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. Advances in Difference Equations (2019)2019:24 Page 6 of 12

∞ k rl tn–l = tl T(n – l, k – l) l! (n – l)! l=0 n=k ∞ k n tn = rlT(n – l, k – l) . (2.13) l n! n=k l=0

By comparing the coefficients on both sides of (2.13), we obtain the following theorem.

Theorem 2.4 For n, k ≥ 0, we have

k n T(r)(n, k)= rlT(n – l, k – l). (2.14) l l=0

From (2.6), we note that

∞ ∞ k l t – t t – t t t e(e 2 –e 2 +rt) = e(e 2 –e 2 )ert = Bel(c) rl k k! l! k=0 k=0 ∞ n n tn = rn–kBel(c) . (2.15) k k n! n=0 k=0

Therefore, by (2.14)and(2.8), we obtain the following theorem.

Theorem 2.5 For n ≥ 0, we have

n k n Bel(c,r) = rlT(n – l, k – l). (2.16) n l k=0 l=0

From (2.1), we have

∞ n t 1 k k T(n, k|x) = e(x– 2 )t et –1 n! k! n=k k 1 k k = (–1)k–je(j+x– 2 )t k! j j=0 ∞ 1 k k k n tn = (–1)k–j j + x – . (2.17) k! j 2 n! n=0 j=0

Therefore, by (2.17), we obtain the following theorem.

Theorem 2.6 For n ≥ 0, we have ⎧ ⎨ 1 k k k n T(n, k|x), if n ≥ k, (–1)k–j j + x – = (2.18) k! j 2 ⎩ j=0 0, if n < k. Kim et al. Advances in Difference Equations (2019)2019:24 Page 7 of 12

Now, we observe that

t t t t – e(e –1+rt) = ertee 2 (e 2 –e 2 ) ∞ k 1 t t k = e( 2 +r)t e 2 – e– 2 k! k=0 ∞ ∞ ∞ k j tj tm = + r T(m, k) 2 j! m! k=0 j=0 m=k ∞ ∞ n n k n–m tn = T(m, k) + r m 2 n! k=0 n=k m=k ∞ n n n k n–m tn = T(m, k) + r . (2.19) m 2 n! n=0 k=0 m=k

Therefore, by (1.10)and(2.19), we obtain the following theorem.

Theorem 2.7 For n ≥ 0, we have n n n k n–m Bel = T(m, k) + r . (2.20) n,r m 2 k=0 m=k

From (2.18), we note that

∞ ∞ n t t (c,r) t 2 – 2 1 k k Bel = erte(e –e ) = e(r– 2 )t et –1 n n! k! n=0 k=0 ∞ k 1 k k = (–1)k–je(j+r– 2 )t k! j k=0 j=0 ∞ ∞ 1 k k k n tn = (–1)k–j j + r – k! j 2 n! n=0 k=0 j=0 ∞ ∞ ∞ tn n tn = T(n, k|r) = T(n, k|r) . (2.21) n! n! k=0 n=k n=0 k=0

Therefore, by comparing the coefficients on both sides of (2.21), we obtain the following theorem.

Theorem 2.8 For n ≥ 0, we have

n (c,r) | Beln = T(n, k r). (2.22) k=0

From (1.2), we have ∞ n t 1 t t k T(n, k) = e 2 – e– 2 + rt – rt n! k! n=0 Kim et al. Advances in Difference Equations (2019)2019:24 Page 8 of 12

k l l (–1) r 1 t t k–l = tl e 2 – e– 2 + rt l! (k – l)! l=0 ∞ k (–1)lrl tn–l = tl T(r)(n – l, k – l) l! (n – l)! l=0 n=k ∞ k n tn = (–1)lrlT(r)(n – l, k – l) . (2.23) l n! n=k l=0

Therefore, by comparing the coefficients on both sides of (2.23), we obtain the following theorem.

Theorem 2.9 For n, k ≥ 0, we have

k n T(n, k)= (–1)lrlT(r)(n – l, k – l). (2.24) l l=0

Now, we observe that

k 1 t t k 1 k t t k–l e 2 – e– 2 + rt = rltl e 2 – e– 2 k! k! l l=0 ∞ k rl tn–l = tl T(n – l, k – l) l! (n – l)! l=0 n=k ∞ k n tn = rlT(n – l, k – l) . (2.25) l n! n=k l=0

Therefore, by (2.1)and(2.25), we obtain the following theorem.

Theorem 2.10 For n, k ≥ 0, we have

k n T(r)(n, k)= rlT(n – l, k – l). (2.26) l l=0

Let m, k ∈ N ∪{0}.Thenwehave

1 t t m 1 t t k e 2 – e– 2 + rt e 2 – e– 2 + rt m! k! m + k 1 t t m+k = e 2 – e– 2 + rt m (m + k)! ∞ m + k tn = T(r)(n, m + k) . (2.27) m n! n=m+k Kim et al. Advances in Difference Equations (2019)2019:24 Page 9 of 12

On the other hand, 1 t t m 1 t t k e 2 – e– 2 + rt e 2 – e– 2 + rt m! k! ∞ ∞ tl tj = T(r)(l, m) T(r)(j, k) l! j! l=m j=k ∞ n–k n tn = T(r)(l, m)T(r)(n – l, k) . (2.28) l n! n=m+k l=m

Therefore, by (2.27)and(2.28), we obtain the following theorem.

Theorem 2.11 For n ≥ m + k, with m, k ≥ 0, we have m + k n–k n T(r)(n, m + k)= T(r)(l, m)T(r)(n – l, k). (2.29) m l l=m

For m, k ≥ 0withm ≥ k,by(2.1), we get

∞ n t 1 t t m–k t t k T(r)(n, m) = e 2 – e– 2 + rt e 2 – e– 2 + rt n! m! n=m k!(m – k)! 1 t t m–k 1 t t k = e 2 – e– 2 + rt e 2 – e– 2 + rt m! (m – k)! k! ∞ ∞ 1 tl tj = T(r)(l, m – k) T(r)(j, k) m l! j! k l=m–k j=k ∞ 1 n–k n tn = T(r)(l, m – k)T(r)(n – l, k) . (2.30) m l n! k n=m l=m–k

By comparing the coefficients on both sides of (2.30), we obtain the following theorem.

Theorem 2.12 For n, m, k ≥ 0, with n ≥ m ≥ k, we have m n–k n T(r)(n, m)= T(r)(l, m – k)T(r)(n – l, k). (2.31) k l l=m–k

Next, we observe that 1 t t m 1 t t k e 2 – e– 2 + rt e 2 – e– 2 + rt m! k! m l k j r 1 t t m–l r 1 t t k–j = tl e 2 – e– 2 tj e 2 – e– 2 l! (m – l)! j! (k – j)! l=0 j=0 ∞ n–k m k n n n – n = 1 1 rl+j l n1 j n=m+k n1=m l=0 j=0 tn × T(n – l, m – l)T(n – n – j, k – j) . (2.32) 1 1 n! Kim et al. Advances in Difference Equations (2019)2019:24 Page 10 of 12

Therefore, by (2.27)and(2.32), we obtain the following theorem.

Theorem 2.13 For n, m, k ≥ 0, with n ≥ m + k, we have m + k T(r)(n, m + k) m n–k m k n1 n n – n1 l+j = r T(n1 – l, m – l)T(n – n1 – j, k – j). (2.33) l n1 j n1=m l=0 j=0

Remark From (2.33)withr = 0, we can derive the following equation: m + k n–k n T(n, m + k)= T(l, m)T(n – l, k), (2.34) m l l=m

where n, m, k ≥ 0withn ≥ m + k.

3 Application A random variable X, taking values 0,1,2,... is said to be a Poisson random variable i ∞ with parameter λ >0ifP(i)=P(X = i)=e–λ λ , i = 0,1,2,.... Thenwehave P(i)= i! i=0 –λ ∞ λi ≥ e i=0 i! = 1. It is easy to show that the Bell polynomials Beln(x), n 0, are connected with the moments of Poisson distribution as follows: n E x =Beln(λ), n ∈ N ∪{0}, λ >0.

Let X be a Poisson random variable with parameter λ >0.Thenwenotethat

∞ n t t t – t E (X + rλ)n = eλe 2 (e 2 –e 2 )eλrt n! n=0 ∞ mt 1 λ t t m = λme 2 e rt e 2 – e– 2 m! m=0 ∞ ∞ n n m n–j tn = λm T(j, m|λr) j 2 n! m=0 n=m j=m ∞ n n n m n–j tn = λm T(j, m|λr) . (3.1) j 2 n! n=0 m=0 j=m

Thus, by (3.1), we get n n n m n–j E (X + rλ)n = λm T(j, m|λr) . j 2 m=0 j=m

From (2.6), we can derive the following equation:

∞ tn Bel(c,r)(λ) n n! n=0 t – t t = eλ(e 2 –1)(e –1)eλ(e –1+rt) Kim et al. Advances in Difference Equations (2019)2019:24 Page 11 of 12

∞ ∞ k l l j tk tm = λl (–1)l–jS k, l| – E (X + rλ)m j 2 2 k! m! k=0 l=0 j=0 m=0 ∞ n k l l n j tn = λl (–1)l–jS k, l| – E (X + rλ)n–k . (3.2) j k 2 2 n! n=0 k=0 l=0 j=0

Thus, we have n k l l n j Bel(c,r)(λ)= λl (–1)l–jS k, l| – E (X + rλ)n–k , n j k 2 2 k=0 l=0 j=0

where X is the Poisson random variable with parameter λ >0,andn ≥ 0.

4Conclusions T. Kim et al. have studied the central factorial polynomials and numbers of the second

kind which are represented by some p-adic integrals on Zp and investigated some proper- ties of these numbers and polynomials. In this paper, we introduced the extended central factorial numbers and polynomials by means of generating functions, which are useful, for example, in obtaining the moments of Poisson random variables. In addition, we gave some identities for the extended central Bell polynomials in terms of those numbers and polynomials. In more detail, in Sect. 2,weinvestigatedsomepropertiesoftheextended central factorial numbers and polynomials in connection with the extended central Bell numbers and polynomials, central factorial numbers and polynomials, and central facto- rial numbers and polynomials of the second kind in Theorems 2.1–2.13.Furthermore,in Sect. 3, we have applied our results to the moments of Poisson distribution.

Acknowledgements The authors would like to express their sincere gratitude to the editor and referees, who gave us valuable comments to improve this paper.

Funding This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2017R1E1A1A03070882).

Competing interests The authors declare that they have no competing interests.

Authors’ contributions All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.

Author details 1Department of Mathematics, Kwangwoon University, Seoul, Republic of Korea. 2Department of Mathematics, Sogang University, Seoul, Republic of Korea. 3Department of Mathematics Education and ERI, Gyeongsang National University, Jinju, Republic of Korea.

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 11 October 2018 Accepted: 15 January 2019

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