Some Identities of Type 2 Degenerate Bernoulli Polynomials of the Second

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The central factorial numbers of the second kind are deﬁned as n (5) xn = ∑ T(n,k)x[k], (see [6]), k=0 or equivalently as ∞ n 1 t − t k t (6) (e 2 − e 2 ) = ∑ T(n,k) , k! n=k n! [0] [n] n n n where x = 1, x = x x + 2 − 1 x + 2 − 2 ··· x − 2 + 1 , (n ≥ 1). It is well known that the Daehee polynomials are deﬁned by n n log(1 +t) x t (7) (1 +t) = ∑ Dn(x) , (see [14, 15]). t k=0 n!

When x = 0, Dn = Dn(0) are called the Daehee numbers.

The Bernoulli polynomials of the second kind of order r are defined by r n n t x (r) t (8) (1 +t) = bn (x) , (see [13]). log(1 +t) ∑ n! k=0 (r) (n−r+1) (r) Note that bn (x)= Bn (x + 1), (n ≥ 0). Here Bn (x) are the ordinary Bernoulli polynomials of order r given by t r n tn (9) ext = B(r)(x) , (see [13, 14, 15, 16]). et − 1 ∑ n n! k=0 It is known that the Stirling numbers of the second kind are defined by ∞ n 1 t k t (10) e − 1 = ∑ S2(n,k) , (see [13]), k! n=k n! and the Stirling numbers of the first kind by ∞ n 1 k t (11) log (1 +t)= ∑ S1(n,k) , (see [13]). k! n=k n! For any nonzero λ ∈ R, the degenerate exponential function is defined by ∞ n x t x λ (12) eλ (t) = (1 + λt) = ∑ (x)n,λ , (see [9]), n=0 n! where (x)0,λ = 1, (x)n,λ = x(x − λ)··· (x − (n − 1)λ), (n ≥ 1). In particular, we let 1 1 λ (13) eλ (t)= eλ (t) = (1 + λt) . In [1, 2], Carlitz introduced the degenerate Bernoulli polynomials which are given by the generating function t ∞ tn (14) ex (t)= β (x) . − λ ∑ n,λ eλ (t) 1 n=0 n! Also, he considered the degenerate Euler polynomials given by ∞ n 2 x E t (15) eλ (t)= ∑ n,λ (x) , (see [1, 2]). eλ (t)+ 1 n=0 n! T. Kim, L.-C. Jang, D. S. Kim, H. Y. Kim 3

Recently, Kim-Kim considered the degenerate central factorial numbers of the second kind given by

k ∞ n 1 1 − 1 t (16) e 2 (t) − e 2 (t) = T (n,k) . k! λ λ ∑ λ n! n=k Note that limλ→0 Tλ (n,k)= T(n,k), (see [12]). In this paper, we introduce the type 2 degenerate Bernoulli polynomials of the second kind and their higher-order analogues, and study some identities and expressions for these polynomials. Speciﬁcally, we obtain a relation between the type 2 degenerate Bernoulli polynomials of the second and the degenerate Bernoulli polynomials of the second, an identity involving higher-order analogues of those polynomials and the degenerate Stirling numbers of second kind, and an expres- sion of higher-order analogues of those polynomials in terms of the higher-order type 2 degenerate Bernoulli polynomials and the degenerate Stirling numbers of the ﬁrst kind.

2. TYPE 2 DEGENERATE BERNOULLIPOLYNOMIALSOFTHESECONDKIND

Let logλ t be the compositional inverse of eλ (t) in (13). Then we have 1 (17) log t = tλ − 1 . λ λ Note that limλ→0 logλ t = logt. Now, we deﬁne the degenerate Daehee polynomials by ∞ n logλ (1 +t) x t (18) (1 +t) = ∑ Dn,λ (x) . t n=0 n!

Note that limλ→0 Dn,λ (x) = Dn(x), (n ≥ 0). In view of (8), we also consider the degenerate Bernoulli polynomials of the second kind of order α given by t α ∞ tn (19) (1 +t)x = b(α)(x) . log (1 +t) ∑ n,λ n! λ n=0 (α) (α) ≥ Note that limλ→0 bn,λ (x)= bn (x), (n 0). From (19), we have α ∞ n λt x− λα (α) t (20) (1 +t) 2 = b (x) . λ − λ ∑ n,λ (1 +t) 2 − (1 +t) 2 ! n=0 n! For α = r ∈ N, and replacing t by e2t − 1 in (20), we get ∞ 1 λt r 1 b(r) (x) (e2t − 1)m = (e2t − 1)re(2x−λr)t ∑ m,λ m! etλ − e−tλ tr m=0 ∞ 2x λ ktk ∞ 1 tm = B∗ − r S (m + r,r)2m+r ∑ k λ k! ∑ 2 m+r m! k=0 m=0 r ∞ n n 2x S (m + r,r) tn (21) = B∗ − r λ n−m 2 2m+r . ∑ ∑ m n−m λ m+r n! n=0 m=0 r ! 4 Some identities on type 2 degenerate Bernoulli polynomials of the second kind

On the other hand, ∞ 1 ∞ ∞ tn (r) 2t − m (r) n ∑ bm,λ (x) (e 1) = ∑ bm,λ (x) ∑ S2(n,m)2 m=0 m! m=0 n=m n! ∞ n n (r) n t (22) = ∑ ∑ bm,λ (x)2 S2(n,m) . n=0 m=0 ! n! From (21) and (22), we have n n n 2x S (m + r,r) (23) b(r) (x)S (n,m)= B∗ − r λ n−m 2 2m+r−n. ∑ m,λ 2 ∑ m n−m λ m+r m=0 m=0 r Now, we deﬁne the type 2 degenerate Bernoulli polynomials of the second kind by −1 ∞ n (1 +t) − (1 +t) x ∗ t (24) (1 +t) = ∑ bn,λ (x) . logλ (1 +t) n=0 n! ∗ ∗ When x = 0, bn,λ = bn,λ (0) are called the type 2 degenerate Bernoulli numbers of the second kind. ∗ ∗ ∗ Note that limλ→0 bn,λ (x)= bn(x), where bn(x) are the type 2 Bernoulli polynomials of the second kind given by −1 ∞ n (1 +t) − (1 +t) x ∗ t (1 +t) = ∑ bn(x) . log(1 +t) n=0 n! From (19) and (24), we note that (1 +t) − (1 +t)−1 t 1 (1 +t)x = (1 +t)x 1 + log (1 +t) log (1 +t) 1 +t λ λ t t = (1 +t)x + (1 +t)x−1 logλ (1 +t) logλ (1 +t) ∞ tn (1) (1) − (25) = ∑ bn,λ (x)+ bn,λ (x 1) . n 0 n! = Therefore, we obtain the following theorem. Theorem 2.1. For n ≥ 0, we have ∗ (1) (1) − bn,λ (x)= bn,λ (x)+ bn,λ (x 1). Moreover, n n n 2x S (m + r,r) b(r) (x)S (n,m)= B∗ − r λ n−m 2 2m+r−n, ∑ m,λ 2 ∑ m n−m λ m+r m=0 m=0 r where r is a positive integer. Now, we observe that −1 ∞ l ∞ m (1 +t) − (1 +t) x ∗ t t (1 +t) = ∑ bl,λ ∑ (x)m logλ (1 +t) l=0 l! m=0 m! ∞ n n n ∗ t (26) = b (x) − , ∑ ∑ l l,λ n l n! n=0 l=0 ! where (x)0 = 1, (x)n = x(x − 1)··· (x − n + 1), (n ≥ 1). From (24) and (26), we get n ∗ n ∗ (27) b (x)= b (x) − , (n ≥ 0). n,λ ∑ l l,λ n l l=0 T. Kim, L.-C. Jang, D. S. Kim, H. Y. Kim 5

For α ∈ R, let us deﬁne the type 2 degenerate Bernoulli polynomials of the second kind of order α by −1 α ∞ n (1 +t) − (1 +t) ∗ t (28) (1 +t)x = b (α)(x) log (1 +t) ∑ n,λ n! λ n=0 ∗(α) ∗(α) When x = 0, bn,λ = bn,λ (0) are called the type 2 degenerate Bernoulli numbers of the second kind of order α. Let α = k ∈ N. Then we have ∞ n −1 k ∗ t (1 +t) − (1 +t) (29) b (k)(x) = (1 +t)x. ∑ n,λ n! log (1 +t) n=0 λ By replacing t by eλ (t) − 1 in (29), we get ∞ k! 1 k 1 − −1 x ∗(k) − l k eλ (t) eλ (t) eλ (t)= ∑ bl,λ (x) (eλ (t) 1) t k! l=0 l! ∞ ∞ n ∗(k) t = ∑ bl,λ (x) ∑ S2,λ (n,l) l=0 n=l n! ∞ n n ∗(k) t (30) = ∑ ∑ bl,λ (x)S2,λ (n,l) , n=0 l=0 ! n! where S2,λ (n,l) are the degenerate Stirling numbers of the second kind given by ∞ n 1 k t (31) (eλ (t) − 1) = ∑ S2,λ (n,k) , (see [10]). k! n=k n! On the other hand, we also have k! 1 k k! 1 k e (t) − e−1(t) ex (t)= e2 (t) − 1 ex−k(t) tk k! λ λ λ tk k! λ λ k! 1 k − x−k = e λ (2t) 1 eλ (t) tk k! 2 ∞ 2m+k tm ∞ tl − = ∑ S2, λ (m + k,k) m k ∑(x k)l,λ 2 + m! l! m=0 k l=0 ∞ n n 2m+k tn m − (32) = ∑ ∑ S2, λ (m + k,k)(x k)n−m,λ . m+k 2 n! n=0 m=0 k ! Therefore, by (30) and (32), we obtain the following theorem . Theorem 2.2. For n ≥ 0, we have n n n l+k ∗(k) l 2 b (x)S2,λ (n,l)= S λ (l + k,k)(x − k)n−l,λ . ∑ l,λ ∑ l+k 2, 2 l=0 l=0 k In particular, n n+k n + k ∗(k) 2 S2, λ (n + k,k)= ∑ bl,λ (k)S2,λ (n,l). 2 k l=0 For α ∈ R, we recall that the type 2 degenerate Bernoulli polynomials of order α are deﬁned by α ∞ n t ∗ t (33) ex (t)= β (α)(x) , (see [6, 12]). − −1 λ ∑ n,λ n! eλ (t) eλ (t)! n=0 6 Some identities on type 2 degenerate Bernoulli polynomials of the second kind

N For k ∈ , let us take α = −k and replace t by logλ (1 +t) in (33). Then we have

−1 k ∞ (1 +t) − (1 +t) ∗ − 1 (1 +t)x = β ( k)(x) (log (1 +t))l log (1 +t) ∑ l,λ l ! λ λ l=0 ∞ ∞ n ∗(−k) t = ∑ βl,λ (x) ∑ S1,λ (n.l) l=0 n=l n! ∞ n n ∗(−k) t (34) = ∑ ∑ βl,λ S1,λ (n.l) , n=0 l=0 ! n! where S1,λ (n,l) are the degenerate Stirling numbers of the ﬁrst kind given by ∞ n 1 k t (35) (logλ (1 +t)) = ∑ S1,λ (n,k) . k! n=k n!

Note here that limλ→0 S1,λ (n,l)= S1(n,l). Therefore, by (26) and (34), we obtain the following theorem. Theorem 2.3. For n ≥ 0 and k ∈ N, we have n ∗(k) ∗(−k) bn,λ (x)= ∑ βl,λ (x)S1,λ (n,l). l=0 We observe that k 1 k 1 1 − 1 k t = (1 +t) 2 − (1 +t) 2 (1 +t) 2 k! k! k 1 1 − 1 k = e 2 (log (1 +t)) − e 2 (log (1 +t)) (1 +t) 2 k! λ λ λ λ ∞ 1 ∞ k tr = T (l,k) (log (1 +t))l ∑ λ l! λ ∑ 2 r! l=k r=0 r ∞ ∞ tm ∞ k tr = T (l,k) S (m,l) ∑ λ ∑ 1,λ m! ∑ 2 r! l=k m=l r=0 r ∞ m tm ∞ k tr = T (l,k)S (m,l) ∑ ∑ λ 1,λ m! ∑ 2 r! m=k l=k r=0 r ∞ n m n k tn (36) = T (l,k)S (m,l) . ∑ ∑ ∑ λ 1,λ m 2 n! n=k m=k l=k n−m! On the other hand, 1 t k 1 tk = (log (1 +t))k k! log (1 +t) k! λ λ ∞ l ∞ m (k) t t = ∑ bl,λ ∑ S1,λ (m,k) l=0 l! m=k m! ∞ n n tn (37) = S (m,k)b(k) . ∑ ∑ 1,λ n−m,λ m n! n=k m=k ! Therefore, by (36) and (37), we obtain the following theorem. T. Kim, L.-C. Jang, D. S. Kim, H. Y. Kim 7

Theorem 2.4. For n,k ≥ 0, we have n m n k n n T (l,k)S (m,l) = S (m,k)b(k) . ∑ ∑ λ 1,λ m 2 ∑ 1,λ n−m,λ m m=k l=k n−m m=k

3. Conclusions

In [1,2], Carlitz initiated study of the degenerate Bernoulli and Euler polynomials. In recent years, many mathematicians have investigated various degenerate versions of some old and new polynomials and numbers, and found quite a few interesting results [3,4,6,10,11]. It is remarkable that studying degenerate versions is not only limited to polynomials but also can be applied to tran- scendental functions. Indeed, the degenerate gamma functions were introduced and studied in [8,9]. In this paper, we introduced the type 2 degenerate Bernoulli polynomials of the second kind and their higher-order analogues, and studied some identities and expressions for these polynomials. Specifically, we obtained a relation between the type 2 degenerate Bernoulli polynomials of the second and the degenerate Bernoulli polynomials of the second, an identity involving higher-order analogues of those polynomials and the degenerate Stirling numbers of second kind, and an expres- sion of higher-order analogues of those polynomials in terms of the higher-order type 2 degenerate Bernoulli polynomials and the degenerate Stirling numbers of the first kind. In addition, we obtained an identity involving the higher-order degenerate Bernoulli polynomials of the second kind, the type 2 Bernoulli polynomials and Stirling numbers of the second kind, and an identity involving the degenerate central factorial numbers of the second kind, the degenerate Stirling numbers of the first kind and the higher-order degenerate Bernoulli polynomials of the second kind.

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

Funding: This research received no external funding.

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1 DEPARTMENT OF MATHEMATICS,KWANGWOON UNIVERSITY,SEOUL 139-701, REPUBLIC OF KOREA E-mail address: [email protected]

GRADUATE SCHOOL OF EDUCATION,KONKUK UNIVERSITY,SEOUL, 05029, REPUBLIC OF KOREA E-mail address: [email protected]

2 DEPARTMENT OF MATHEMATICS,SOGANG UNIVERSITY,SEOUL 121-742, REPUBLIC OF KOREA E-mail address: [email protected]

4 DEPARTMENT OF MATHEMATICS,KWANGWOON UNIVERSITY,SEOUL 139-701, REPUBLIC OF KOREA, E-mail address: [email protected]