Extended central factorial polynomials of the second kind

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1 Kim et a. Advances in Difference Equations 09 09:4 R E S E A R C H Open Access Extended centra factoria poynomias of the second ind Taeyun Kim,DaeSanKim,Gwan-WooJang and Jongyum Kwon 3* * Correspondence: math6@gnu.ac.r 3 Department of Mathematics Education and ERI, Gyeongsang Nationa University, Jinu, Repubic of Korea Fu ist of author information is avaiabe at the end of the artice Abstract In this paper, we consider the extended centra factoria poynomias and numbers of the second ind, and investigate some properties and identities for these poynomias and numbers. In addition, we give some reations between those poynomias and the extended centra Be poynomias. Finay, we present some appications of our resuts to moments of Poisson distributions. MSC: B75; B83 Keywords: Extended centra factoria poynomias of the second ind; Extended centra Be poynomias Introduction For n 0, the centra factoria numbers of the second ind are defined by x n Tn, x [], n 0, see [,, 0 ],. where x [] xx + x + x +,, x[0]. By., we see that the generating function of the centra factoria numbers of the second ind is given by e t e t Tn, tn see [ 4, ].. Here the definition of Tn, isextendedsothattn, 0forn <.Thisagreementwi be appied to a simiar situations without further mention. Then, by., we have Tn, n, n, 0..3 The Authors 09. This artice is distributed under the terms of the Creative Commons Attribution 4.0 Internationa License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the origina authors and the source, provide a in to the Creative Commons icense, and indicate if changes were made.

2 Kim eta. Advances in Difference Equations 09 09:4 Page of Let us reca that the Stiring poynomias of the second ind are defined by ext e t S n, x tn see [0, 6, 7],.4 where is a nonnegative integer. When x 0,S n, S n, 0, n, 0, are the Stiring numbers of the second ind given by x n S n, x, n 0see[, 4, 5, ], where x 0,x xx x +,. From.4, we note that S n, x n S, x n + x n, n, 0..5 The Be poynomias are given by the generating function e xet Be n x tn see [5 ]..6 Then, from.4and.6, we get Be n x S n, mx m see [0]..7 In [7], the extended Stiring poynomias of the second ind are defined by ext e t +rt S,r n, x tn,.8 where n, N {0} and r R. When x 0,S,r n, S,r n, 0, n, 0, are caed the extended Stiring numbers of the second ind. Note that S,0 n, S n, ands,0 n, xs n, x. From.4and.8, we note that S,r n, 0 n r S n,, n, 0see[7]..9 It is nown that the extended Be poynomias are defined by e xet +rt Be n,r x tn see [3, 8, 9]..0 For x,be n,r Be n,r are caed the extended Be numbers.

3 Kim eta. Advances in Difference Equations 09 09:4 Page 3 of Then, from.0,we get Be n,r x x m S,r n, m see[7].. Recenty, the centra Be poynomias were defined by Kim as Be c n xtn exe t e t see [ 3].. For x,be c n Bec n arecaedthecentrabenumbers. Thus, by., we get Be c n x x Tn,, n 0see[7]..3 The purpose of this paper is to consider the extended centra factoria poynomias and numbers of the second ind, and investigate some properties and identities for these poynomias and numbers. In addition, we give some reations between those poynomias and the extended centra Be poynomias. Finay, we present some appications of our resuts to moments of Poisson distributions. Extended centra factoria poynomias of the second ind Motivated by.8, we define the extended centra factoria poynomias of the second ind by ext e t e t + rt T r n, x tn,. where N {0} and r R. Whenx 0,T r n, T r n, 0, n, 0, are caed the extended centra factoria numbers of the second ind. Note here that, when r 0, Tn, xt 0 n, xandtn, T 0 n, are respectivey the centra factoria poynomias of the second ind and the centra factoria numbers of the second ind. From., we note that T r n, x tn m! xm t m T r, t! n t T r, x n n.. Therefore, by comparing the coefficients on both sides of., we obtain the foowing theorem. Theorem. For n, 0, wehave T r n, x n T r, x n..3

4 Kim eta. Advances in Difference Equations 09 09:4 Page 4 of From., we note that x t e e t + rt x T r n, tn x T r n,..4 On the other hand, x t e e t + rt e xe t e t +rt e xe t e t e xrt Be c xt r m x m tm m! n Be c xrn x n Now, we define the extended centra Be poynomias by..5 e xe t e t +rt Be c,r n x tn..6 For x,be c,r n Be c,r n are caed the extended centra Be numbers. Therefore, by combining.4.6, we obtain the foowing theorem. Theorem. For n 0, we have Be c,r n x In particuar, Be c,r n x T r n, Remar By.6, we get n Be c xrn x n..7 n Be c rn T r n,..8 Be c,r n x tn ext e xe t e t +r t x ext e t e t +r t

5 Kim eta. Advances in Difference Equations 09 09:4 Page 5 of T r n, x tn x x T r n, x..9 Comparing the coefficients on both sides of.9, we get Be c,r n x In particuar, x T r n, x, n 0..0 Be c,r n T r n,,. and, invoing.3, Be c, n x Tn, xx Therefore, by.0., we obtain the foowing coroary. n T, x n +.. Coroary.3 For n 0, we have Be c,r n x In particuar, x T r n, x. and Be c,r n T r n, Be c, n x From., we note that T r n, tn n T, x n r t e t e t r! t t e e t!

6 Kim eta. Advances in Difference Equations 09 09:4 Page 6 of 0 r! t 0 Tn, n! n r Tn,..3 By comparing the coefficients on both sides of.3, we obtain the foowing theorem. Theorem.4 For n, 0, wehave T r n, 0 n r Tn,..4 From.6, we note that e e t e t +rt e e t e t e rt n Be c t r n Be c r t! Therefore, by.4 and.8, we obtain the foowing theorem...5 Theorem.5 For n 0, we have Be c,r n 0 From., we have n r Tn,..6 Tn, x tn ex t e t e +x t + x n..7 Therefore, by.7, we obtain the foowing theorem. Theorem.6 For n 0, we have + x n Tn, x, if n, 0, if n <..8

7 Kim eta. Advances in Difference Equations 09 09:4 Page 7 of Now, we observe that e et +rt e rt e e t e t e t e +rt t e e t m t + r! m Tm, tm m! m n Tm, m n m + r n n m Tm, m + r..9 Therefore, by.0 and.9, we obtain the foowing theorem. Theorem.7 For n 0, we have Be n,r m From.8, we note that Be c,r n ert e e n n m Tm, m + r..0 t e t er t e t e +r t + r n Tn, r tn Tn, r tn.. Therefore, by comparing the coefficients on both sides of., we obtain the foowing theorem. Theorem.8 For n 0, we have Be c,r n Tn, r.. From., we have Tn, tn t e e t + rt rt

8 Kim eta. Advances in Difference Equations 09 09:4 Page 8 of r t t e e t + rt!! 0 r 0! 0 t n T r n, n! r T r n,..3 Therefore, by comparing the coefficients on both sides of.3, we obtain the foowing theorem. Theorem.9 For n, 0, wehave Tn, 0 n r T r n,..4 Now, we observe that t e e t + rt 0 0 r! t r t e t e t 0 Tn, n! n r Tn,..5 Therefore, by.and.5, we obtain the foowing theorem. Theorem.0 For n, 0, wehave T r n, 0 n r Tn,..6 Let m, N {0}.Thenwehave t e e t + rt m t e e t + rt m! m + t e e t + rt m+ m m +! m + m nm+ T r n, m + tn..7

9 Kim eta. Advances in Difference Equations 09 09:4 Page 9 of On the other hand, t e e t + rt m t e e t + rt m! T r, m t T r, t!! m nm+ n m n T r, mt r n, Therefore, by.7 and.8, we obtain the foowing theorem. Theorem. For n m +, with m, 0, we have m..8 m + n n T r n, m + T r, mt r n,..9 m For m, 0withm,by., we get nm T r n, m tn t e e t + rt m t e e t + rt m! m! m! m m m n nm m! T r, m t! m t e e t + rt m t e e t + rt T r, t! n T r, m T r n,..30 By comparing the coefficients on both sides of.30, we obtain the foowing theorem. Theorem. For n, m, 0, with n m, we have m T r n, m Next, we observe that n m n T r, m T r n,..3 t e e t + rt m t e e t + rt m! m r! t t e e t m r m!! t t e e t! 0 n m n n n n r + nm+ n m 0 n Tn, m Tn n, tn..3

10 Kim eta.advances in Difference Equations 09 09:4 Page 0 of Therefore, by.7 and.3, we obtain the foowing theorem. Theorem.3 For n, m, 0, with n m +, we have m + T r n, m + m n m n n n n r + Tn, m Tn n,..33 n m 0 n Remar From.33with r 0, we can derive the foowing equation: m + n n Tn, m + T, mtn,,.34 m m where n, m, 0withn m +. 3 Appication A random variabe X, taing vaues 0,,,... is said to be a Poisson random variabe λ λi with parameter λ >0ifPi PX i e i!, i 0,,,... Then we have i0 Pi e λ i0 λi. It is easy to show that the Be poynomias Be i! n x, n 0, are connected with the moments of Poisson distribution as foows: E [ x n] Be n λ, n N {0}, λ >0. Let X be a Poisson random variabe with parameter λ >0.Thenwenotethat E [ X + rλ n] Thus, by 3., we get E [ X + rλ n] eλe m t e t e t e λrt λ m e mt λ m m! eλrt e t e nm m n λ m m t m n T, m λr n m λ m T, m λr T, m λr From.6, we can derive the foowing equation: Be c,r n λ tn e λe t e t e λet +rt n. m n m n. 3.

11 Kim eta.advances in Difference Equations 09 09:4 Page of 0 Thus, we have Be c,r n λ λ S, t E [ X + rλ m] tm m! 0 0 n λ S, E [ X + rλ n ]. 3. n λ S, E [ X + rλ n ], where X is the Poisson random variabe with parameter λ >0,andn 0. 4 Concusions T. Kim et a. have studied the centra factoria poynomias and numbers of the second ind which are represented by some p-adic integras on Z p and investigated some properties of these numbers and poynomias. In this paper, we introduced the extended centra factoria numbers and poynomias by means of generating functions, which are usefu, for exampe, in obtaining the moments of Poisson random variabes. In addition, we gave some identities for the extended centra Be poynomias in terms of those numbers and poynomias. In more detai, in Sect., weinvestigatedsomepropertiesoftheextended centra factoria numbers and poynomias in connection with the extended centra Be numbers and poynomias, centra factoria numbers and poynomias, and centra factoria numbers and poynomias of the second ind in Theorems..3.Furthermore,in Sect. 3, we have appied our resuts to the moments of Poisson distribution. Acnowedgements The authors woud ie to express their sincere gratitude to the editor and referees, who gave us vauabe comments to improve this paper. Funding This wor was supported by the Nationa Research Foundation of Korea NRF grant funded by the Korea government MEST No. 07REAA Competing interests The authors decare that they have no competing interests. Authors contributions A authors contributed equay to the manuscript and typed, read, and approved the fina manuscript. Author detais Department of Mathematics, Kwangwoon University, Seou, Repubic of Korea. Department of Mathematics, Sogang University, Seou, Repubic of Korea. 3 Department of Mathematics Education and ERI, Gyeongsang Nationa University, Jinu, Repubic of Korea. Pubisher s Note Springer Nature remains neutra with regard to urisdictiona caims in pubished maps and institutiona affiiations. Received: October 08 Accepted: 5 January 09 References. Butzer, P.L., Schmidt, M., Star, E.L., Vogt, L.: Centra factoria numbers; their main properties and some appications. Numer. Funct. Ana. Optim. 05 6, Caritz, L., Riordan, J.: The divided centra differences of zero. Can. J. Math. 5, Charaambides, Ch.A.: A centra factoria numbers and reated expansions. Fibonacci Q. 95, Comtet, L.: Nombres de Stiring généraux et fonctions symétriques. C. R. Acad. Sci. Paris Sér. A B 75,

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