Generalized Degenerate Bernoulli Numbers and Polynomials Arising

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As to the generalized degenerate Bernoulli polynomials, we represent them in terms of the degenerate Stirling polynomials of the second kind. For the rest of this section, we recall the necessary facts that are needed throughout this paper. For any λ ∈ R, the degenerate exponential functions are defined by ∞ n x t 1 (1) eλ (t)= ∑ (x)n,λ , eλ (t)= eλ (t), (see [6,9]), n=0 n! x xt where (x)0,λ = 1, (x)n,λ = x(x − λ)··· (x − (n − 1)λ), (n ≥ 1). Note that lim eλ (t)= e . λ→0 Let logλ (t) be the compositional inverse function of eλ (t) with logλ eλ (t) = eλ logλ (t) = t. Then we have ∞ n n−1 t (2) logλ (1 +t)= ∑ λ (1)n,1/λ , (see [7]). n=1 n! In [7], the degenerate Stirling numbers of the first kind are defined by n (3) (x)n = ∑ S1,λ (n,l)(x)l,λ , (n ≥ 0), l=0 where (x)0 = 1, (x)n = x(x − 1)(x − 2)··· (x − n + 1), (n ≥ 1). As the inversion formula of (3), the degenerate Stirling numbers of the second kind are defined by n (4) (x)n,λ = ∑ S2,λ (n,k)(x)k, (n ≥ 0), (see [7]). k=0 From (3) and (4), we note that ∞ n 1 k t (5) log (1 +t) = S (n,k) , k! λ ∑ 1,λ n! n=k and ∞ n 1 k t (6) e (t) − 1 = S (n,k) , (k ≥ 0), (see [7]). k! λ ∑ 2,λ n! n=k It is well known that the Gauss hypergeometric function is given by ∞ k a, b haikhbik x (7) 2F1 x = , (see [1,2,12]), c ∑ hci k! k=0 k where hai0 = 1, haik = a(a + 1)··· (a + k − 1), (k ≥ 1). The Pfaff’s transformation formula is given by a, b a, c − b x (8) F x = (1 − x)−a F , (see [1,2]), 2 1 c 2 1 c x − 1

and the Euler’s transformation formula is given by a, b c−a−b c − a, c − b (9) 2F1 x = (1 − x) 2F1 x , (see [1,2]). c c

n The Eulerian number k is the number of permutation {1,2,3,..., n} having k permutation ascents. The Eulerian numbers are given explicitly by the ﬁnite sum n k+1 n + 1 (10) = ∑(−1) j (k − j + 1)n, (n,k ≥ 0,n ≥ k), k j=0 j GENERALIZED DEGENERATE BERNOULLI NUMBERS AND POLYNOMIALS 3 and n n (11) ∑ = n!, (see [4,5]). k=0 k For n,m ≥ 0, we have n−m n n − k n−k−m (12) = ∑ S2(n,k) (−1) k!, (see [5]), m k=0 m and n n x + k (13) xn = ∑ , (see [4,5]). k=0 k n Recently, the degenerate Stirling polynomials of the second kind are deﬁned by ∞ n 1 k t (14) e (t) − 1 ex (t)= S (n,k|x) , (k ≥ 0), (see [8]). k! λ λ ∑ 2,λ n! n=k Thus, by (14), we get n n (15) S2,λ (n,k|x) = ∑ S2,λ (l,k)(x)n−l,λ , (see [8]), l=k l n n = ∑ S2,λ (l,k)(x)n−l,λ , (n ≥ 0). l=0 l

For x = 0, S2,λ (n,k)= S2,λ (n,k|0), (n,k ≥ 0,n ≥ k), are called the degenerate Stirling numbers of the second kind. Carlitz introduced the degenerate Bernoulli polynomials given by ∞ n t x t (16) eλ (t)= ∑ βn,λ (x) , (see [3]). eλ (t) − 1 n=0 n!

When x = 0, βn,λ = βn,λ (0), (n ≥ 0), are called the degenerate Bernolli numbers.

2. GENERALIZED DEGENERATE BERNOULLI NUMBERS By (1) and (2), we get ∞ t 1 1 n (17) = λ n−1(1) e (t) − 1 e (t) − 1 e (t) − 1 ∑ n,1/λ n! λ λ λ n=1 ∞ k λ (1)k+1,1/λ 1 k = · e (t) − 1 ∑ k + 1 k! λ k=0 ∞ k ∞ n λ (1)k+1,1/λ t = ∑ ∑ S2,λ (n,k) k=0 k + 1 n=k n! ∞ n k n λ (1)k+1,1/λ t = ∑ ∑ S2,λ (n,k) . n=0 k=0 k + 1 n! Therefore, by (16) and (17), we obtain the following theorem. Theorem 1. For n ≥ 0, we have n k λ (1)k+1,1/λ βn,λ = ∑ S2,λ (n,k). k=0 k + 1 4 TAEKYUNKIM 1,†, DAE SAN KIM 2*, LEE-CHAE JANG 3**, HYUNSEOK LEE 1,‡, AND HANYOUNG KIM 1,††

Replacing t by logλ (1 +t) in (16), we get ∞ log (1 +t) 1 k (18) λ = β log (1 +t) e (log (1 +t)) − 1 ∑ k,λ k! λ λ λ k=0 ∞ ∞ tn = ∑ βk,λ ∑ S1,λ (n,k) k=0 n=k n! ∞ n tn = ∑ ∑ S1,λ (n,k)βk,λ . n=0 k=0 n! On the other hand, by (2), we get ∞ n logλ (1 +t) 1 1 n−1 t (19) = logλ (1 +t) = ∑ λ (1)n,1/λ eλ (logλ (1 +t)) − 1 t t n=1 n! ∞ n n λ (1)n+1,1/λ t = ∑ . n=0 n + 1 n! Therefore, by (18) and (19), we obtain the following theorem. Theorem 2. For n ≥ 0, we have n 1 n ∑ S1,λ (n,k)βk,λ = λ (1)n+1,1/λ . k=0 n + 1 From (16) and (17), we note that ∞ n ∞ t 1 1 n (20) β = λ n−1(1) e (t) − 1 ∑ n,λ n! e (t) − 1 ∑ n,1/λ n! λ n=0 λ n=1 ∞ n n n (−1) (1)n+1,1/λ λ n! (1 − e (t)) = ∑ λ n=0 (n + 1)! n! ∞ h1 − λi h1i (1 − e (t))n = ∑ n n λ n=0 h2in n! 1 − λ, 1 = 2F1 1 − e (t) . 2 λ

In view of (20), we may consider the generalized degenerate Bernoulli numbers given in terms of Gauss hypergeometric function by 1 − λ, 1 ∞ tn (21) F 1 − e (t) = β (p) , 2 1 p + 2 λ ∑ n,λ n! n=0

Z (0 ) where p ∈ with p ≥−1. When p = 0, βn,λ = βn,λ , (n ≥ 0). Let us take p = −1 in (21). Then we have ∞ n (−1) t 1 − λ, 1 β = 2F1 1 − e (t) ∑ n,λ n! 1 λ n=0 ∞ ∞ (λ − 1)n n λ − 1 n (22) = e (t) − 1 = e (t) − 1 ∑ n! λ ∑ n λ n=0 n=0 ∞ n λ−1 t = eλ (t) = ∑ (λ − 1)n,λ . n=0 n! GENERALIZED DEGENERATE BERNOULLI NUMBERS AND POLYNOMIALS 5

By comparing the coefﬁcients on the both sides of (22), we get (−1) (23) βn,λ = (λ − 1)n,λ , (n ≥ 0). From (21), we note that ∞ tn 1 − λ, 1 ∞ h1 − λi h1i (1 − e (t))k (24) β (p) = F 1 − e (t) = k k λ ∑ n,λ n! 2 1 p + 2 λ ∑ hp + 2i k! n=0 k=0 k ∞ k λ (1)k+1,1/λ k! 1 k = (p + 1)! e (t) − 1 ∑ (p + k + 1)! k! λ k=0 ∞ k ∞ n λ (1)k+1,1/λ k! t = (p + 1)! ∑ ∑ S2,λ (n,k) k=0 (p + k + 1)! n=k n! ∞ n k n λ (1)k+1,1/λ t = S (n,k) . ∑ ∑ p+k+1 2,λ n! n=0 k=0 p+1 Therefore, by comparing the coefﬁcients on both sides of (24), we obtain the following theorem. Theorem 3. For n ≥ 0 and p ≥−1, we have n λ k(1) (p) k+1,1/λ S n k βn,λ = ∑ p+k+1 2,λ ( , ). k=0 p+1 From (6), we get

∞ n k t 1 k 1 k (25) S (n,k) = e (t) − 1 = (−1)k−lel (t) ∑ 2,λ n! k! λ k! ∑ l λ n=k l=0 ∞ k n 1 k k−l t = ∑ ∑ (−1) (l)n,λ . n=0 k! l=0 l n! By (25), we get

k k k!S (n,k), if n ≥ k, (26) (−1)k−l (l) = 2,λ ∑ n,λ 0, otherwise. l=0 l Let △ be a difference operator with △ f (x)= f (x + 1) − f (x). Then we have n n △n f (x)= ∑ (−1)n−k f (x + k). k=0 k From (26), we have k (27) k!S2,λ (n,k)= △ (0)n,λ , (n,k ≥ 0,n ≥ k). In light of (12), we may consider the degenerate type Eulerian numbers given by n−m k n−m n k n − k △ (0)n,λ (28) (−1) = ∑ λ (1)k+1,1/λ . mλ k=0 m k! By (27) and (28), we get n−m n−m n k n − k (29) (−1) = ∑ λ (1)k+1,1/λ S2,λ (n,k). mλ k=0 m 6 TAEKYUNKIM 1,†, DAE SAN KIM 2*, LEE-CHAE JANG 3**, HYUNSEOK LEE 1,‡, AND HANYOUNG KIM 1,††

We observe that n n n−k k n−k k n − k m (30) ∑ λ (1)k+1,1/λ S2,λ (n,k)(t + 1) = ∑ λ (1)k+1,1/λ S2,λ (n,k) ∑ t k=0 k=0 m=0 m n n−m k n − k m = ∑ ∑ λ (1)k+1,1/λ S2,λ (n,k) t m=0 k=0 m n n = ∑ (−1)n−m tm. m=0 mλ From (30) and Theorem 3, we note that n −1 (p) k p + k + 1 (31) βn,λ = ∑ λ (1)k+1,1/λ S2,λ (n,k) k=0 k n 1 k p k = (p + 1) ∑ λ (1)k+1,1/λ S2,λ (n,k) t (1 −t) dt k=0 Z0 1 n n−k k n p t = (p + 1) ∑ λ (1 −t) t (1)k+1,1/λ S2,λ (n,k) 1 + dt Z0 k=0 1 −t 1 n n t k = (p + 1) (1 −t)nt p ∑ (−1)n−k dt Z0 k=0 kλ 1 −t n n 1 = (p + 1) ∑ (−1)n−k (1 −t)n−kt p+kdt k=0 kλ Z0 n n (n − k)!(p + k)! = (p + 1) ∑ (−1)n−k k=0 kλ (p + n + 1)! p + 1 n n p + n −1 = ∑ (−1)n−k . n + p + 1 k=0 kλ p + k Therefore, by (31), we obtain the following theorem. Theorem 4. For n, p ≥ 0, we have n −1 (p) p + 1 n n−k p + n βn,λ = ∑ (−1) . n + p + 1 k=0 kλ p + k n Let r be a positive integer. The unsigned r-Stirling number of the ﬁrst kind k r is the number of permutations of the set [n]= {1,2,3,...,n} with exactly k disjoint cycles in such a way that the n numbers 1,2,3,...,r are in distinct cycles, while the r-Stirling number of the second kind k r counts the number of partitions of the set [n] into k non-empty disjoint subsets in such a way that the numbers 1,2,3,...,r are in distinct subsets. In [13], Kim-Kim-Lee-Park introduced the n n unsigned degenerate r-Stirling numbers of the ﬁrst kind k r,λ as a degenerate version of k r and r n n the degenerate -Stirling number of the second kind k r,λ as a degenerate version of k r. It is known that the degenerate r-Stirling numbers of the second kind are given by n n + r (32) (x + r)n,λ = ∑ (x)k, (n ≥ 1). k=0 k + rr,λ From (32), we note that ∞ n 1 k n + r t (33) e (t) − 1 er (t)= , (k ≥ 0, r ≥ 1). k! λ λ ∑ k + r n! n=k r,λ GENERALIZED DEGENERATE BERNOULLI NUMBERS AND POLYNOMIALS 7

By the Euler’s transformation formula in (9), we get ∞ tn 1 − λ, 1 (34) β (p) = F 1 − e (t) ∑ n,λ n! 2 1 p + 2 λ n=0 ∞ k p+λ hp + 1 + λiλ hp + 1ik (1 − eλ (t)) = eλ (t) ∑ k=0 hp + 2iλ k! ∞ k+p k p + 1 λ h1ip+k+1,1/λ (−1) k = e (t) − 1 ep+k(t) λ ph1i ∑ p + k + 1 k! λ λ p+1,1/λ k=0 ∞ k+p ∞ m ∞ l p + 1 λ h1ip+k+1,1/λ k m + p t t = p ∑ (−1) ∑ ∑(k)l,λ λ h1ip+1,1/λ k=0 p + k + 1 m=k k + p p,λ m! l=0 l! ∞ m k+p m ∞ l p + 1 λ h1ip+k+1,1/λ k m + p t t = ∑ p ∑ (−1) ∑(k)l,λ m=0 λ h1ip+1,1/λ k=0 p + k + 1 k + p p,λ m! l=0 l! ∞ n n p + 1 m (−λ)k m + p tn = ∑ ∑ ∑ h1ip+k+1,1/λ (k)n−m,λ , n=0 m=0 mh1ip+1,1/λ k=0 p + k + 1 k + p p,λ n! where hxi0,λ = 1, hxin,λ = x(x + λ)··· (x + (n − 1)λ), (n ≥ 1). Therefore, we obtain the following theorem. Theorem 5. For n ≥ 1 and p ≥ 0, we have n m k (p) p + 1 n (−λ) m + p βn,λ = ∑ ∑ h1ip+k+1,1/λ (k)n−m,λ . h1ip+1,1/λ m=0 k=0 m p + k + 1 k + p p,λ Note that n m (p) p + 1 n k (p + k)! m + p lim βn,λ = ∑ ∑ (−1) (k)n−m. λ→0 p! m=0 k=0 m p + k + 1 k + p p From Theorem 3, we have ∞ n ∞ n k n t λ (1)k+1,1/λ t (35) β (p) = S (n,k) ∑ n,λ n! ∑ ∑ p+k+1 2,λ n! n=0 n=0 k=0 p+1 ∞ k λ (1)k+1,1/λ 1 k = e (t) − 1 ∑ p+k+1 k! λ k=0 p+1 ∞ p!k! 1 k = (p + 1) λ k(1) e (t) − 1 ∑ (k + p + 1)! k+1,1/λ k! λ k=0 ∞ k k 1 (−1) λ (1)k+1,1/λ k = (p + 1) 1 − e (t) (1 − x)pxkdx ∑ k! λ Z k=0 0 ∞ 1 k λ − 1 k p k = (p + 1) ∑(−1) 1 − eλ (t) (1 − x) x dx k Z0 k=0 1 λ−1 = (p + 1) (1 − x)p 1 − x(1 − e (t)) dx. Z λ 0 Therefore, we obtain the following theorem. Theorem 6. For p ≥ 0, we have ∞ n 1 t λ−1 β (p) = (p + 1) (1 − x)p 1 − x(1 − e (t)) dx. ∑ n,λ n! Z λ n=0 0 8 TAEKYUNKIM 1,†, DAE SAN KIM 2*, LEE-CHAE JANG 3**, HYUNSEOK LEE 1,‡, AND HANYOUNG KIM 1,††

3. GENERALIZED DEGENERATE BERNOULLI POLYNOMIALS In this section, we consider the generalized degenerate Bernoulli polynomials which are derived from the Gauss hypergeometric function. In light of (21), we deﬁne the generalized degenerate Bernoulli polynomials by

∞ n (p) t 1 − λ, 1 x (36) β (x) = 2F1 1 − e (t) e (t). ∑ n,λ n! p + 2 λ λ n=0

(p) (p) When x = 0, βn,λ (0)= βn,λ , (n ≥ 0). Thus, by (36), we get

∞ tn 1 − λ, 1 (37) β (p)(x) = F 1 − e (t) ex (t) ∑ n,λ n! 2 1 p + 2 λ λ n=0 ∞ l ∞ m (p) t t = ∑ βl,λ ∑ (x)m,λ l=0 l! m=0 m! ∞ n n n (p) t = ∑ ∑ βl,λ (x)n−l,λ . n=0 l=0 l n!

Therefore, by comparing the coefﬁcients on both sides of (37), we obtain the following theorem.

Theorem 7. For n ≥ 0, we have

n (p) n (p) βn,λ (x)= ∑ βl,λ (x)n−l,λ . l=0 l

From (36), we note that

∞ n d (p) t 1 − λ, 1 d x β (x) = 2F1 1 − e (t) e (t) ∑ dx n,λ n! p + 2 λ dx λ n=1

1 1 − λ, 1 x = log(1 + λt)2F1 1 − e (t) e (t) λ p + 2 λ λ

∞ l−1 l ∞ m 1 (−1) λ l (p) t = ∑ t ∑ βm,λ (x) λ l=1 l m=0 m! (p) ∞ n (−λ)l−1 n!β (x) tn = ∑ ∑ n−l,λ n=1 l=1 l (n − l)! n!

Thus, we have

n d (p) l−1 n (p) βn,λ (x)= ∑(−λ) (l − 1)! βn−l,λ (x). dx l=1 l

Proposition 8. For n ≥ 1, we have

n d (p) l−1 n (p) βn,λ (x)= ∑(−λ) (l − 1)! βn−l,λ (x). dx l=1 l GENERALIZED DEGENERATE BERNOULLI NUMBERS AND POLYNOMIALS 9

By (14), we easily get

∞ n t 1 k S (n,k|x) = e (t) − 1 ex (t) ∑ 2,λ n! k! λ λ n=k k 1 k k−l l+x = ∑ (−1) eλ (t) k! l=0 l ∞ k n 1 k−l t = ∑ ∑(−1) (l + x)n,λ . n=0 k! l=0 n! Thus we have

1 k k S (n,k|x), if n ≥ k, (38) (−1)k−l (l + x) = 2,λ ∑ n,λ 0, otherwise. k! l=0 l From (38), we note that 1 S (n,k|x)= △k(x) , (n ≥ k). 2,λ k! n,λ Lemma 9. For n,k ≥ 0 with n ≥ k, we have 1 S (n,k|x)= △k(x) , (n ≥ k). 2,λ k! n,λ Now, we observe that

∞ n ∞ t (p + 1)!k! 1 k (39) β (p)(x) = λ k(1) e (t) − 1 ex (t) ∑ n,λ n! ∑ (p + k + 1)! k+1,1/λ k! λ λ n=0 k=0 ∞ k ∞ n (1)k+1,1/λ λ t = S (n,k|x) ∑ p+k+1 ∑ 2,λ n! k=0 p+1 n=k ∞ n k n (1)k+1,1/λ λ t = S (n,k|x) . ∑ ∑ p+k+1 2,λ n! n=0 k=0 p+1 Therefore, by (39), we obtain the following theorem.

Theorem 10. For n ≥ 0, we have

n k (p) (1)k+1,1/λ λ βn,λ (x)= ∑ p+k+1 S2,λ (n,k|x). k=0 p+1 Remark 11. Let p be a nonnegative integer. Then, by Theorem 7 and (36), we easily get

n (p) n (p) βn,λ (x + y)= ∑ βk,λ (x)(y)n−k,λ , (n ≥ 0), k=0 k n−1 (p) (p) n (p) βn,λ (x + 1) − βn,λ (x) = ∑ βk,λ (x)(1)n−k,λ , (n ≥ 1), k=0 k n (p) n (p) n−k βn,λ (mx) = ∑ βk,λ (x)(m − 1) (x)n−k,λ/m−1, (n ≥ 0,m ≥ 2). k=0 k 10 TAEKYUN KIM 1,†, DAE SAN KIM 2*, LEE-CHAE JANG 3**, HYUNSEOK LEE 1,‡, AND HANYOUNG KIM 1,††

4. CONCLUSION In recent years, degenerate versions of many special polynomials and numbers have been investigated by means of various different tools including generating functions, combinatorial methods, umbral calculus, p-adic analysis, differential equations, special functions, probability theory and analytic number theory. Studying degenerate versions of some special polynomials and numbers, which was initiated by Carlitz in [3], has yielded many interesting arithmetic and combinatorial results (see [7-13 and references therein]) and has potential to find many applications in diverse areas. A new family of p-Bernoulli numbers and polynomials, which reduce to the classical Bernoulli numbers and polynomials for p = 0, was introduced by Rahmani in [14] by means of the Gauss hypergeometric function. Motivated by that paper, we were interested in finding a degenerate version of those numbers and polynomials. Indeed, the generalized degenerate Bernoulli numbers and polynomials, which reduce to the Carlitz degenerate Bernoulli numbers and polynomials for p = 0, were introduced again in terms of the Gauss hypergeometric function. In addition, the degenerate type Eulerian numbers was introduced as a degenerate version of Eulerian numbers. In this paper, we expressed the generalized degenerate Bernoulli numbers in terms of the degenerate Stirling numbers of the second kind, of the degenerate type Eulerian numbers, of the degenerate p-Stirling numbers of the second kind and of an integral on the unit interval. In dddi- tion, we represented the generalized degenerate Bernoulli polynomials in terms of the degenerate Stirling polynomials of the second kind. It is one of our future projects to continue pursuing this line of research. Namely, by studying degenerate versions of some special polynomials and numbers, we want to find their applications in mathematics, science and engineering.

Acknowledgements The authors would like to thank the reviewers for their valuable comments and suggestions and Jangjeon Institute for Mathematical Science for the support of this research.

Funding Not applicable.

Availability of data and materials Not applicable.

Competing interests The authors declare that they have no conﬂicts of interest.

Authors’ contributions TK and DSK conceived of the framework and structured the whole paper; DSK and TK wrote the paper; HL typed; LCJ, and HYK checked the results of the paper; DSK and TK completed the revision of the paper. All authors have read and approved the ﬁnal version of the manuscript.

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1 DEPARTMENT OF MATHEMATICS,KWANGWOON UNIVERSITY,SEOUL 139-701, REPUBLIC OF KOREA Email address: [email protected]† , [email protected]‡ , [email protected]

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

3 GRADUATE SCHOOL OF EDUCATION,KONKUK UNIVERSITY,SEOUL 143-701, REPUBLIC OF KOREA Email address: [email protected]**