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Copyright © 2017, the Authors. Published by Atlantis Press.This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
Comparing Properties of Rising Factorial Function with Properties of Falling Factorial Function
Weida Qin and Wusheng Wang
School of Mathematics and Statistics, Hechi University
Guangxi, Yizhou 546300, P. R. China
Keywords: Falling factorial function; Rising factorial function; Comparing; Property 2000 MSC 26D10, 26D15, 26D20, 45A99
Abstract. This paper firstly introduces the definition of the falling factorial polynomial and the
definition of the rising factorial polynomial, and then proves their properties by analysis methods, finally concludes similar properties of rising factorial function with falling factorial function and
different properties of rising factorial function with falling factorial function.
Introduction
The factorial operation is encountered in many areas of mathematics, notably in combinatorics, algebra, and mathematical analysis. The falling factorial polynomial (sometimes called the
descending factorial, falling sequential product, lower factorial) is defined:
1
0, , .
nn
jx x y x R n N
(1)
Remark 1. From the definition of the falling factorial polynomial, we see that 0
1;x 1 2
; 1 ;x x x x x and
0;
nx when
, 2, 1 ;x n and we have
1
0
1! , , \ , 2, 1
1
nn
j
xxx x j n n N x n R
nx n
(2)
Where denotes the special gamma function.
The rising factorial polynomial (sometimes called the Pochhammer function, Pochhammer
polynomial, ascending factorial, rising sequential product, upper factorial) is defined
1
0, , .
nn
jx x j x R n N
(3)
Remark 2. From the definition of the rising factorial polynomial, we have 0 1;x 1x x
and 2 1 ,x x x
and 0nt when , 2, 1 ;x
and we have
1
0
1! , \ , 2, 1,0 , .
nn
j
x nx nx x j n x R n N
nx
(4)
Remark 3. From the definitions of the falling and rising factorial polynomials, we have
1 1 .n n nnx x n x
(5)
Preliminary Definitions and Properties
Extending the two above definitions from an integer n to an arbitrary real number y, the power
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7th International Conference on Education, Management, Computer and Society (EMCS 2017)Advances in Computer Science Research (ACSR), volume 61
function is defined by [3, 4, 6].
11, , , , \ , 2, 1, ,
1
y x nxx for x R x y R
nx y
(6)
, , , \ , 2, 1,0y
x yx for y R x R
x
(7)
We assume that 0
yx when , 2, 1 ;x y and 0 0, 0y yx when , 2, 1 .x
Remark 4. Using the properties of the Gamma function it is easily seen that 0
yx when
1, 1,x x y , and 1,yx when 0, 0,x x y
We will list some of the properties of the falling factorial function with their proofs.
Lemma 1. ([3], Theorem 2.1.). Assume that the following factorial functions are well defined. 1
,y y
x yx
(8)
1
1
y y kky
x xy k
(9)
1,
y yx y x x
(10)
1 ,x
x x (11)
, , 1,
y yx r x r y x
(12)
,0 1,z
yz yx x z
(13)
,yy z z
x x z x
(14)
where 1 .y t y t y t
Proof. The proof of (8). From (6), we have
2 1
2 1
y x xx
x y x y
1 1 1 1
1 1 1 1
x x x y x
x y x y
11
.1 1
yxy yx
x y
(15)
The proof of (9). From (8), we get
11 1
1.
1
y y y y kk k ky
x x yx xy k
(16)
The proof of (10). From (6), we have
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Advances in Computer Science Research (ACSR), volume 61
1 1 1
.11 1
y yx x y xx x y x
x yx y
(17)
The proof of (11). From (6), we obtain
11 .
1
x xx x
x x
(18)
The proof of (12). By Euler's infinite product.
1
11
1.
1
x
n
nx
xx
n
(19)
For , 1,x r y x , we have
1
11
1 11 11 1
11 1 11 1
x
y
x yn
x yx x y n nx
xx y x
n n
1
11 1
1
1 1
y
n
x y nx y n
x x n
1
11 1 1
1 1
y
n
y y
x n x n
1
11 1 1
1 1
y
n
y y
r n r n
.
yr (20)
The proof of (13). From the log-convexity property of the gamma function.
1
1 ,0 1,z z
za z b a b z
(21)
We obtain
1 1
1 1 1 1
yz x xx
x yz z x y z x
1
1.
1 1
zy
z z
xx
x y x
(22)
The proof of (14). From (6), we have
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Advances in Computer Science Research (ACSR), volume 61
1 1
1 1
y z x x zx
x z y x z
1 1.
1 1
y zx z xx z x
x z y x z
(23)
These complete the proofs.
Main Results
We will list some of the properties of the rising factorial function with their proofs. Theorem 1. ([3], Theorem 2.1.). Assume that the following factorial functions are well defined.
1;y yx yx (24)
1
1
k y y ky
x xy k
(25)
1y yx y x x (26)
2;x
xx
x
(27)
, , ,y yx r x r y x (28)
,0 1,z
yz yx x y (29)
;yy z zx x z x
(30)
Where 1 .x t x t x t
Proof. The proof of (24)
11
1
yy yx y x y
x x xx x
1 1 1 1x y x y x x y
x x
1
1.y
x yy yx
x
(31)
This completes the proof. The proof of (25)
1 1 1
11 1 .
1
k y k y k y y k y ky
x x y x y y y k x xy k
(32)
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Advances in Computer Science Research (ACSR), volume 61
This completes the proof. The proof of (26). From (7), we have
1
1.y y
x y x yx x y x y x
x x
(33)
This completes the proof. The proof of (27). From (7), we have
2.x
xx
x
(34)
This completes the proof.
The proof of (28). By Euler's infinite product.
1
11
1.
1
x
n
nx
xx
n
(35)
For , ,x r y x , we have
1
11 1
11 1
x y
y
xn
xx y x n nx
x yx x y
n n
1 1
1 11 1
1
1 1
y y
n n
x nx n n
y yx y x y n
x x n
1
11
1
1 1
y
n
n
y y
x x n
1
11
1
1 1
y
n
n
y y
r r n
yr (36)
This completes the proof. The proof of (29). From the log-convexity property of the gamma function.
1
1 ,0 1,z z
za z b a b z
(37)
We obtain
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Advances in Computer Science Research (ACSR), volume 61
1yz
z x y z xx yzx
x x
1
.
z zz
yx y x
xx
(38)
The proof of (30). From (7), we have
y z
x y z x zx
x x z
.
y zx z y x z
x z xx z x
(39)
These complete the proofs.
Conclusion
Similar properties of rising factorial function with falling factorial function.
1 1, ;y y y yx yx x yx
1 1, .
1 1
y y kk k y y ky y
x x x xy k y k
Different properties of rising factorial function with falling factorial
function 1 1, ;y y y yx y x x x y x x
21 , ;
x xx
x x xx
, , , ;
y y y yx r x r x r y x
, ,0 1;z z
yz y yz yx r x x z
, .y yy z z y z zx x z x x x z x
Acknowledgments
This research was supported by National Natural Science Foundation of China (Project
No.11561019, 11161018) and Natural Science Foundation of Guangxi Autonomous Region of China (No. 2016GXNSFAA380090) and Hechi Universiry master's degree awarded in 2016 to
build the project fund(No.2016YT003).
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