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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 [email protected] 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 , , . n n j x x y x Rn N (1) Remark 1. From the definition of the falling factorial polynomial, we see that 0 1; x 1 2 ; 1; x xx xx and 0; n x when ,2,1; x n and we have 1 0 1 ! , , \ , 2, 1 1 n n j x x x x j n n Nx n R n x 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 , , . n n j x x j x Rn N (3) Remark 2. From the definition of the rising factorial polynomial, we have 0 1; x 1 x x and 2 1, x xx and 0 n t when ,2,1; x and we have 1 0 1 ! , \ , 2, 1,0 , . n n j x n x n x x j n x R n N n x (4) Remark 3. From the definitions of the falling and rising factorial polynomials, we have 1 1 . n n n n x 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 369 7th International Conference on Education, Management, Computer and Society (EMCS 2017) Advances in Computer Science Research (ACSR), volume 61

Comparing Properties of Rising Factorial ... - Atlantis Press(39) y z. x z y x z x z x x z x * * * * These complete the proofs. Conclusion. Similar properties of rising factorial function

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Page 1: Comparing Properties of Rising Factorial ... - Atlantis Press(39) y z. x z y x z x z x x z x * * * * These complete the proofs. Conclusion. Similar properties of rising factorial function

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

[email protected]

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

369

7th International Conference on Education, Management, Computer and Society (EMCS 2017)Advances in Computer Science Research (ACSR), volume 61

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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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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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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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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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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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