The Correlation Properties and Theorems of Backward Difference

Home , Factorial

2018 International Conference on Chemical, Physical and Biological (ICCPB2018) The Correlation Properties and Theorems of Backward Difference

1st Wusheng Wang* 2nd Xuehua Ma School of Mathematics and Statistics, Hechi University, Yizhou, School of Mathematics and Statistics, Hechi University, Yizhou, Guangxi, P. R. China Guangxi, P. R. China [email protected] [email protected]

Abstract—Difference is an important concept in calculus, which is widely used in mathematics, computer, engineering and ecology. This paper is made up of three parts: the first part is the introduction, which mainly introduces the current status of the research on the difference operation; the second part introduces the related properties and theorem about forward difference operation, including linear character, difference of the product , difference of quotient, the high order difference, Leibniz formula, discrete Rolle theorem, the difference mean value theorem, factorial power difference, discrete Taylor formula, etc; the third part is the result of the study, in which the properties and theorems of backward difference are obtained by analoging the basic properties and theorems of the forward difference, and the proof of them is given.

Keywords—Forward Difference, Backward Difference, Discrete Rolle theorem, Difference Mean Value Theorem, Discrete Taylor Formula

I. INTRODUCTION Suppose that a function f(x) is defined on natural number set N. Define the forward difference of the function f(x) by f(x) = f(x + 1) f(x), and the backward difference of the function f(x) by f(x) = f(x) f(x 1) (see [1,2]). Difference can be regarded as the discretization of derivative of continuous function, and the process of many variables can be described by difference∆ (see [3]).− Many scholars have made some meaningful achievements in∆ difference calculus.− − In 2005, Sun and Hu [2] compare the difference operator with the differential operator, and study the basic properties and conclusions of the difference operator and its inverse operator. In 2010, Chen [4] proposed a new definition of fractional order sum, and studied the basic properties of fractional order difference and fractional order sum. In 2014, Zhou and Chao etc. discussed the basic knowledge of difference calculus, difference sum, difference mean value theorem and Newton-Lebnitz formula. This paper firstly discusses the properties and theorems of the forward difference operation, the discrete Rolle theorem, the difference mean value theorem and the discrete Taylor formula. Then the relative properties and theorems of the backward difference are obtained.

II. THE PROPERTIES OF FORWARD DIFFERENCE In this paper, we use the following notation Natural number set: N {n|n = 0,1, … }; partial natural number set: N(a) {a,a+1,a+2,…|a N}; Set of finite natural numbers: N(a, b) {a, a + 1, … , b|a N, b N, a < b}. ≜ ≜ ∈ From the definition≜ of the forward∈ difference,∈ we have the following property of forward difference. For the functions f(x), g(x) defined on N, and real numbers c, d, we have (see [1-5])

cf(x) + dg(x) = c f(x) + d g(x); (1)

∆� f(x)g(x) =� f(x∆)g(x + 1)∆+ f(x) g(x); (2)

( ) ( ) ( ) ( ) ( ) ∆� = � ∆ ; ∆ ( ) ( ) ( ) (3) f x ∆f x g x −f x ∆g x ∆ �g x � g x g x+1 f(x) = ( 1) C f(x r + k); (4) k k r r r=0 k ∆ af(x) +∑bg(x−) = a f(x−) + b g(x); (5) k k k ∆ � f(x)g(x) �= ∆ f(x∆ + k i) g(x) = g(x + k i) f(x), (6) k k k i k−i k k i k−i i=0 i=0 where k is any natural number. ∆ � � ∑ �i �∆ − ∆ ∑ �i �∆ − ∆ In calculus, differential mean value theorem has important applications [6]. For functions defined on N, we can get some similar results.

Definition 1 [2] Suppose that the function x(k) is defined on N(a, b). If x(a) = 0, a is called the node of x(k). If k > a, and x(k) = 0,or x(k 1)x(k) < 0, k is called the node of x(k). x(k) N(1, m) x(k) P Lemma 1 [2]− (The discrete Rolle theorem) Suppose that the function is defined on , if has nodes,then x(k) has Q nodes in N(1, m 1) such that Q P 1. m x(k) N(a, b), ∆ Lemma 2m [2] (The mean value− theorem of mforward≥ m −difference) Soppose that is defined on then there is a c N(a + 1, b 1), such that one of the following inequalities holds

( ) ( ) ∈ − x(c) x(c 1), (7) x b −x a ∆ ≤ b−a ( )≤( ∆) − x(c) x(c 1). (8) x b −x a Definition 2 [2] Descending factorial∆ ≥ powerb−a function≥ ∆ is defined− by

x( ) x(x 1)(x 2) … x (n 1) = (x j), (9) n n−1 j=0 ( ) ≜ − − � − − � ∏ − x = (x + j) , (10) ( )( )…( ) −n 1 n −1 ≜ x+n x+n−1 x+1 ∏j=0 where n is any natural number, x( ) = 1. 0 Lemma 3 [2] For any n, k N, n > k, we have

∈x( ) = nx( ), (11) n n−1 ∆ x( ) = n(n 1) … (n k + 1)x( ). (12) k n n−k Lemma 4 [2] If f(x) is a polynomial∆ of degree− n for− x, then

( ) ( ) ( ) f(x) = f(0) + f(0)x( ) + x( ) + + x( ) + x( ). (13) 2 ! (n−1 )! n ! 1 ∆ f 0 2 ∆ f 0 n−1 ∆ f 0 n ∆ 2 ⋯ n−1 n Lemma 5 [2] (Discrete Taylor formula) Suppose that the function x(k) is defined on N(k ). For any n, k N, n 1, we have

0 ( )( ) ∈ ≥ x(k) = x(k ) + (k j 1)( ) x(j). (14) ! j ( )! n−1 k−k0 j 1 k−n n−1 n ∑j=0 j ∆ 0 n−1 ∑j=k0 − − ∆ The correlation properties and theorems of backward difference From the definition of the forward difference, we have the following property of forward difference. For the functions f(x), g(x) defined on N, and real numbers c, d, we have the following properties

cf(x) + dg(x) = c f(x) + d g(x), (15)

∇� f(x)g(x) =� f(x)∇ g(x) + ∇f(x)g(x 1) = g(x) f(x) + g(x)f(x 1), (16)

( ) ( ) ( ) ( ) ( ) ∇� = � ∇ ∇ − ∇ ∇ − ( ) ( ) ( ) , (17) f x ∇f x g x −f x ∇g x ∇ �g x � g x g x−1 f(x) = ( 1) C f(x r), (18) k k r r r=0 k ∇ af(x∑) + bg−(x) = a −f(x) + b g(x), (19) k k k ∇ � f(x)g(x) =� ∇ f(x) ∇ g(x i) = g(x) f(x i) . (20) k k k i k−i k k i k−i i=0 i=0 Theroem 1 (The discrete Rolle theorem)∇ � Suppose� ∑ that� thei �∇ function∇ x(k)− is defined∑ on�i N�∇(0, m).∇ If x(k) −has P nodes, then x(k) has Q nodes in N(1, m) such that Q P 1. m m m = 1 x(k) m N(0,1)m x(k) x = 1. x(0) = 0 and x(1) = 0, x(k) ∇ Proof. When . has two points in≥ − , is defined only at If then has two nodes, and x(1) = x(1) x(0) = 0, x(k) has one node. The relation Q P 1 holds. In other cases, x(k) has one node at most, the relation Q P 1 is also set up.∇ ∇ − ∇ m ≥ m − m r, Q P 1 r N. Suppose that the conclusion ofm the≥ theoremm − is set up for i.e. holds for any ≤ r ≥ r − ∈

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Consider m = r + 1 cases. If P = P , then Q Q P 1 P 1, Q P 1 holds.

r +r+1 1 r r+1P =r P +r 1 > P r+1P = 0 r+1 Qr+1 P 1 If is a node, then ≥ ≥ − ≥ . If − , then≥ − . P 1. r+1 r r r m+1 m+1 When Consider the following several cases. ≥ − x(r) = x(r + 1) = 0, P = P + 1 and Q = Q + 1 Q P 1 Q P 1 If m ≥ then , from we see . x(r + 1) = 0, x(r) > 0r+1 r a r+1 r rand ar r, P =r+1 P ,Q r+1 P 1 = P 1. When . Suppose that is the maximum node, ≥ − then ≥ − If x(a) = 0, then a < r and x(a + 1) > 0. Otherwise, from x(r) > 0, x(k) has nodes in a + 1 k r, a is not the maximum node of x(k) for k r. So x(a + 1) > 0, x(r + 1) < 0. x(k) has one node ≤in a + 1 a k r r +a 1≥ at aleast.− So rQ− Q + 1 P = P 1. If x(r + 1) = 0, x(r) < 0, then x(r + 1) < 0. We also can obtain ≤Q ≤ P 1. ≤ ∇ ∇ ∇ ≤ ≤ r+1 ≥ x(r + 1)x(r) < 0. Q P 1. a When≥ r r+1 − We also can obtain ∇ r+1 ≥ r+1 − r+1 r+1 x(k) N(a, b), Theorem 2 (The mean value theorem of backward≥ difference)− Soppose that is defined on then there is a c N(a + 1, b 1), such that one of the following inequalities hold

( ) ( ) ∈ − x(c + 1) x(c); (21) x b −x a ∇ ≤ b−a ≤( ∇) ( ) x(c + 1) x(c). (22) x b −x a Theorem 3 For any n, k N,∇ n > k, we≥ haveb−a ≥ ∇

∈ x( ) = nx( ), (23) ��n�� n−1�� x( ) = n(n 1) … (n k + 1)x( ), (24) k ��n�� n−k�� where x( ) x(x + 1)(x +∇ 2) (x + n− 1), x( ) −= 1. �� n 0 ， Theorem 4 ≜If the function f(x) is⋯ a polynomial− of degree n in x then

( ) ( ) ( ) f(x) = f(0) + f(0)x( ) + x( ) + + x( ) + x( ). (25) 2 ! (n−1 )! n ! ��1�� ∇ f 0 ��2�� ∇ f 0 ��n−1�� ∇ f 0 ��n�� ∇ 2 ⋯ n−1 n Theorem 5 (Discrete Taylor formula) Suppose that the function x(k) is defined on N(k n + 1). For any k N(k n + 1), n 1, we have 0 − ∈ 0 − ≥ ( ) x(k) = x(k ) + (k j + 1) x(j). (26) ! ȷ̅ ( )! n−1 k−k0 j 1 k n−1�� n ∑j=0 j ∇ 0 n−1 ∑j=k0+1 − ∇ Proof. When n = 1. x(k) = x(k ) + x(j) = x(k). k Suppose that0 Discrete∑j=k 0Taylor+1 ∇ formula holds for natural number n, i.e.

( ) x(k) = x(k ) + (k j + 1) x(j). (27) ! ȷ̅ ( )! n−1 k−k0 j 1 k n−1�� n ∑j=0 j ∇ 0 n−1 ∑j=k0+1 − ∇ Let denotes the difference on j. By Eq. 16 , we have

j ∇ (k j) x(j) = (k j + 1) x(j) + (k j) x(j). (28) n� n n� n+1 n� n j j From Eq. 28, we obtain ∇ � − ∇ � − ∇ ∇ − ∇

(k j) x(j) = (k k ) x(k ) (k j + 1) x(j). (29) k n� n n� n k n� n+1 j=k0+1 j 0 0 j=k0+1 From Eq. 23, we get ∑ ∇ − ∇ − − ∇ − ∑ − ∇

(k j) = n(k j + 1) . (30) n� n−1�� j From Eq. 30, we obtain ∇ − − −

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(k j + 1) x(j) = (k j) x(j). (31) k n−1�� n 1 k n� n j=k0+1 j=k0+1 j So ∑ − ∇ − n ∑ ∇ − ∇

(k j + 1) x(j) = (k j) x(j) ( )! ! 1 k n−1�� n 1 k n� n n−1 ∑j=k0+1 − ∇ − n ∑j=k0+1 ∇j − ∇ = (k k ) x(k ) + (k j + 1) x(j) . (32) ! 1 n� n k n� n+1 0 0 j=k0+1 Substituting Eq. 32 in Eq. 27, wen have� − ∇ ∑ − ∇ �

( ) x(k) = x(k ) + ((k k ) x(k ) + (k j + 1) x(j)) ! ȷ̅ ! n−1 k−k0 j 1 n� n k n� n+1 j=0 j 0 n 0 0 j=k0+1 ( ) ∑ ∇ − ∇ ∑ − ∇ = x(k ) + (k j + 1) x(j). (33) ! ȷ̅ ! n k−k0 j 1 k n� n+1 ∑j=0 j ∇ 0 n ∑j=k0+1 − ∇ Eq. 33 is the discrete Taylor formula Eq. 26 for natural number n + 1. By mathematical induction, the theorem 5 is proved.

III. SUMMARY This paper introduces the current status of the research on the difference operation, and the related properties and theorem about forward difference operation, including linear character, difference of the product, difference of quotient, the high order difference, Leibniz formula, discrete Rolle theorem, the difference mean value theorem, factorial power difference, discrete Taylor formula, etc. Finally, we discuss the properties and theorems of backward difference by analoging the basic properties and theorems of the forward difference, and give the proof of them.

ACKNOWLEDGEMENTS 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).

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