Differential equations associated with higher-order Bernoulli numbers of the second kind

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1 Goba Journa of Pure and Appied Mathematics. ISS Voume 2, umber 3 (206), pp Research India Pubications Differentia equations associated with higher-order Bernoui numbers of the second ind Dae San Kim Department of Mathematics, Sogang University, Seou 2-742, Repubic of Korea. E-mai: dsim@sogang.ac.r Taeyun Kim Department of Mathematics, Kwangwoon University, Seou 39-70, Repubic of Korea. E-mai: tim@w.ac.r Jin-Woo Par Department of Mathematics Education, Daegu University, Gyeongsan-si, Gyeongsangbu-do, 72-74, Repubic of Korea. E-mai: a04700@nu.ac.r Jong-Jin Seo Department of Appied Mathematics, Puyong ationa University, Busan, Repubic of Korea. E-mai: seo20@pnu.ac.r Abstract The purpose of this paper is to derive differentia equations associated with the generating function of higher-order Bernoui numbers of the second ind. In addition, we find some new and interesting identities invoving those numbers arising from our differentia equations. AMS subject cassification: 05A9, B83, 34A30. Keywords: higher-order Bernoui numbers of the second ind, differentia equation. Corresponding author.

2 2504 Dae San Kim, Taeyun Kim, Jin-Woo Par, and Jong-Jin Seo. Introduction For a given positive integer r, the Bernoui poynomias of order r are given by the generating function t r e t e xt = n=0 B (r) n (x)tn, (see [, 2, 8, 0]). n! In the specia case, for x = 0, B n (r) = B n (r) (0) are caed the Bernoui numbers of order r. For a given positive integer r, the Bernoui poynomias of the second ind of order r are defined by the generating function t r ( + t) x = b n (r) og( + t) (x)tn n!. When x = 0, b n (r) = b n (r) (0) are caed the Bernoui numbers of the second ind of order r (see [3, 9, ]). In [4, 5], authors introduced new methods of using diffferentia equations in order to obtain some interesting identities reated to Bernoui numbers of the second ind and Frobenius-Euer numbers of higher order. This idea of using differentia equations turned out to be very usefu toos for studying specia poynomias and mathematica physics (see [3, 4, 5, 6, 7]). In this paper, we derive differentia equations associated with the generating function of higher-order Bernoui numbers of the second ind. In addition, we find some new and interesting identities invoving those numbers arising from our differentia equations. n=0 2. Some identities of higher-order Bernoui numbers of second ind arising from noninear differentia equations Throughout this paper, a derivatives wi be taen with respect to t. For r Z >0,weet r F = F(t,r) = = (og( + t)) r. (2.) og( + t) Then, by (2.), we get F () (t, r) = r( + t) F(t,r + ), (2.2) F (2) (t, r) = r( + t) 2 F(t,r + )+ <r> 2 ( + t) 2 F(t,r + 2), (2.3) F (3) (t, r) = 2r( + t) 3 F(t,r + ) 3 <r> 2 ( + t) 3 F(t,r + 2) <r> 3 ( + t) 3 F(t,r + 3). (2.4)

3 Higher-order Bernoui numbers of the second ind 2505 where <x> n = x(x + ) (x + n ), (n ), and <x> 0 =. So, we are ed to put F () (t, r) = ( ) ( + t) a i ()<r> i F(t,r + i), (2.5) where =, 2,... Taing the derivative with respect to t of (2.5), we have F (+) (t, r) = ( ) ( )( + t) a i ()<r> i F(t,r + i) + ( ) ( + t) a i ()<r> i F () (t, r + i) = ( ) + ( + t) a i ()<r> i F(t,r + i) + ( ) ( + t) a i ()<r> i ( (r + i))( + t) F(t,r + i + ) (2.6) As initia conditions, we have = ( ) + ( + t) a i ()<r> i F(t,r + i) + ( ) + ( + t) a i ()<r> i+ F(t,r + i + ) = ( ) + ( + t) a i ()<r> i F(t,r + i) + + ( ) + ( + t) i=2 a i ()<r> i F(t,r + i). F () (t, r) = ( + t) ra ()F (t, r + ) = r( + t) F(t,r + ). (2.7) Thus, by (2.7), we get a () =. F (2) (t, r) = ( + t) 2 {a (2)rF (t, r + ) + a 2 (2) <r> 2 F(t,r + 2)} = ( + t) 2 {rf(t,r + )+ <r> 2 F(t,r + 2)}. (2.8)

4 2506 Dae San Kim, Taeyun Kim, Jin-Woo Par, and Jong-Jin Seo Thus, by (2.8), we get a (2) = a 2 (2) =. (2.9) Aso, comparing the above, we obtain a ( + ) = a (), a + ( + ) = a (), (2.0) and a i ( + ) = a i () + a i (), (2 i ). (2.) From (2.0), we have a ( + ) = a () = ( )a ( ) = = ( ) 2a (2) =! (2.2) and a + ( + ) = a () = =a () =. (2.3) For i = 2 in (2.), we have a 2 ( + ) = a () + a 2 () = a () + (a ( ) + ( )a 2 ( )) = a () + a ( ) + () 2 a 2 ( ) = a () + a ( ) + () 2 (a ( 2) + ( 2)a 2 ( 2)) = a () + a ( ) + () 2 a ( 2) + () 3 a 2 ( 2) = 2 = () a ( ) + () a 2 (2) = () a ( ), where (x) r = x(x ) (x r + ), (r ), and (x) 0 =. Simiary to the i = 2 case, for i = 3, 4 in (2.), we have 2 (2.4) a 3 ( + ) = () a 2 ( ). (2.5) 3 a 4 ( + ) = () a 3 ( ). (2.6)

5 Higher-order Bernoui numbers of the second ind 2507 Thus, we can deduce that, for 2 i, a i ( + ) = i+ ow, we give expicit expressions for a i (j). From (2.2) and (2.4), we have () a i ( ). (2.7) where a 2 ( + ) = () a ( ) = () (n )! 2 =! =! =!H, a 3 ( + ) = () a 2 ( ) 2 ( = () (n )!H 2 =! H ( H, =! =!H,2, H,j = H,j ) + H 2, + + H, H j,j,(2 j ), j and H, = H = Thus, we deduce that for 2 i, ) (2.8) (2.9) a i ( + ) =!H,i. (2.20)

6 2508 Dae San Kim, Taeyun Kim, Jin-Woo Par, and Jong-Jin Seo ote that H, =!. So, (2.20) aso vaid for i = +. We aso define H,0 = for a. ow,wehave the foowing theorem. Theorem 2.. The foowing famiy of differentia equations F () = ( ) ( + t) ( )! H,i <r> i (og( + t)) i F, ( =, 2,...) have a soution r F = F(t,r) =, og( + t) where H,0 =, for a, H, = H = + + +, H,j = H,j + H 2,j + + H j,j,(2 j ). j (2.2) 3. Appications Reca that the Bernoui numbers of the second ind of order r are defined by the generating function t r = t og( + t)!. (3.) From Theorem 2., we note that d ( ) r dt og( + t) = ( ) ( + t) ( )! ow, we observe that r = og( + t) t r = r+i H,i <r> i. og( + t) t r! = t r! +r = r t ( + r)! + +r + =r t ( + r)!. t r! (3.2) (3.3)

7 Higher-order Bernoui numbers of the second ind 2509 Thus, by (3.3), we get d r = dt og( + t) = r +r () t ( + r)! + = +r () t ( + r)!. (3.4) From (3.4), we note that By (3.2), we get d ( t r+ dt og( + t) = = r r = d ( t r+ dt og( + t) +r () ) r t +r ( + r)! + = ( r) t! + =+r ) r = ( ) ( + t) ( )! +r () t +r ( + r)! ( r) t!. H,i <r> i t i t r+i. og( + t) (3.5) (3.6) ow, ( ) ( + t) ( )! = ( ) ( )! m=0 b m (r+i) t m m! = ( ) ( )! = ( ) ( )! = i +i ( H,i <r> i t i t og( + t) H,i <r> i t i H,i <r> i t i H,i <r> i ( + i ) r+i ( ) ( + ) t s s=0 ) ( ) ( + ) b (r+i) s +i! ( ) ( + ) b (r+i) s t ( + i )! s! ts

8 250 Dae San Kim, Taeyun Kim, Jin-Woo Par, and Jong-Jin Seo = ( ) ( )! ( + i = ( ) ( )! = i +i ) ( ) ( + ) b (r+i) ( + i = ( ) ( )!! + i=max{,} H,i <r> i +i +i ) ( ) ( + ) b (r+i) i=max{,} t ( + i )! H,i <r> i +i +i t! ( + i )!! ( ) <r> i ( + i )! H,i b (r+i) t +i Therefore, by (3.5), (3.6) and (3.7), we obtain the foowing theorem.!. (3.7) Theorem 3.. For = 0,, 2,... and =, 2, 3,..., we have the foowing: (a) for with 0 r or + r, = ( ) ( )!! ( r) i=max{,} +i ( + i )! H,i b (r+i) +i. (b) for r + r, we have References i=max{,} +i + ( ) <r> i + ( ) <r> i ( + i )! H,i b (r+i) +i = 0. [] D. Ding and J. Yang, Some identities reated to the Aposto-Euer and Aposto- Bernoui poynomias, Adv. Stud. Contemp. Math., 20 (200), no., 7 2. [2] S. Gaboury, R. Trembay, B.J. Fugère, Some expicit formuas for certain new casses of Bernoui, Euer and Genocchi poynomias, Proc. Jangjeon Math. Soc., 7 (204), 5 23.

9 Higher-order Bernoui numbers of the second ind 25 [3] D. S. Kim and T. Kim, Some identities for Bernoui numbers of the second ind arising from a non-inear differentia equation, Bu. Korean Math. Soc., 52 (205), [4] T. Kim, Identities invoving Frobenius-Euer poynomias arising from non-inear differentia equations, J. umber Theory, 32 (202), no. 2, [5] T. Kim and D. S. Kim, A note on non-inear Changhee differentia equations, Russ. J. Math. Phys., 23 (206), no., [6] T. Kim and D. S. Kim, Identities invoving degenerate Euer numbers and poynomias arising from non-inear differentia equations, J. oninear Sci. App., 9 (206), [7] H. I. Kwon, T. Kim and J. J. Seo, Some identities of modified degenerate Euer numbers arising from ordinary differentia equations, Int. J. Math. Ana., 0 (206), no. 2, [8] H. Ozden, I.. Cangu, andy. Simse, Remars on q-bernoui numbers associated with Daehee numbers, Adv. Stud. Contemp. Math., 8 (2009), no., [9] T. R. Prabhaar and S. Gupta, Bernoui poynomias of the second ind and genera order, Indian J. Pure App. Math., (980), [0] Y. Simse, Generating functions of the twisted Bernoui numbers and poynomias associated with their interpoation functions,, Adv. Stud. Contemp. Math., 6 (2008), no. 2, [] H. Wang and G. Liu, An expicit formua for higher order Bernoui poynomias of the second ind, Integers, 3 (203), 7 pages.

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