), available at
http://jipam.vu.edu.au/ .
[10] A. Petojević: The {Ki(z)}
i=1 functions, Rocky Mountain Journal of Mathematics,
Vol. 36, No. 5, (2006), 1637-1650.
[11] Ž. Mijajlović, B. Malešević: Differentially transcendental functions, accepted
in Bulletin of the Belgian Mathematical Society − Simon Stevin 2007, available at
http://arxiv.org/abs/math.GM/0412354 .
[12] Ž. Mijajlović, B. Malešević: Analytical and differential – algebraic properties of
Gamma function, to appear in International Journal of Applied Mathematics &
Statistics
J.Rassias (ed.), Functional Equations, Integral Equations, Differen-
tial Equations & Applications, http://www.ceser.res.in/ijamas/cont/fida.html
Special Issues dedicated to the Tri-Centennial Birthday Anniversary of L. Euler, 2007.,
available at http://arxiv.org/abs/math.GM/0605430 .
University of Belgrade, (Received : 04/01/2007 )
Faculty of Electrical Engineering, (Accepted : 05/25/2007 )
P.O.Box 35-54, 11 120 Belgrade, Serbia
malesh@eunet.yu, malesevic@etf.bg.ac.yu
http://elib.mi.sanu.ac.yu/pages/browse_journals.php
http://gauss.elfak.ni.ac.yu
http://pefmath.etf.bg.ac.yu/
http://jipam.vu.edu.au/
http://arxiv.org/abs/math.GM/0412354
http://www.ceser.res.in/ijamas/cont/fida.html
http://arxiv.org/abs/math.GM/0605430

  In the article [Petojevic 2006], A. Petojevi\' c verified useful properties
of the $K_{i}(z)$ functions which generalize Kurepa's [Kurepa 1971] left
factorial function. In this note, we present simplified proofs of two of these
results and we answer the open question stated in [Petojevic 2006]. Finally, we
discuss the differential transcendency of the $K_{i}(z)$ functions.

A NOTE ABOUT THE {Ki(z)}
FUNCTIONS
Branko J. Malešević
In the article [10], A. Petojević verified useful properties of the Ki(z) functions
which generalize Kurepa’s [1] left factorial function. In this note, we present
simplified proofs of two of these results and we answer the open question stated
in [10]. Finally, we discuss the differential transcendency of the Ki(z) functions.
A. Petojević [7, p. 3.] considered the family of functions:
vMm(s; a, z) =
(−1)k−1
z +m+ 1− k
L[s; 2F1(a, k − z,m+ 2; 1− t)],(1)
for ℜ(z) > v−m−2, where v∈N is a positive integer; m∈{−1, 0, 1, 2, . . .} is an integer;
s, a, z are complex variables; L[s;F (t)] is Laplace transform and 2F1(a, b, c;x) is
the hypergeometric function (|x| < 1). D- .Kurepa has considered in the articles
[1, p. 151.] and [2, p. 297.] a complex function defined by the integral:
K(z) =
tz − 1
dt,(2)
for ℜ(z)>0. Especially, forKurepa’s functionK(z), it is true thatK(z)=1M0(1; 1, z),
for ℜ(z)>0, according to [10]. For various of values of parameters v,m, s, a, z from (1),
different special functions, as presented in [10], are obtained. A. Petojević has con-
sidered in the article [10, p. 1640.] the following sequence of functions:
Ki(z) =
1M0(1; 1, z + i− 1)− 1M0(1; 1, i− 1)
1M−1(1; 1, i)
for i∈N and ℜ(z)>−i. On the basis of the definition in (3), the following represen-
tation via Kurepa’s function is true:
Ki(z) =
(i−1)!
K(z + i− 1)−K(i− 1)
for i∈N and ℜ(z)>−i+1. Note that K(0)=0 [2, p. 297.] and therefore K1(z)=K(z)
for ℜ(z)>0. Analytical and differential–algebraic properties of Kurepa’s function
K(z) are considered in articles [1− 12] and in many other articles. On the basis
of well-known statements for Kurepa’s function K(z), using representation (4), in
many cases we can get simple proofs for analogous statements for Ki(z) functions.
For example, it is a well-known fact that it is possible to analytically continue Ku-
repa’s function to a meromorphic function with simple poles at integer points z = −1
and z = −m, (m ≥ 3) [2, p. 303.], [3, p. 474.]. Residues of Kurepa’s function at
these poles have the following form [2]:
Research partially supported by the MNTRS, Serbia, Grant No. 144020.
http://arxiv.org/abs/0704.0068v2
2 Branko J. Malešević
z = −1
K(z) = −1 and res
z = −m
K(z) =
(−1)k−1
, (m≥3).(5)
For Kurepa’s function K(z) the infinite point is an essential singularity [3]. Hence,
on the basis of (4), each function Ki(z) is meromorphic with simple poles at integer
points z=−i and z=−(i+m), (m≥2). On the basis of (4) we have:
z = −(i+m)
Ki(z) =
(i−1)!
· res
z = −(i+m)
K(z + i− 1) = 1
(i−1)!
· res
z = −(m+1)
K(z),(6)
where m = 0 or m≥2. Hence:
z = −i
Ki(z)=−
(i−1)!
and res
z = −(i+m)
Ki(z)=
(i−1)!
(−1)k−1
, (m≥2).(7)
For each Ki(z) function the infinite point is an essential singularity. Therefore, we get
Theorem 3.3. from [10]. Next, it is a well-known fact that for Kurepa’s function
the following asymptotic relation K(x) ∼ Γ(x) is true for real x such that x → ∞
and where Γ(x) is the gamma function [2, p. 299.]. Hence, for fixed i ∈N and real
x>−i+1, on the basis of (4), we get:
Ki(x)
Γ(x+ i− 1)
(i−1)!
· K(i+ x− 1)−K(i− 1)
Γ(x+ i− 1)
(i−1)!
Ki(x)
Γ(x+ i)
(i−1)!
· K(i+ x− 1)−K(i− 1)
(x+ i− 1)Γ(x+ i− 1)
0.(9)
Therefore, we get Theorem 3.6. from [10]. Next we give a solution to the open
problem stated in Question 3.7. in [10]. Namely, the following formula in the article
[8, p. 35.] is given:
K(z) =
Ei(1) + iπ
(−1)zΓ(1 + z)Γ(−z,−1)
,(10)
for values z ∈ C\{−1,−2,−3,−4, . . .} and i =
−1. In the previous formula Ei(z)
and Γ(z, a) are exponential integral and incomplete gamma function respectively [8].
Then, for fixed i∈N and values z∈C\{−i,−i− 1,−i− 2,−i− 3, . . .}, on the basis of
(4) and (10), we get:
Ki(z) =
(i−1)!
K(z + i− 1)−K(i− 1)
Ei(1) + iπ
e(i−1)!
(−1)z+i−1Γ(1 + z + i− 1)Γ(−z − i+ 1,−1)
e(i−1)!
−Ei(1) + iπ
e(i−1)!
− (−1)
i−1Γ(i)Γ(−i+ 1,−1)
e(i−1)!
= (−1)ie−1
Γ(1− i,−1)− (−1)z Γ(1− i− z,−1)Γ(i+ z)
(i−1)!
Therefore, the affirmative answer for Question 3.7. from [10] is true for complex
values z∈C\{−i,−i− 1,−i− 2,−i− 3, . . .}.
A note about the {Ki(z)}∞i=1 functions 3
Finally, at the end of this note let us emphasize one differential–algebraic fact for
the sequence of functions Ki(z). On the basis of the formula (17) from the article
[10], we can conclude that each Ki(z) function satisfies the following recurrence re-
lation (i−1)!Ki(z + 1)− (i−1)!Ki(z) = Γ(z + i). The previous relation can be used
to verify the differential transcendency of these functions as discussed in [11, 12].
Therefore, we can conclude that each Ki(z) function is a differential transcendental
function, i.e. it satisfies no algebraic differential equation over the field of complex ra-
tional functions.
REFERENCES
[1] D- . Kurepa: On the left factorial function !n, Mathematica Balkanica 1 (1971), 147−153.
[2] D- . Kurepa: Left factorial function in complex domain, Mathematica Balkanica 3 (1973),
297 − 307.
[3] D. Slavić: On the left factorial function of the complex argument, Mathematica Balkan-
ica 3 (1973), 472− 477.
[4] A. Ivić, Ž. Mijajlović: On Kurepa problems in number theory, Publications de
l’Institut Mathématique, SANU Beograd, 57, (71) (1995), 19 − 28, available at
http://elib.mi.sanu.ac.yu/pages/browse journals.php .
[5] G.V. Milovanović: Expansions of the Kurepa function, Publications de l’Institut
Mathématique, SANU Beograd 57 (71) (1995), 81− 90, available at home page
http://gauss.elfak.ni.ac.yu .
[6] G.V. Milovanović, A. Petojević: Generalized factorial functions, numbers and poly-
nomials, Mathematica Balkanica 16 (2002), 113− 130.
[7] A. Petojević: The function vMm(s; a, z) and some well-known sequences, Journal of
Integer Sequences, Article 02.1.6, Vol. 5 (2002).
[8] B. Malešević: Some considerations in connection with Kurepa’s function, Univerzitet u
Beogradu, Publikacije Elektrotehničkog Fakulteta, Serija Matematika, 14 (2003), 26−36,
available at http://pefmath.etf.bg.ac.yu/ .
[9] B. Malešević: Some inequalities for Kurepa’s function, Journal of Inequalities in
Pure and Applied Mathematics, Vol. 5, Issue 4, Article 84, (2004), available at
http://jipam.vu.edu.au/ .
[10] A. Petojević: The {Ki(z)}
i=1 functions, Rocky Mountain Journal of Mathematics,
Vol. 36, No. 5, (2006), 1637-1650.
[11] Ž. Mijajlović, B. Malešević: Differentially transcendental functions, accepted
in Bulletin of the Belgian Mathematical Society − Simon Stevin 2007, available at
http://arxiv.org/abs/math.GM/0412354 .
[12] Ž. Mijajlović, B. Malešević: Analytical and differential – algebraic properties of
Gamma function, to appear in International Journal of Applied Mathematics &
Statistics
J.Rassias (ed.), Functional Equations, Integral Equations, Differen-
tial Equations & Applications, http://www.ceser.res.in/ijamas/cont/fida.html
Special Issues dedicated to the Tri-Centennial Birthday Anniversary of L. Euler, 2007.,
available at http://arxiv.org/abs/math.GM/0605430 .
University of Belgrade, (Received : 04/01/2007 )
Faculty of Electrical Engineering, (Accepted : 05/25/2007 )
P.O.Box 35-54, 11 120 Belgrade, Serbia
malesh@eunet.yu, malesevic@etf.bg.ac.yu
http://elib.mi.sanu.ac.yu/pages/browse_journals.php
http://gauss.elfak.ni.ac.yu
http://pefmath.etf.bg.ac.yu/
http://jipam.vu.edu.au/
http://arxiv.org/abs/math.GM/0412354
http://www.ceser.res.in/ijamas/cont/fida.html
http://arxiv.org/abs/math.GM/0605430
