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Extended Central Factorial Polynomials of the Second Kind Taekyun Kim1,Daesankim2,Gwan-Woojang1 and Jongkyum Kwon3*

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Extended Central Factorial Polynomials of the Second Kind Taekyun Kim1,Daesankim2,Gwan-Woojang1 and Jongkyum Kwon3* Kim et al. Advances in Difference Equations (2019)2019:24 https://doi.org/10.1186/s13662-019-1963-1 R E S E A R C H Open Access Extended central factorial polynomials of the second kind Taekyun Kim1,DaeSanKim2,Gwan-WooJang1 and Jongkyum Kwon3* *Correspondence: [email protected] Abstract 3Department of Mathematics Education and ERI, Gyeongsang In this paper, we consider the extended central factorial polynomials and numbers of National University, Jinju, Republic the second kind, and investigate some properties and identities for these polynomials of Korea and numbers. In addition, we give some relations between those polynomials and Full list of author information is available at the end of the article the extended central Bell polynomials. Finally, we present some applications of our results to moments of Poisson distributions. MSC: 11B75; 11B83 Keywords: Extended central factorial polynomials of the second kind; Extended central Bell polynomials 1 Introduction For n ≥ 0, the central factorial numbers of the second kind are defined by n xn = T(n, k)x[k], n ≥ 0, (see [1, 2, 20–22]), (1.1) k=0 [k] k k ··· k ≥ [0] where x = x(x + 2 –1)(x + 2 –2) (x – 2 +1),k 1, x =1. By (1.1), we see that the generating function of the central factorial numbers of the sec- ond kind is given by ∞ n 1 t t k t e 2 – e– 2 = T(n, k) (see [1–4, 21]). (1.2) k! n! n=k Here the definition of T(n, k)isextendedsothatT(n, k)=0forn < k.Thisagreementwill be applied to all similar situations without further mention. Then, by (1.2), we have 1 k k k n T(n, k)= (–1)k–j j – , n, k ≥ 0. (1.3) k! j 2 j=0 © The Author(s) 2019. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Kim et al. Advances in Difference Equations (2019)2019:24 Page 2 of 12 Let us recall that the Stirling polynomials of the second kind are defined by ∞ n 1 k t ext et –1 = S (n, k|x) (see [10, 16, 17]), (1.4) k! 2 n! n=k where k is a nonnegative integer. When x =0,S2(n, k)=S2(n, k|0), n, k ≥ 0, are the Stirling numbers of the second kind given by n n x = S2(n, k)(x)k, n ≥ 0(see[12, 14, 15, 21]), k=0 where (x)0 =1,(x)k = x(x –1)···(x – k +1),k ≥ 1. From (1.4), we note that n n S (n, k|x)= S (l, k)xn–l 2 l 2 l=k 1 k k = (–1)k–j(j + x)n, n, k ≥ 0. (1.5) k! j j=0 The Bell polynomials are given by the generating function ∞ n t t ex(e –1) = Bel (x) (see [5–11]). (1.6) n n! n=0 Then, from (1.4)and(1.6), we get n m Beln(x)= S2(n, m)x (see [10]). (1.7) m=0 In [17], the extended Stirling polynomials of the second kind are defined by ∞ n 1 k t ext et –1+rt = S (n, k|x) , (1.8) k! 2,r n! n=k where n, k ∈ N ∪{0} and r ∈ R. When x =0,S2,r(n, k)=S2,r(n, k|0), n, k ≥ 0, are called the extended Stirling numbers of the second kind. Note that S2,0(n, k)=S2(n, k)andS2,0(n, k|x)=S2(n, k|x). From (1.4)and(1.8), we note that k n S (n, k)= rlS (n – l, k – l), n, k ≥ 0(see[17]). (1.9) 2,r l 2 l=0 It is known that the extended Bell polynomials are defined by ∞ n t t ex(e –1+rt) = Bel (x) (see [13, 18, 19]). (1.10) n,r n! n=0 For x =1,Beln,r =Beln,r(1) are called the extended Bell numbers. Kim et al. Advances in Difference Equations (2019)2019:24 Page 3 of 12 Then, from (1.10), we get n m Beln,r(x)= x S2,r(n, m)(see[17]). (1.11) m=0 Recently, the central Bell polynomials were defined by Kim as ∞ n t t – t Bel(c)(x) = ex(e 2 –e 2 ) (see [1–3]). (1.12) n n! n=k (c) (c) For x =1,Beln =Beln (1)arecalledthecentralBellnumbers. Thus, by (1.12), we get n (c) k ≥ Beln (x)= x T(n, k), n 0(see[17]). (1.13) k=0 The purpose of this paper is to consider the extended central factorial polynomials and numbers of the second kind, and investigate some properties and identities for these poly- nomials and numbers. In addition, we give some relations between those polynomials and the extended central Bell polynomials. Finally, we present some applications of our results to moments of Poisson distributions. 2 Extended central factorial polynomials of the second kind Motivated by (1.8), we define the extended central factorial polynomials of the second kind by ∞ n 1 t t k t ext e 2 – e– 2 + rt = T(r)(n, k|x) , (2.1) k! n! n=k where k ∈ N ∪{0} and r ∈ R.Whenx =0,T(r)(n, k)=T(r)(n, k|0), n, k ≥ 0, are called the extended central factorial numbers of the second kind. Note here that, when r =0, T(n, k|x)=T(0)(n, k|x)andT(n, k)=T(0)(n, k) are respectively the central factorial polyno- mials of the second kind and the central factorial numbers of the second kind. From (2.1), we note that ∞ ∞ ∞ tn 1 tl T(r)(n, k|x) = xmtm T(r)(l, k) n! m! l! n=k m=0 l=k ∞ n n tn = T(r)(l, k)xn–l . (2.2) l n! n=k l=k Therefore, by comparing the coefficients on both sides of (2.2), we obtain the following theorem. Theorem 2.1 For n, k ≥ 0, we have n n T(r)(n, k|x)= T(r)(l, k)xn–l. (2.3) l l=k Kim et al. Advances in Difference Equations (2019)2019:24 Page 4 of 12 From (2.1), we note that ∞ ∞ ∞ k n x t t k t e 2 – e– 2 + rt = xk T(r)(n, k) k! n! k=0 k=0 n=k ∞ n tn = xkT(r)(n, k) . (2.4) n! n=0 k=0 On the other hand, ∞ k t t t t x t t k 2 – 2 2 – 2 e 2 – e– 2 + rt = ex(e –e +rt) = ex(e –e )exrt k! k=0 ∞ ∞ tk tm = Bel(c)(x) rmxm k k! m! k=0 m=0 ∞ n n tn = Bel(c)(x)rn–kxn–k . (2.5) k k n! n=0 k=0 Now, we define the extended central Bell polynomials by ∞ n t – t t ex(e 2 –e 2 +rt) = Bel(c,r)(x) . (2.6) n n! n=0 (c,r) (c,r) For x =1,Beln =Beln (1) are called the extended central Bell numbers. Therefore, by combining (2.4)–(2.6), we obtain the following theorem. Theorem 2.2 For n ≥ 0, we have n (c,r) k (r) Beln (x)= x T (n, k) k=0 n n = Bel(c)(x)rn–kxn–k. (2.7) k k k=0 In particular, n n n Bel(c,r) = Bel(c)rn–k = T(r)(n, k). (2.8) n k k k=0 k=0 Remark By (2.6), we get ∞ n t t – t Bel(c,r)(x) = extex(e 2 –e 2 +(r–1)t) n n! n=0 ∞ 1 t t k = xk ext e 2 – e– 2 +(r –1)t k! k=0 Kim et al. Advances in Difference Equations (2019)2019:24 Page 5 of 12 ∞ ∞ tn = T(r–1)(n, k|x) xk n! k=0 n=k ∞ n tn = xkT(r–1)(n, k|x) . (2.9) n! n=0 k=0 Comparing the coefficients on both sides of (2.9), we get n (c,r) k (r–1) | ≥ Beln (x)= x T (n, k x), n 0. (2.10) k=0 In particular, n (c,r) (r–1) | Beln = T (n, k 1), (2.11) k=0 and, invoking (2.3), n n n n Bel(c,1)(x)= T(n, k|x)xk = T(l, k)xn–l+k. (2.12) n l k=0 k=0 l=k Therefore, by (2.10)–(2.12), we obtain the following corollary. Corollary 2.3 For n ≥ 0, we have n (c,r) k (r–1) | Beln (x)= x T (n, k x). k=0 In particular, n (c,r) (r–1) | Beln = T (n, k 1) k=0 and n n n Bel(c,1)(x)= T(l, k)xn–l+k. n l k=0 l=k From (2.1), we note that ∞ n k t 1 k t t k–l T(r)(n, k) = rltl e 2 – e– 2 n! k! l n=0 l=0 k l r 1 t t k–l = tl e 2 – e– 2 l! (k – l)! l=0 Kim et al.
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