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Some Identities of Type 2 Degenerate Bernoulli Polynomials of the Second

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Some Identities of Type 2 Degenerate Bernoulli Polynomials of the Second SOME IDENTITIES ON TYPE 2 DEGENERATE BERNOULLI POLYNOMIALS OF THE SECOND KIND TAEKYUN KIM 1, LEE-CHAE JANG2,∗, DAE SAN KIM3, AND HAN YOUNG KIM4 ABSTRACT. In recent years, many mathematicians studied various degenerate versions of some spe- cial polynomials of which quite a few interesting results were discovered. 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 expression 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. Keywords: type 2 degenerate Bernoulli polynomials of the second kind; degenerate central fac- torial numbers of the second kind. 1. INTRODUCTION As is known, the type 2 Bernoulli polynomials are defined by the generating function t ∞ tn (1) ext = B∗(x) , (see [7]). t − −t ∑ n e e n=0 n! From (1), we note that x + 1 (2) B∗(x)= 2n−1B , (n ≥ 0), n n 2 where Bn(x) are the ordinary Bernoulli polynomials given by t ∞ tn ext = B (x) . t − ∑ n e 1 n=0 n! arXiv:1908.06587v1 [math.NT] 19 Aug 2019 Also, the type 2 Euler polynomials are given by ∞ n xt 2 xt ∗ t (3) e secht = t −t e = ∑ En (x) , (see [7]). e + e n=0 n! Note that x + 1 (4) E∗(x)= 2nE , (n ≥ 0), n n 2 where En(x) are the ordinary Euler polynomials given by ∞ n 2 xt t t e = ∑ En(x) , (see [3, 4, 5, 8, 10, 11]). e + 1 n=0 n! 1 2 Some identities on type 2 degenerate Bernoulli polynomials of the second kind 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 gener- ating 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 polynomi- als. 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 BERNOULLI POLYNOMIALS OF THE SECOND KIND 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.
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