Some Identities for Umbral Calculus Associated with Partially Degenerate Bell Numbers and Polynomials

Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 10 (2017), 2966–2975 Research Article

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Some identities for umbral calculus associated with partially degenerate Bell numbers and polynomials

Taekyun Kima,b, Dae San Kimc, Hyuck-In Kwonb, Seog-Hoon Rimd,∗ aDepartment of Mathematics, College of Science Tianjin Polytechnic University, Tianjin 300160, China. bDepartment of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea. cDepartment of Mathematics, Sogang University, Seoul 121-742, Republic of Korea. dDepartment of Mathematics Education, Kyungpook National University, Taegu 702-701, Republic of Korea.

Communicated by Y. J. Cho

Abstract In this paper, we study partially degenerate Bell numbers and polynomials by using umbral calculus. We give some new identities for these numbers and polynomials which are associated with special numbers and polynomial. c 2017 All rights reserved. Keywords: Partially degenerate Bell polynomials, umbral calculus. 2010 MSC: 05A19, 05A40, 11B83.

1. Introduction

The Bell polynomials (also called the exponential polynomials and denoted by φn(x)) are deﬁned by the generating function (see [4, 10, 16])

n x(et−1) t e = Beln(x) . (1.1) ∞ n! n= X0 From (1.1), we note that 2 3 2 Bel0(x) = 1, Bel1(x) = x, Bel2(x) = x + x, Bel3(x) = x + 3x + x, 4 3 2 5 4 3 2 Bel4(x) = x + 6x + 7x + x, Bel5(x) = x + 10x + 25x + 15x + x, ··· .

When x = 1, Beln = Beln(1) are called the Bell numbers. As is well-known, the Stirling numbers of the second kind are deﬁned by the generating function (see [12, 14, 16])

n t m t e − 1 = m! S2(n, m) , (m > 0). ∞ n n=m ! X

∗Corresponding author Email addresses: [email protected] (Taekyun Kim), [email protected] (Dae San Kim), [email protected] (Hyuck-In Kwon), [email protected] (Seog-Hoon Rim) doi:10.22436/jnsa.010.06.11 Received 2017-03-25 T. Kim, D. S. Kim, H.-I. Kwon, S.-H. Rim, J. Nonlinear Sci. Appl., 10 (2017), 2966–2975 2967

The Stirling numbers of the ﬁrst kind are deﬁned as (see [1–17])

n l (x)n = S1(n, l)x (n > 0), l= X0 where (x)0 = 1, (x)n = x(x − 1) ··· (x − n + 1), (n > 1). From (1.1), we note that (see [10, 12, 16]) n l Beln(x) = S2(n, l)x , (n > 0). l= X0 The Bernoulli polynomials are given by the generating function as follows (see [1–17])

n t xt t t e = Bn(x) . (1.2) e − 1 ∞ n! n= X0 When x = 0, Bn = Bn(x) are called Bernoulli numbers. From (1.2), we note that (see [7, 11, 12, 16])

n n B (x) = B xn−l. n l l l= X0 Note that d B (x) = nB (x), (n 1). dx n n−1 > Recently, Kim-Kim considered the partially degenerate Bell polynomials which are given by the generating function (see [13])

1 n x (1+λt) λ −1 t e = Beln,λ(x) . (1.3) ∞ n! n= X0

When x=1, Beln,λ = Beln,λ(1) are called the partially degenerate Bell numbers. Note that limλ→0 Beln,λ(x) = Beln(x), (n > 0). Let C be the complex number ﬁeld and let F be the set of all formal power series in variable t over C with

k t F = f(t) = ak ak ∈ C . ∞ k! k= X0 Let P = C[x] and let P∗ be the vector space of all linear functionals on P. The action of a linear functional L ∈ P∗on a polynomial p(x) is denoted by hL | p(x)i, which is linearly extended by the rule 0 0 0 0 0 tk hcL + c L | p(x)i = chL | p(x)i + c hL | p(x)i, where c, c ∈ C. For f(t) = k=0 ak k! ∈ F, the action hf(t) | ·i of the linear functional f(t) on P is deﬁned by (see [11, 16]) P∞ n hf(t) | x i = an for all n > 0. (1.4)

By (1.4), we easily get (see [11, 14, 16])

k n ht | x i = n!δn,k, (n, k > 0), (1.5)

k tk where δn,k is the Kronecker symbol. Let fL(t) = k=0hL | x i k! . Then, from (1.5), we have ∞ Pn n hfL(t) | x i = hL | x i. T. Kim, D. S. Kim, H.-I. Kwon, S.-H. Rim, J. Nonlinear Sci. Appl., 10 (2017), 2966–2975 2968

∗ Additionally, the map L 7−→ fL(t) is a vector space isomorphism from P onto F. Henceforth, F denotes both the algebra of the formal power series in t and the vector space of all linear functionals on P, and an element f(t) of F will be thought of as both a formal power series and a linear functional. We call F umbral algebra. The umbral calculus is the study of umbral algebra (see [11, 14, 16]). Let f(t)(6= 0) ∈ F. Then the order of f(t) is the smallest positive integer k for which the coefﬁcient of tk does not vanish. The order of f(t) is denoted by of(t). Let f(t), g(t) ∈ F with of(t) = 1 and o g(t) = 0. Then there exists a unique sequence Sn(x) of polynomials such that D E D E k k g(t)f(t) Sn(x) = g(t) f(t) Sn(x) = n!δn,k, (n, k > 0), (see [16]). The sequence Sn(x) is called the Sheffer sequence for g(t), f(t) , for which we write Sn(x) ∼ g(t), f(t) . tk Let f(t) = k=0 ak k! ∈ F and p(x) ∈ P. Then, by (1.5), we get P∞ k k k t k x f(t) = hf(t) | x i , p(x) = ht | p(x)i . (1.6) ∞ k! ∞ k! k= k= X0 X0 From (1.6), we have

d k p(k)(0) = htk | p(x)i = h1 | p(k)(x)i, p(k)(x) = p(x). (1.7) dx By (1.7), we easily get

tkp(x) = p(k)(x), eytp(x) = p(x + y), and heyt | p(x)i = p(y). (1.8)

It is well-known that n 1 xf¯(t) t Sn(x) ∼ g(t), f(t) ⇐⇒ e = Sn(x) , (see [16]), (1.9) g f¯(t) ∞ n! n= X0 where f¯(t) is the compositional inverse of f(t) satisfying f¯ f(t) = f f¯(t) = t. Let f1(t), f2(t), ··· , fm(t) ∈ F. Then we have n n i1 im hf1(t) ··· fm(t) | x i = hf1(t) | x i · · · hfm(t) | x i. i , ··· , im i +···+i =n 1 1 Xm Let f(t) be the linear functional such that y hf(t) | p(x)i = p(u)du (1.10) Z0 for all polynomials p(x). Then, from (1.6), we have

k k+1 hf(t) | x i k y k 1 yt f(t) = t = t = e − 1 . (1.11) ∞ k! ∞ (k + 1)! t k= k= X0 X0 Thus, by (1.10) and (1.11), we get Deyt − 1 E y eyt − 1 x+y p(x) = p(u)du, p(x) = p(u)du. (1.12) t t Z0 Zx For Sn(x) ∼ g(t), f(t) and rn(x) ∼ h(t), l(t) , we have (see [11, 14, 16])

Sn(x) = Cn,krk(x), (1.13) k= X0 T. Kim, D. S. Kim, H.-I. Kwon, S.-H. Rim, J. Nonlinear Sci. Appl., 10 (2017), 2966–2975 2969 where k 1 D h f¯(t) nE Cn,k = l f¯(t) x . (1.14) k! g f¯(t) In this paper, we study partially degenerate Bell numbers and polynomials by using umbral calculus. We give some new identities for these numbers and polynomials which are associated with special numbers and polynomials.

2. Some identities for the partially degenerate Bell numbers and polynomials From (1.3), we recall that

1 n x (1+λt) λ −1 t e = Beln,λ(x) . ∞ n! n= X0 Note that

n 1 n t x (1+λt) λ −1 x(et−1) t lim Beln,λ(x) = lim e = e = Beln(x) . (2.1) ∞ λ→0 n! λ→0 ∞ n! n= n= X0 X0 By (2.1), we get

lim Beln,λ(x) = Beln(x), (n > 0). λ→0 From (1.9), we note that

λ 1 n (1 + t) − 1 x (1+λt) λ −1 t Beln,λ(x) ∼ 1, ⇐⇒ e = Beln,λ(x) . (2.2) λ ∞ n! n= X0 Now, we observe that

n 1 m m t x (1+λt) λ −1 x 1 Beln,λ(x) = e = (1 + λt) λ − 1 ∞ n! ∞ m! n= m= X0 X0 m m x 1 ( +λt) = e λ log 1 − 1 ∞ m! m= X0 k m −k 1 = x S2(k, m)λ log(1 + λt) ∞ ∞ k! m= k=m X0 X k ! (2.3) k m −k 1 = S2(k, m)x λ log(1 + λt) ∞ k! k= m= X0 X0 k ! n m −k n t = S2(k, m)x λ S1(n, k)λ ∞ ∞ n! k= m= n=k X0 X0 X n k ! n m n−k t = S2(k, m)S1(n, k)x λ . ∞ n! n= k= m= X0 X0 X0 Comparing the coefﬁcients on both sides of (2.3), we obtain the following theorem. Theorem 2.1. For n > 0, we have n k n n ! n−k m n−k m Beln,λ(x) = S2(k, m)S1(n, k)λ x = S2(k, m)S1(n, k)λ x . k= m= m= k=m X0 X0 X0 X T. Kim, D. S. Kim, H.-I. Kwon, S.-H. Rim, J. Nonlinear Sci. Appl., 10 (2017), 2966–2975 2970

By (1.12), we get

et − 1 x+1 Bel (x) = Bel (u)du t n,λ n,λ Zx n k x+1 n−k m = S2(k, m)S1(n, k)λ u du k= m= x X0 X0 Z n k 1 = S (k, m)S (n, k)λn−k (x + 1)m+1 − xm+1 2 1 m + 1 (2.4) k= m= X0 X0 n k m m + 1 1 = S (k, m)S (n, k)λn−kxl l m + 1 2 1 k= m= l= X0 X0 X0 n n k ! m + 1 1 = S (k, m)S (n, k)λn−k xl. l m + 1 2 1 l= k=l m=l X0 X X From (1.2) and (1.9), we have

t n e − 1 t xt t Bn(x) ∼ , t ⇐⇒ t e = Bn(x) . (2.5) t e − 1 ∞ n! n= X0 Thus, by (2.5), we get

t n Bn(x) = x , (n > 0). (2.6) et − 1 From (2.4) and (2.6), we can derive the following equation (2.7): n k m m + 1 n−k 1 t l Beln λ(x) = S (k, m)S (n, k)λ x , l 2 1 m + 1 et − 1 k= m= l= X0 X0 X0 (2.7) n k m m + 1 n−k 1 = S (k, m)S (n, k)λ Bl(x). l 2 1 m + 1 k= m= l= X0 X0 X0 Therefore, by (2.7), we obtain the following theorem.

Theorem 2.2. For n > 0, we have n k m m + 1 n−k 1 Beln λ(x) = S (k, m)S (n, k)λ Bl(x). , l 2 1 m + 1 k= m= l= X0 X0 X0 By (1.8) and (2.7), we get n k m m + 1 n−k 1 tBeln λ(x) = S (k, m)S (n, k)λ tBl(x) , l 2 1 m + 1 k= m= l= X0 X0 X0 n k m m + 1 n−k l = S (k, m)S (n, k)λ Bl− (x) l 2 1 m + 1 1 k= m= l= X1 X1 X1 n k m−1 m + 1 n−k l + 1 = S (k, m)S (n, k)λ Bl(x) l + 1 2 1 m + 1 k= m= l= X1 X1 X0 n k m−1 m = S (k, m)S (n, k)λn−kB (x). l 2 1 l k= m= l= X1 X1 X0 T. Kim, D. S. Kim, H.-I. Kwon, S.-H. Rim, J. Nonlinear Sci. Appl., 10 (2017), 2966–2975 2971

From (1.12), we have

eyt − 1 x+y Bel (x) = Bel (u)du t n,λ n,λ Zx n k m x+y m + 1 n−k 1 = S (k, m)S (n, k)λ Bl(u)du l 2 1 m + 1 k= m= l= x X0 X0 X0 Z n k m m + 1 1 = S (k, m)S (n, k)λn−k (2.8) l 2 1 (m + 1)(l + 1) k=0 m=0 l=0 X X X × Bl+1(x + y) − Bl+1(x)

n k m l n−k m + 1 l + 1 λ l+1−j = S (k, m)S (n, k) y Bj(x). l j 2 1 (m + 1)(l + 1) k= m= l= j= X0 X0 X0 X0 Thus, by (2.8), we get

Beln,λ(x + y) − Beln,λ(x) yt = e − 1 Beln,λ(x)

n k m l n−k m + 1 l + 1 λ l+1−j = S (k, m)S (n, k) y tBj(x) l j 2 1 (m + 1)(l + 1) k= m= l= j= X0 X0 X0 X0 n k m l n−k m + 1 l + 1 jλ l+1−j = S (k, m)S (n, k) y Bj− (x) (2.9) l j 2 1 (m + 1)(l + 1) 1 k= m= l= j= X1 X1 X1 X1 n k m l−1 n−k m + 1 l + 1 (j + 1)λ l−j = S (k, m)S (n, k) y Bj(x) l j + 1 2 1 (m + 1)(l + 1) k= m= l= j= X1 X1 X1 X0 n k m l−1 n−k m + 1 l λ l−j = S (k, m)S (n, k) y Bj(x). l j 2 1 m + 1 k= m= l= j= X1 X1 X1 X0 Therefore, by (2.9), we obtain the following theorem.

Theorem 2.3. For n > 1, we have

n k m l−1 n−k m + 1 l λ l−j Beln λ(x + y) − Beln λ(x) = S (k, m)S (n, k) y Bj(x). , , l j 2 1 m + 1 k= m= l= j= X1 X1 X1 X0 Let

Pn = p(x) ∈ C[x] deg p(x) 6 n , (n > 0).

For p(x) ∈ Pn, we assume that

p(x) = alBell,λ(x). (2.10) l= X0 From (2.2), we have

D λ m E (t+1) −1 (x) = n δ (m n ) λ Beln,λ ! m,n, , > 0 . (2.11) T. Kim, D. S. Kim, H.-I. Kwon, S.-H. Rim, J. Nonlinear Sci. Appl., 10 (2017), 2966–2975 2972

By (2.10) and (2.11), we get n n D λ m E D λ m E (t+1) −1 p(x) = a (t+1) −1 (x) = a l δ = m a λ l λ Bell,λ l ! l,m ! m. (2.12) l= l= X0 X0 From (2.12), we have m 1 D (t+1)λ−1 E am = p(x) , (m > 0). (2.13) m! λ Therefore, by (2.10) and (2.13), we obtain the following theorem.

Theorem 2.4. For p(x) ∈ Pn, (n > 0), we have n

p(x) = amBelm,λ(x), m= X0 where m 1 D (t+1)λ−1 E am = p(x) . m! λ Let us take p(x) = Bn(x) ∈ Pn, (n > 0). Then, by Theorem 2.4, we get n

Bn(x) = amBelm,λ(x), (2.14) m= X0 where m m −m 1 D (1+t)λ−1 E λ m m−lD λl E am = Bn(x) = (−1) (1 + t) Bn(x) m! λ m! l l= X0 −m m λ m λl m−lD k E = (−1) t Bn(x) m! ∞ l k l= k= X0 X0 (2.15) −m m n λ m λl m−l = (−1) (n)kBn−k m! l k l= k= X0 X0 −m m n λ m λl n m−l = (−1) Bn−kk!. m! l k k l= k= X0 X0 We have another expression for am.

−mD 1 λ m E am = λ (1 + t) − 1 Bn(x) m! m −mD 1 λ log(1+t) E = λ e − 1 Bn(x) m! n −m kD 1 k E = λ S (k, m)λ (log(1 + t)) Bn(x) 2 k! k=m (2.16) X n n j k−mD t E = S (k, m)λ S (j, k) Bn(x) 2 1 j! k=m j=k X X n n n = S (k, m)S (j, k)λk−m B . 2 1 j n−j k=m j=k X X Therefore, by (2.14), (2.15), and (2.16), we obtain the following theorem. T. Kim, D. S. Kim, H.-I. Kwon, S.-H. Rim, J. Nonlinear Sci. Appl., 10 (2017), 2966–2975 2973

Theorem 2.5. For n > 0, we have n m n ! m λl n −m m−l k! Bn(x) = λ (−1) Bn−k Belm λ(x) l k k m! , m=0 l=0 k=0 X X X  n n n n = S (k, m)S (j, k)λk−m B Bel (x).  2 1 j n−j m,λ m= k=m j=k X0 X X Let us consider the following two Sheffer sequences:

(1+t)λ−1 n Beln,λ(x) ∼ 1, λ , x ∼ (1, t), (n > 0). (2.17)

Then, by (1.13), (1.14), and (2.17), we get

n m Beln,λ(x) = Cn,mx , (2.18) m= X0 where m 1 D 1 nE Cn m = (λt + 1) λ − 1 x , m! m 1 m m−lD l nE = (−1) (λt + 1) λ x m! l l=0 Xm 1 m l (2.19) = (−1)m−l λ λkhtk | xni m! l ∞ k l=0 k=0 Xm X n! m l = λ (−1)m−lλn. m! l n l= X0 Therefore, by (2.18) and (2.19), we obtain the following theorem.

Theorem 2.6. For n > 0, we have n m l ! n 1 m λ m−l m Beln λ(x) = n!λ (−1) x . , m! l n m= l= X0 X0 Remark 2.7. Note that l n λ l(l − λ)(l − 2λ) ··· (l − (n − 1)λ) λ = , (n > 0). (2.20) n n! By (2.20), we easily get

l n n λ l lim λ = , (n > 0). λ→0 n n! From Theorem 2.6, we have n m ! 1 m n m−l m lim Beln,λ(x) = l (−1) x λ→0 m! l m= l= X0 X0 n n 1 m n m m = ∆ 0 x = S (n, m)x = Beln(x), m! 2 m= m= X0 X0 T. Kim, D. S. Kim, H.-I. Kwon, S.-H. Rim, J. Nonlinear Sci. Appl., 10 (2017), 2966–2975 2974 where ∆ is the difference operator with ∆f(x) = f(x + 1) − f(x). t For (x)n ∼ 1, e − 1 , we have

m n x log(1+t) m 1 m t e = x log(1 + t) = x S1(n, m) ∞ m! ∞ ∞ n! m= m= n=m X0 X0 X n ! n n m t t = S1(n, m)x = (x)n . ∞ n! ∞ n! n= m= n= X0 X0 X0 Let us consider the following two Sheffer sequences:

(1+t)λ−1 t Beln,λ(x) ∼ 1, λ , (x)n ∼ (1, e − 1). (2.21) From (1.13), (1.14), and (2.21), we have n Beln,λ(x) = Cn,m(x)m, (n > 0). (2.22) m= X0 where 1 m 1 D (λt+1) λ −1 nE Cn m = e − 1 x , m! k 1 D 1 nE = S2(k, m) (λt + 1) λ − 1 x ∞ k! k=m X n k S2(k, m) k k−lD l nE = (−1) (λt + 1) λ x k! l k=m l= X X0 n k S (k, m) k l (2.23) = 2 (−1)k−l λ λjhtj | xni k! l ∞ j k=m l= j= X X0 X0 n k S (k, m) k l = 2 (−1)k−l λ λnn! k! l n k=m l= X X0 n k S (k, m)k l = n!λn 2 λ (−1)k−l. k! l n k=m l= X X0 Therefore, by (2.22) and (2.23), we obtain the following theorem.

Theorem 2.8. For n > 0, we have n n k l ! n S2(k, m) k λ k−l Beln λ(x) = n!λ (−1) (x)m. , k! l n m= k=m l= X0 X X0 On the other hand,

1 m 1 D (λt+1) λ −1 nE Cn m = e − 1 x , (2.24) , m! 1 m 1 1 e (λt+1) λ −1 − 1 = e (λt+1) λ −1 − 1 × · · · × e (λt+1) λ −1 − 1

m−times     | tl1 {z tlm } =  Bell1,λ  × · · · ×  Bellm,λ  (2.25) ∞ l ! ∞ lm! l = 1 l = X1 1 Xm 1 T. Kim, D. S. Kim, H.-I. Kwon, S.-H. Rim, J. Nonlinear Sci. Appl., 10 (2017), 2966–2975 2975

  k tk =  Bel ··· Bel  .  l1,λ lm,λ ∞ l1, ··· , lm k! k=m l1+···+lm=k X l1,···X,lm>1

From (2.24) and (2.25), we have

1 m 1 D (λt+1) λ −1 nE Cn m = e − 1 x , m!   n 1 k 1 =  Bel ··· Bel  htk|xni  l1,λ lm,λ m! l1, ··· , lm k! (2.26) k=m l1+···+lm=k X l1,···X,lm>1 1 n = Bell1,λ ··· Bellm,λ. m! l1, ··· , lm l1+···+lm=n l1,···X,lm>1

Therefore, by (2.22) and (2.26), we get   n 1 n Bel (x) =  Bel ··· Bel  (x) . n,λ  l1,λ lm,λ m m! l1, ··· , lm m=0 l1+···+lm=n X l1,···X,lm>1

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