Revisit nonlinear differential equations associated with Bernoulli numbers of the second kind

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1 Global Journal of Pure Applied Mathematics. ISSN Volume 2, Number 2 (206), pp Research India Publications Revisit nonlinear differential equations associated with Bernoulli numbers of the second ind Taeyun Kim Department of Mathematics, Kwangwoon University, Seoul 39-70, Republic of Korea. tim@w.ac.r Dae San Kim Department of Mathematics, Sogang University, Seoul 2-742, Republic of Korea. dsim@sogang.ac.r Hyuc In Kwon Department of Mathematics, Kwangwoon University, Seoul 39-70, Republic of Korea. sura@w.ac.r Jong Jin Seo Department of Applied Mathematics, Puyong National University, Busan, Republic of Korea. seo20@pnu.ac.r Abstract In this paper, we revisit nonlinear differential equations which are derived from the generating function of the Bernoulli numbers of the second ind. In addition, we give explicit new identities for the Bernoulli numbers higher-order Bernoulli numbers of the second ind. AMS subject classification: 05A9, B83, 34A34. Keywords: Bernoulli numbers of the second, nonlinear differential equation. Corresponding author.

2 894 Taeyun Kim et al.. Introduction For r N, it is well nown that the higher order Bernoulli numbers of the second ind are defined by the generating function From (), we note that t r log ( + t) b (r) n n0 b (r) n t n, (see [, 6, ]). () n! B (n r+) n (), (n 0), where B n (r) (x) are Bernoulli polynomials of order r which are given by the generating function t r e t e xt B (r) tn n (x), (see [2, 4, 7]). n! n0 In particular, for r, b n b n () are called the Bernoulli numbers of the second ind. In [2], Bayad Kim studied the following nonlinear differential equations y N (N )! N a (N) y, ( N), (2) d where y () y (t). dt For y y (t) qe t, Bayad-Kim gave explicit formula for Apostol-Bernoulli ± Apostol-Euler numbers polynomials arising from (2). Recently, Kim Kim introduced the following nonlinear differential equations arising from the generating function of Bernoulli numbers of the second in order to obtain explicit identities formulas of the Bernoulli numbers of the second ind as follows: where F (N) (t) ( )N ( + t) N N+ j2 (j )! (N )!H N,j 2 F j, (see [6]), (3) H N,0 for all N H N, H N N, H N,j H N,j N + H N 2,j 2 N + + H 0,j, H 0,j 0 (2 j N).

3 NDE for Bernoulli numbers of the second ind 895 Several authors have studied nonlinear differential equations arising from the generating functions of special numbers polynomials (see [, 2, 3, 4, 5, 8, 6, 7,, 9, 0, 2, 3, 4]). In this paper, we revisit nonlinear differential equations which are derived from the generating functions of the Bernoulli numbers of the second ind. In addition, we give explicit new identities for the Bernoulli numbers of the second ind higher-order Bernoulli numbers of the second ind. 2. Revisit nonlinear differential equations associated with Bernoulli numbers of the second ind Let Then, by (), we get F F (t) log ( + t). (4) F () df (t) dt log ( + t) 2 + t + t F 2. (5) From (5), we can derive the following equations: F 2 ( + t) F (), (6) 2!F 3 ( ) 2 { ( + t) F () + ( + t) 2 F (2)}, (7) 3!F 4 ( ) 3 { ( + t) F () + 3 ( + t) 2 F (2) + ( + t) 3 F (3)}. (8) Thus, we are led to put N!F N+ ( ) N N a i (N)( + t) i F (i), (N 0,, 2,...), (9) d i where F (i) F (t). dt Taing derivatives on both sides of (9), we have { N } N (N + )!F N F () ( ) N a i (N) i ( + t) i F (i) + a i (N)( + t) i F (i+). (0)

4 896 Taeyun Kim et al. Thus, by (5) (0), we get { N (N + )!F N+2 ( ) N+ ia i (N)( + t) i F (i) + { N ( ) N+ ia i (N)( + t) i F (i) + } N a i (N)( + t) i+ F (i+) N+ i2 On the other h, by replacing N by N + in (9), we get N+ (N + )!F N+2 ( ) N+ () } a i (N)( + t) i F (i) a i (N + )( + t) i F (i). (2) Comparing the coefficients on both sides of () (2), we have In addition, by (6) (9), we get Thus, by (5), we see a (N + ) a (N), a N+ (N + ) a N (N), (3) a i (N + ) ia i (N) + a i (N), (2 i N). (4) ( + t) F () F 2 a ()( + t) F (). (5) From (7) (9), we can derive the following equation: a (). (6) ( ) 2 { ( + t) F () + ( + t) 2 F (2)} 2!F 3 (7) ( ) 2 { a (2)( + t) F () + a 2 (2)( + t) 2 F (2)}. By comparing the coefficients on the both sides of (7), we get From (3), we note that a (2), a 2 (2). (8) a (N + ) a (N) a (N ) a (), (9). a N+ (N + ) a N (N) a N (N ) a (). (20)

5 NDE for Bernoulli numbers of the second ind 897 For i 2, 3, 4 in (4), we have a 2 (N + ) a 3 (N + ) a 4 (N + ) So, we can deduce that, for 2 i N, a i (N + ) N N 2 N 3 N i+ Now, we give explicit expressions for a i (N + ). By (2), we get a 2 (N + ) a 3 (N + ) N 0 N a 4 (N + ) So, we deduce that, for 2 i N, a i (N + ) N i+ i 0 N i+ i a (N ), (2) 3 a 2 (N ), (22) 4 a 3 (N ). (23) i a i (N ). (24) 2 2 N, (25) 3 2 a 2 (N 2 ) N N N N , (26) 4 3 a 3 (N 3 ) (27) N N N i+ i 2 0 Therefore, we obtain the following theorem i i (i ) i 2 2. (28)

6 898 Taeyun Kim et al. Theorem 2.. For N N, the nonlinear differential equations N!F N+ ( ) N have a solution F F (t) a i (N) N i i 0 N i i i 2 0 N a i (N)( + t) i F (i) (N, 2, ) log ( + t), where a (N) N i i 2 0 i i (i ) i 2 2, (2 i N). Now, we recall that the Bernoulli numbers of the second ind, b ( 0), are defined by the generating function t log ( + t) t b!. Also, the Bernoulli numbers of the second ind of order r, b (r) (r N), are given by the generating function t r b (r) t log ( + t)!. From (4), we note that F F (t) (29) t log ( + t) t t b! t b +! t b + t +! + t. For a positive integer i,wehave d i F (i) dt log ( + t) b + t i + ( i)! + ( )i i!t i i b +i+ t + i +! + ( )i i!t i. (30)

7 NDE for Bernoulli numbers of the second ind 899 From Theorem 2., we note that N!t N+ F N+ t N+ N! (3) log ( + t) N! b n (N+) t n n!. n0 From Theorem 2., we can derive the following equation: ( ) N t N+ N a i (N)( + t) i F (i) (32) N ( ( ) N t N+ t l ) b +i+ t a i (N) (i) l l! + i +! + ( )i i!t i. l0 Now, we observe that ( ) N t N+ N a i (N) t l (i) l l! l0 N ( ) N t N+ a i (N) n0 n nn+ nn+ n n0 N ( n a i (N)( ) N n N n N N N b +i+ t + i +! ( n ) (i) n ) (i) n b +i+ t n + i + n! b +i+ t n+n+ + i + n! n N a i (N)( ) N (i) n N b +i+ n N ( ) N (i) n N (n) N+ a i (N) t n (33) + i + (n N )! b +i+ t n + i + n!,

8 900 Taeyun Kim et al. ( ) N t N+ N a i (N) t l (i) l l! ( )i i!t i (34) l0 N ( ) N a i (N) (i) l l! ( )i i!t N i+l N ( ) N i0 ( ) N n0 n0 min{n,n} i0 l0 a N i (N) min{n,n} i0 l0 (N i) l l! ( )N i (N i)!t i+l a N i (N)(N i) n i (n i)! ( )N i (N i)!t n ( ) i (N i) n i (n) i (N i)!a N i (N) tn n!. Therefore, by Theorem 2., (3), (32), (33) (34), we obtain the following theorem. Theorem 2.2. For n 0, N, we have min{n,n} n N ( ) i (N i) n i a N i (N), if 0 n N, i i i0 N ) N a N i (N) i b (N+) n References ( n ( ) i (N i) n i i i0 n + ( ) N (N + ) N + n N N ( n N ) (i) n N a i (N) b +i+ + i + if n N +. [] T. Agoh K. Dilcher, Recurrence relations for Nörlund numbers Bernoulli numbers of the second ind, Fibonacci Quart. 48 (200), no., [2] A. Bayad T. Kim, Higher recurrences for Apostol-Bernoulli-Euler numbers, Russ. J. Math. Phys. 9 (202), no., [3] E. Beggs, Differential holomorphic differential operators on noncommutative algebras, Russ. J. Math. Phys. 22 (205), no. 3, [4] L. Carlitz, Some remars on the multiplication theorems for the Bernoulli Euler polynomials, Glas. Mat. Ser. III 6(36) (98), no.,

9 NDE for Bernoulli numbers of the second ind 90 [5] D. Kang, J. Jeong, S.-J. Lee, S.-H. Rim, A note on the Bernoulli polynomials arising from a non-linear differential equation, Proc. Jangjeon Math. Soc. 6 (203), no., [6] D. S. Kim T. Kim, Some identities for Bernoulli numbers of the second ind arising from a non-linear differential equation, Bull. Korean Math. Soc. 52 (205), no. 6, [7] D. S. Kim, T. Kim,Y. H. Kim, S. H. Lee, Some arithmetic properties of Bernoulli Euler numbers, Adv. Stud. Contemp. Math. (Kyungshang) 22 (202), no. 4, [8] D. S. Kim, N. Lee, J. Na, K. H. Par, Abundant symmetry for higher-order Bernoulli polynomials (I), Adv. Stud. Contemp. Math. (Kyungshang) 23 (203), no. 3, [9] T. Kim D. S. Kim, Identities involving degenerate Euler numbers polynomials arising from non-linear differentiable equations, J. Nonlinear Sci. Appl. 9 (206), [0], A note on nonlinear Changhee differential equations, Russ. J. Math. Phys. 23 (206), no., 5. [] T. Kim, D. S. Kim, D. V. Dolgy, J.-J. Seo, Bernoulli polynomials of the second ind their identities arising from umbral calculus, J. Nonlinear Sci. Appl. 9 (206), [2] T. Kim, T. Komatsu, S.-H. Lee, J.-J. Seo, q-poly-cauchy numbers associated with Jacson integral, J. Comput. Anal. Appl. 8 (205), no. 4, [3] J.-W. Par, New approach to q-bernoulli polynomials with weight or wea weight, Adv. Stud. Contemp. Math. (Kyungshang) 24 (204), no., [4] P. T. Young, A p-adic formula for the Nörlund numbers for Bernoulli numbers of the second ind, Congr. Numer. 20 (200),