Theory of Generalized k-difference Operator and Its Application in Number Theory

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1 Internationa Journa of Mathematica Anaysis Vo. 9, 2015, no. 19, HIKARI Ltd, Theory of Generaized -Difference Operator and Its Appication in Number Theory V. Chandrasear Department of Mathematics, S.K.P. Engineering Coege Tiruvannamaai , Tami Nadu, S. India G. Britto Antony Xavier Department of Mathematics, Sacred Heart Coege Tirupattur , Tami Nadu, S. India R. Vijayaraj Department of Mathematics, Arunai Engineering Coege Tiruvannamaai , Tami Nadu, S. India Copyright c 2014 V. Chandrasear, G. Britto Antony Xavier and R.Vijiyaraj. This is an open access artice distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the origina wor is propery cited. Abstract In this paper, we define the generaized -difference operator and present the discrete version of Leibnitz theorem according to the generaized -difference operator. Aso, by defining its inverse, we obtain the sum of product of arithmetic and geometric progressions in the fied of Numerica Anaysis. Mathematics Subject Cassification: 39A70, 47B39, 97N40 Keywords: Factoria -Difference Operator, Generaized Factoria, Poynomia

2 956 V. Chandrasear, G. Britto Antony Xavier and R. Vijayaraj 1 Introduction The theory of Difference equations is based on the operator defined as u) = u + 1) u), N = {0, 1, 2,...}. 1) Even though many authors 1, 7 9 have suggested the definition of the difference operator as u) = u + ) u), 0, ), 2) no significant progress too pace on this ine. But, in , by taing the definition of as given in 2), the theory of difference equations was deveoped in a different direction and many interesting resuts were obtained in Number Theory 4. Jerzy Popenda 2, whie discussing the behavior of soutions of particuar type of difference equation, defined α as α u) = u + 1) αu). 3) This definition of α is being ignored for a ong time. But, recenty, M.M.S.Manue, V.Chandrasear and G.Britto Antony Xavier 5, 6, have generaized the definition of α by α) defined as α) u) = u + ) αu) 4) for the rea vaued function u) and 0, ) and aso obtained the soutions of certain type of generaized α-difference equations, in particuar generaized cairaut s α-difference equation, generaized Euer α-difference equation and formua for the sum of severa types of arithmetic-geometric progressions in Number Theory using the generaized α-bernoui poynomia, which is a soution of the generaized α-difference equation as u + ) αu) = n n 1, for n N1). With this bacground, in this paper, we define the generaized -difference operator and obtain some resuts, reations and theorems. Aso, we derive the formua for sum of product of factorias and arithmetic progressions in the fied of Numerica Anaysis using the inverse of generaized -difference operator. Throughout this paper, the foowing assumptions are used: i) N = ){0, 1, 2, 3,...}, ii) N j) = {j, j +, j + 2,...}, n iii) r v) n! = r!n r)!, iv) 1 is the integer part of and ) u) = ) u) ) ) ) uj), j is the upper integer part of.

3 -Difference operator and its appication in number theory Generaized -Difference Operator and its Reation with and E In this section, we present some basic definitions and preiminary resuts which wi be usefu for further subsequent discussions. Definition 2.1 Let u) be a rea vaued function defined on 0, ). Then the generaized -difference operator ) is defined as ) u) = u + ) u), 0, ), 0, ). 5) Remar 2.2 i) If α is a constant, then the generaized α-difference operator α) assumes the reation α) u) = u + ) αu). ii) Repacing by ± n in 5), we obtain ) u ± n) = u ± n + ) u ± n), for appropriate choosing n N. The foowing emma is the reation between ), and E. Lemma 2.3 i) If E is the usua shift operator, then E u) = ) + )u) = 1 + ) u) = 1 + )u). 6) ii) If m is positive integer, then + m) )u) = m i=0 ) m i u). 7) i iii) If c 1, c 2 are any two non-zero scaars and u), v) are any two rea vaued functions defined on 0, ), then ) c 1 u) + c 2 v) = c 1 ) u) + c 2 ) v). iv) If 0, ) and n is a positive integer, then n )u) = n v) If is, i = 1, 2,...n are positive reas, then ) n ) r u + n r)). 8) r n))u) = n + i ))u). 9) i=1 vi) If n is a positive integer, then n) u) = 1 + ) n )u). 10)

4 958 V. Chandrasear, G. Britto Antony Xavier and R. Vijayaraj The foowing is the discrete version of Leibnitz theorem according to ). Theorem 2.4 If u) and v), 0, ), are any two rea vaued functions, then n )u)v) = n Proof. Define the shift operator E 1 and E 2 as ) n n r r r )u) ) n r v + r) ). E 1u)v) = u + )v) and E 2u)v) = u)v + ). 11) Hence, we obtain Let us denote E = E 1E 2. 12) ) ) 1 = E 1 and ) ) 2 = E 2 13) for the functions u) and v), respectivey. This impies ) = E 1E 2. From 6), we obtain ) = ) ) 1 E 2 ) 2, where E 2 1 = ) 2 2, Theorem 2.5). Hence, n )u)v) = ) 2 + ) ) 1 E 2 n u)v). 14) The proof foows by Binomia Theorem, 11), 13) and 14). Lemma 2.5 If 0, ) and n is a positive integer, then E n u) = n ) n n r r r )u). 15) Proof. The proof foows from 6) and Binomia Theorem. The foowing is discrete version of generaized Binomia theorem invoving ). Theorem 2.6 If m and n are any two positive integres, then + n) m = n r ) ) n r 1) n r n+t r + n t) m. 16) r t t=0 Proof. The proof foows by taing u) = m in 15) and using 8).

5 -Difference operator and its appication in number theory 959 Lemma 2.7 Let u), 0, ), be a rea vaued function and 0, ). Then x r ur) = e x x ) r! r e u0). 17) Proof. From the shift operator, E r u0) = ur) and binomia theorem, we find e x E x r ur) u0) =. 18) r! r The proof foows from 6) and 18). The foowing theorem is the generaized version of Montmorte s theorem with reference to ). Theorem 2.8 If the series u)x, expressed as =0 u)x = =0 =0 Proof. From the shift operator E, we have N converges, then it can be x ) u0). 19) 1 x ) +1 u)x = =0 x E u0) = 1 x E ) 1 u0). =0 19) foows from 6) and the above reation. 3 The Generaized Factoria In this section, we define the generaized factoria and obtain the reation between the generaized -difference operator and generaized factoria. Definition 3.1 If 0, ) and, ), then the Generaized Factoria denoted by )! is defined as where j =. )! = ) 2) j, 20) Theorem 3.2 Let 0, ) and, ). Then Proof. The proof foows from 5) and 20). ) )! = )! 21)

6 960 V. Chandrasear, G. Britto Antony Xavier and R. Vijayaraj Lemma 3.3 Let n) = ) 2)... n 1)) be the generaized poynomia factoria. Then ) n) Proof. The proof foows from 5) and n 1) = n 1) 1 ) n 1)). 22) = ) 2) n). Theorem 3.4 If m and n are any two positive integer, then m ) n+m i) m i r t m i t=1 m ) ) n = n + m i) r t t=1 r m+1 i i=1 r m+1 i =1 r m+1 i n+m 1) m r p p=1 Proof. The proof foows by induction on m. ) +1 Coroary 3.5 If n is any positive integer, then ) ) ) ) n n +1 =. 23) t=1 n t ) n t t. 24) Proof. The proof foows by substituting m = 1 in 23). Note that when is mutipe of, ) ) = 0 and +1) = 0, hence bothsides of 24) are zero. 4 Inverse of the Operator ) In this section, we define the inverse of the operator ) and obtaining some theorems invoving the inverse of generaized -difference operator. Definition 4.1 The inverse of the generaized -difference operator is denoted by ) is defined as foows. If )v) = u), then ) u) = v) ) ) vj), 25) where vj) is constant for a N j). Theorem 4.2 If 0, ) and, ), then ) u) j = ) r 1) u r). 26)

7 -Difference operator and its appication in number theory 961 Proof. The proof foows from Definition 4.1, and the reation ) ) r 1) u r) = u). Theorem 4.3 Let λ 1, 2 and p) be a function in. Then ) r 1) λ r P r) { } λ = λ 1) λ+ ) 2 λ 1) + λ+2 2 ) 2 3 λ 1) + λ+3 3 ) 3 4 λ 1) + P ). 4 j Proof. For the function F ), we find hence, we obtain 27) ) λ F ) = λ λ E )F ) = λ P ), 28) λ E ) 1 P ) = F ). 29) From 28) and Definition 4.1, we find ) λ P )) = λ ) F ) ) λ j F j), and hence by 29), we obtain ) λ P )) = λ λ E ) 1 ) P ) ) λ j λ E j) 1 P j). 30) The proof now foows from 26), 30) and Binomia theorem. 5 Appications In section, we estabish the formua for the sum of product of factorias and arithmetic progression and arithmetic geometric progression in number theory as an appication of ). Lemma 5.1 Let 0, ) and, ). Then Proof. The proof foows from 21) and Definition 4.1. ) )! = )!. 31)

8 962 V. Chandrasear, G. Britto Antony Xavier and R. Vijayaraj Theorem 5.2 Let 0, ) and, ). Then ) r 1) r) )! = 1 ) )! ) j )!. 32) Proof. The proof foows by taing u) = )! in 26) and 31). Exampe 5.3 Substituting = 70, = 3 and j = 1 in 32), we get 67 3 )! + 67) 1) )! + 67) 2) )! + 67) 22) )! = ) 3)! 70 3) 3 )!1) 3 )! = X Theorem 5.4 If n is any positive integer, then +1) n nc t n t t ) t=1 Proof. The proof foows by 24) and Definition 4.1. Theorem 5.5 If n is any positive integer, then n r) r+1) ) r 1) nc t n t t t=1 Proof. The proof foows from 33) and Definition 4.1. Exampe 5.6 In 34), by taing n = 2, we find that 2 r) r+1) ) r 1) nc t n t t t=1 = ) ) n. 33) = ) ) = ) ) When, = 61, = 3 and j = 1, we get 58) 20) 3 58) 0) 3 357) + 55) 19) 3 58) 1) 3 339) + + 1) 1) 3 58) 19) 3 15) = 61 3) 20) ) 20) = X Theorem 5.7 Let J = ) r 1) r) r). Then, for a N J), = ) ) ) n. 34) j 2. J j ). 35) j

9 -Difference operator and its appication in number theory 963 Proof. From the equation 5) and Definition 4.1, we have ) ) The proof foows from 26) and 36). = ). 36) Exampe 5.8 In 35), substituting = 62, = 3, j = 2 and J = 1, we get 59) 0) 3 59) 20) ) 1) 3 56) 19) ) 2) 3 53) 18) ) 20) 3 1) 0) 3 6 Concusion = )21) 3 59) 20) 3 1) 1) 3 = X By seecting arge vaue for and sma vaue for > 0, one can find the sum of severa series easiy using the Theorem mentioned above. References 1 R. P. Agarwa, Difference Equations and Inequaities, Marce Deer, New Yor, Jerzy Popenda and Bazij Smanda, On the Osciation of Soutions of Certain Difference Equations, Demonstratio Mathematica, XVII1), 1984), M. Maria Susai Manue, G. Britto Antony Xavier and E.Thandapani, Theory of Generaized Difference Operator and Its Appications, Far East Journa of Mathematica Sciences, 202) 2006), M. Maria Susai Manue, G. Britto Antony Xavier and E. Thandapani, Generaized Bernoui Poynomias Through Weighted Pochhammer Symbos, Far East Journa of Appied Mathematics, 263) 2007), M. Maria Susai Manue, V. Chandrasear and G. Britto Antony Xavier, Soutions and Appications of Certain Cass of Difference Equations, Internationa Journa of Appied Mathematics, 246) 2011), M. Maria Susai Manue, V. Chandrasear and G. Britto Antony Xavier, Theory Of Generaized α-difference Operator and its Appication in Number Theory, Advances in Differentia Equations and Contro Processes, 92) 2012),

10 964 V. Chandrasear, G. Britto Antony Xavier and R. Vijayaraj 7 Ronad E. Micens, Difference Equations, Van Nostrand Reinhod Company, New Yor, Saber N. Eaydi, An Introduction to Difference Equations, Third edition, Springer, USA, Water G. Keey, Aan C. Peterson, Difference Equations, An Introduction with Appications, Academic Press Inc., Received: December 15, 2014; Pubished: March 23, 2015