Some identities involving Changhee polynomials arising from a differential equation 1

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1 Global Journal of Pure and Applied Mathematics. ISS Volume, umber 6 (6), pp Research India Publications Some identities involving Changhee polynomials arising from a differential equation Jongyum Kwon Department of Mathematics educations and RIS, Gyeongsang ational University, JinJu, 588, Republic of Korea. YiJin Choi Gyeongnam Science High School, Jinju, 56, Republic of Korea. MinSeo Jang Gyeongnam Science High School, Jinju, 56, Republic of Korea. SiOn Yang Gyeongnam Science High School, Jinju, 56, Republic of Korea. MinSu Seong Gyeongnam Science High School, Jinju, 56, Republic of Korea. This wor was supported by the Korea Foundation for the Advancement of Science and Creativity (KOFAC) grant funded by the Ministry of Science, ICT and Future Planning (MSIP). Corresponding author.

2 Abstract In this paper, we derive a family of differential equations from the generating functions of the Changhee polynomials and study the solutions of these differential equations. Also we give some new and interesting identities and formulas related to Euler polynomials for the Changhee polynomials of higher order by using our non-linear differential equations. AMS subject classification: 5A9, B37, 34A3. Keywords: Changhee polynomials, non-linear differential equation.. Introduction The classical Euler polynomials E n (x) are defined by generating functions as follows: e t + ext E n (x) tn, (see [,3,5-9,9]). (.) In the special case x, E n () E n for n,,... are called the n-th Euler numbers. The Euler polynomials and numbers have numerous important applications in combinatorics, number theory and numerical analysis. Several researchers have been studied these polynomials (see [,3,5-9,9]). The higher-order Euler polynomials E (r) (x) are defined by generating functions as follows: ( ) r e t e xt + n E (r) n (x)tn,(see [9, 7]). (.) In the special case x, E n (r) () E(r) n for n,,...are called the n-th higher-order Euler numbers. The Changhee polynomials are defined by generating functions as follows: + t ( + t)x Ch n (x) tn, (see [,,,8]), (.3) when x, Ch n () Ch n for n,,...are called Changhee numbers. Recently, several authors have studied for the differential equation related to various special polynomials (see [4,-6,9]). In this paper, we use the developed method by T. Kim and D.S. Kim (see []) and derive a family of non-linear differential equations from (.3) and study the solutions of these differential equations. Then we give some new and interesting identities and formulas related to Euler polynomials for the Changhee polynomials by using our differential equations.

3 Some identities involving Changhee polynomials The identities and formulas for Changhee polynomials of differential equations Throughout this paper, we put F F(t) + t + ( + t) ( ) n ( + t) n. (.4) Differentiate both sides in (.4) with respect to t to get ( F (t) e log(+t)) + t F F. (.5) ow we repeat the process from (.5). F FF ( )!F 3, F (3) ( ) 3 3!F 4, F (4) ( ) 4 4!F 5, F (5) ( ) 5 5!F 6. (.6) Continuing same process, we get F () ( )!F +, for all (.7) Since then we get F () F F(t) + t Ch n t n, ( d ) F ( d dx dx n ( ) + t ) Ch n t n t n Ch n (n )! Ch n+ t n. (.8) (.9)

4 486 Jongyum Kwon, et al. ow, we consider that the higher-order Changhee numbers are defined by the generating function to be ( ) r Ch (r) t n n + t. (.) Then, by (.7), (.9) and (.), we get t n Ch n+ ( )! + Ch (+) t n n. (.) Therefore, by (.), we obtain the following theorem. Theorem.. For each n,wehave Indeed, Ch n+ ( )! Ch (+) n. (.) + t + ( + t) + e log(+t) E m (log( + t))m m! E m S (n, m) tn nm ( n ) E m S (n, m) t n. (.3) Hence, by (.3), we obtain the following theorem. Theorem.. For each n,wehave n+ E m S (n +,m) ( ) ( )! Ch (+) n. (.4)

5 Some identities involving Changhee polynomials 486 From the higher-order Changhee number, we note that ( ) r ( + t + e log(+t) ) r E m (r) (log( + t))m m! E m (r) S (n, m) tn nm ( n ) E m (r) S (n, m) t n. (.5) Therefore, by (.) and (.5), we obtain the following theorem. Theorem.3. For each n,wehave Ch n+ ( )! n E (+) m S (n, m). (.6) ow, we consider that differential equation for Changhee polynomials. We set Differentiate both side in (.7), we get so that Therefore, we have F F(t) + t + e F (t) df dt elog(+t) ( + e log(+t) ) + t ( + elog(+t) ) + ( + e log(+t) ) + t F( + F) + t, Differentiate both sides in (.) with respect to t gives log(+t), (.7) (.8) ( + t)f F + F. (.9) F F + ( + t)f. (.) FF F + F + ( + t)f F + ( + t)f. (.)

6 486 Jongyum Kwon, et al. Multiplying ( + t) on the both sides of (.) and using (.), we get ( + t)ff ( + t)f + ( + t) F F( F + F ) ( + t)f + ( + t) F F 3 F + ( + t)f + ( + t) F F + 4( + t)f + ( + t) F (.) Repeating the wor on (.), we have 6F F F + 4F + 4( + t)f + ( + t)f + ( + t) F (3) 6F + 6( + t)f + ( + t) F (3) 6( + t)f F 6( + t)f + 6( + t) F + ( + t) 3 F (3) 6F ( F + F ) 6( + t)f + 6( + t) F + ( + t) 3 F (3) so that 6F 4 6F 3 + 6( + t)f + 6( + t) F + ( + t) 3 F (3) 6F + 8( + t)f + 9( + t) F + ( + t) 3 F (3) Continuing this process, we are led to set, for all,, 3,...!F + ( d ) F(t). where F () dt Differentiating (.3), we obtain ( + t) F (), (.3)!( + )F F ( + )!( + t)f F { ( + t) F () + ( + t) F (+)} + ( + t) F () + ( + t) F () ( + t) ( + ) F () + ( + t) + F (+),

7 Some identities involving Changhee polynomials 4863 and by (.9) ( + )!F + ( + )!F ( + t) + ( + ) ( + t) ( F (+) ( + t) F () + + ( + t) + F (+) ( + ) F + ( + ) + ( + t) ( ( + ) F ) F () ( + t) ( ( + t) F () ) F () + ( + t) + ( + t) { ( + + ) + ) F () F (+) + } F () + ( + t) + F (+) (.4) On the other hand, by replacing by ( + ) in (.3) and comparing to (.4), we get the followings. ( + )!F + ( + ) F + ( + t) + F (+) + ( + t) { ( + + ) + } F () Therefore, we have and for,,...,, a (+) a (+) ( + ), a (+) + ( + + ) (.5) (.6) + (.7) We see a () a () from (.), so that, from (.6) we have the followings easily. a (+) ( + ) ( + )a ( ) ( + )!, a (+) + a() (.8)

8 4864 Jongyum Kwon, et al. Also, for,,...,, from (.7) we get a (+) ( + + ) + ( + + ) { ( + )a ( ) ( + + ) <> a ( ) ( + + ) <>{ ( + )a ( ) + ( + + )a ( ) + a ( ) } + a () + ( + + )a ( ) + a ( ) + ( + + ) <3> a ( ) + n + } ( + + ) < n> a (n) (.9) where <i> ( ) ( i + ). Furthermore, a (+) ( + + ) <+ > a () + n n n n ( + + ) < n > a (n ) n n n n n n n n 3 3 Repeating this process, a (+) ( + + ) < n > a (n ) ( + + ) < n > (n + ) <n n > a (n ) (n + + )(n + ) ( + + )<+ n > a (n ) n (n + + )(n + ) (n + )(n + ) ( + + )<+ n 3> a (n 3) 3 n n m n m m (n + + )(n + ) (n + )(n + ) (n m + (m 3))(n m + (m )) ( + + ) <+(m ) nm> a (n m) m

9 Some identities involving Changhee polynomials 4865 n n n n (n + + )(n + ) (n + )(n + ) (n + 3)(n + ) ( + + )<+( ) n> a (n ) n n (n + + )(n + ) (n + )(n + ) (n + 3)(n + ) ( + + )<++> (n + )(n + ) n m ( + + )! (n m + m + )(n m + m + ) m n m m where n +. Therefore, we obtain the following theorem. Theorem.4. For each,,..., the nonlinear differential equations!f + ( + t) F (), (.3) have a solution F(t), where a()!, and for,,...,, + t with n. References ( + )! n m m n m m (n m + m + )(n m + m + ), [] A. Bayad, T. Kim, Identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang) (), no., [] D.V. Dolgy, T. Kim, J.J. Seo, Symmetric identities for Changhee polynomilas arising from the fermionic p-adic integral on Z p,adv. Stud. Contemp. Math. 6 (6), no., [3] K. W. Hwang, D. V. Dolgy, D. S. Kim, T. Kim, S. H. Lee, Some theorems on Bernoulli and Euler numbers, Ars Combin., 9(3), [4] D. S. Kim, T. Kim, Some identities for Bernoulli numbers of the second ind arising from a nonlinear differential equation, Bull. Korean Math. Soc., 5(5),.

10 4866 Jongyum Kwon, et al. [5] D. S. Kim, T. Kim, Y. H. Kim, D. V. Dolgy, A note on Eulerian polynomials associated with Bernoulli and Euler numbers and polynomials, Adv. Stud. Contemp. Math. (), no.3, [6] T. Kim, q-euler numbers and polynomials associated with p-adic q-integrals, J. onlinear Math. Phys., 4 (7), 5 7. [7] T. Kim, Symmetry p-adic invariant integral on Z p for Bernoulli and Euler polynomials, J. Diff. Equ. Appl., 4(8), [8] T. Kim, Symmetry p-adic invariant on Z p for Bernoulli and Euler polynomials,j. Difference. Equ., 4 (8), [9] T. Kim, ew approach to q-euler polynomials of higher-order, Russ. J. Math. Phys., 7 (), 8 5. [] T. Kim, Identities involving Frobenius-Euler polynomials arising from non-linear differential equation, J. of umber Theory, 3(), [] T. Kim, D. V. Dolgy, D. S. Kim, J.J. Seo, Differential equations for Changhee polynomials and their applications, J. onlinear Sci. Appl., 9(6), [] T. Kim, D. S. Kim, A note on nonlinear Changhee differential equations, Russ. J. Math. Phys., 3(6), [3] T. Kim, D. S. Kim, J.J. Seo, Differential equations associated with degenerate Bell polynomials, Inter. J. Pure and Appl. Math., 8(6), no. 3, [4] T. Kim, D. S. Kim, J.J. Seo, H.I. Kwon, Differential equations associated with λ- Changhee polynomials, J. onlinear Sci. Appl., 9(6), [5] T. Kim, T. J.J. Seo, Revisit nonlinear differential equations arising from the generating functions of degenerate Bernoulli numbers, Adv. Stud. Com. Math., 6(6), no. 3, [6] H.I. Kwon, T. Kim, T. J.J. Seo, A note on Daehee numbers arising from differential equations, Glob. J. Pure and Appl. Math., (6), no. 3, [7] Q.-M. Luo, Apostol-Euler polynomials of higher order and Gaussian hypergeometric function, Taiwanese J. Math., (6), [8] J.-W. Par, on the twisted q-changhee polynomials of higher order, J. Comput. Anal. Appl. (6), no., [9] S. H. Rim, J. H. Jeong, J. W. Par, Some identities involving Euler polynomials arising from a non-linear differential equation, Kyungpoo Math. J., 53(3),