A NOTE ON RECURRENCE FORMULA FOR VALUES OF THE EULER ZETA FUNCTIONS ζ E (2n) AT POSITIVE INTEGERS. 1. Introduction

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1 Bull. Korean Math. Soc. 5 (4), No. 5, htt://dx.doi.org/.434/bkms A NOTE ON RECURRENCE FORMULA FOR VALUES OF THE EULER ZETA FUNCTIONS ζ E (n) AT POSITIVE INTEGERS Hui Young Lee and Cheon Seoung Ryoo n s. Abstract. The Euler zeta function is defined by ζ E (s)= The urose of this aer is to find formulas of the Euler zeta function s values. In this aer, for s N we find the recurrence formula of ζ E (s) using the Fourier series. Also we find the recurrence formula of (n ) s, where s ( N).. Introduction ( ) n The Euler zeta function is defined by ζ E (s) = n (see [3, 4]). In s this aer we investigate the recurrence formula of the Euler zeta function for s = n with Fourier series. By this result we can find ζ E (n) for all n N. For s C, the Riemann zeta function or the Euler-Riemann zeta function, ζ(s) is defined by ζ(s) = n s (s C), (see [5, 6]) which converges when the real art of s is greater than. R. Aéry roved that the number ζ(3) is irrational. But it is still an oen roblem to rove irrationality of ζ(k +) for the long time. As well known secial values, for any ositive even number n, ζ(n) = ( ) n+b n(π) n, (see []) (n)! where B n are Bernoulli numbers. For negative integers, one has for n. ζ( n) = B n+ n+ Received Aril, 3; Revised November, 3. Mathematics Subject Classification. Primary 4B5, B68, S4, S8. Key words and hrases. zeta function, Euler zeta function, Fourier series. This work was financially suorted by Hannam University. 45 c 4 Korean Mathematical Society

2 46 H. Y. LEE AND C. S. RYOO The constants E n in the Taylor series exansion F(t) = e t + = n= E n t n n! for t < π (see [3, 4, 6, 8]), are well known as the Euler number. The Euler olynomials are also defined by e t + ext = e E(x)t = E n (x) tn n!, t < π n= with the usual convention about relacing E(x) n by E n (x) (see [, 3, 4, 5, 6, 7, 8]). The recurrence formula of the Euler numbers is { (E +) n if n =, +E n = if n. In [3, 4], T. Kim find the relation of the Euler zeta function and the Euler numbers. From this result, one can obtain values of the Euler zeta function. But in our work, one has the recurrence formula of the Euler zeta function itself and as a result one has values of the Euler zeta function without Euler numbers in generally. Throughout this aer n P r denotes the ermutation, where n P r = n(n )(n ) (n r+).. Fourier series Definition (Fourier Series). Let f(x) be function on (,) with f(x) = a + ( a k cos kπx +b ksin kπx ) (see [9]). Then it is called the Fourier series and coefficients are a = a n = nπx f(x)cos dx, b n = nπx f(x)sin dx. f(x)dx, Note that the Fourier series of f on (,) converges to f(x) at oints of continuity and converges to f(x+)+f(x ) at the oint of discontinuities, where f(x+) is the limit of right side at x, f(x ) is the limit of left side at x. Remark. From -test we know that { absolutely convergence if >, n = if. If =, then n = ζ() = π 6. Also n = (n) + (n ) = 4 ζ()+ (n ).

3 A NOTE ON RECURRENCE FORMULA 47 Hence, we get the following: (n ) =3 4 n = 3 π ζ() = 4 8. Let us consider f(x) on (,). If one takes that f is an even function on (,), then one has coefficients as below: a = a n = b n = f(x)dx = f(x)dx, f(x)cos nπx dx = f(x)sin nπx dx =. Then the Fourier series of f on (,) is given by f(x)cos nπx dx, () f(x) = a + a n cos nπx, where a = f(x)dx, a n = nπx f(x)cos dx. Then it is called the cosine series (see [9]). Let f be an odd function on (,). Then coefficients are as below; () a = a n = b n = f(x)dx =, f(x)cos nπx dx =, f(x)sin nπx dx = Then the Fourier series of f on (,) is given by (3) f(x) = b n sin nπx, where b n = f(x)sin nπx dx. nπx f(x)sin dx. Then it is called the sine series (see [9]). Examle. The sine series of Square wave function is given by { if π < x <, f(x) = if x < π. Above the Square wave function is an odd function and one has the sine series as below: f(x) = b n sinnx, where b n = π π sinnxdx = π ( ( ) n n ).

4 48 H. Y. LEE AND C. S. RYOO Hence Thus, we note that = f( π ) =4 π f(x) = 4 π Therefore one has as below: (4) n n sin(n )x. n sin π = 4 π n = π 4. It is used as the initial value for (n ) in Section 4. s 3. Euler-ζ-function The Euler-ζ-function is well known as below: (5) ζ E (s) = n s, s C (see [4]). n. Examle 3. Let f(x) = x, < x < which is an even function. Then (6) f(x) = a + a n cos nπx, where a = x dx = 8 3, a n = Hence one has the following cosine series: f(x) = ( ) n π n x cos nπx dx = 6( )n (nπ). cos nπx, < x <. Taking x = in f(x), we get ζ() = π 6. Taking x = in f(x), we get ζ E () = π, which is used as the initial value for ζ E(n) in Section 4. Let f(x) = x 3, < x <. Then the sin series of f(x) on (,) is given by (7) f(x) = b n sin nπx, where b n = x3 sin nπx dx = 4 ( )n nπ + ( )n 5 3 n 3 π. Then, if we take x = 3 in (7), we get easily the following: (n ) 3 = π3 5.

5 A NOTE ON RECURRENCE FORMULA 49 Let f(x) = x 4, < x <. Then the cosine series of f(x) on (,) is given by (8) f(x) = a + a n cos nπx, where a = 5 5, a n = ( )n 7 easily the following: n π ( )n 8 3 n 4 π 4 ζ E (4) = n 4 =. Then, if we take x = in (8), we get 7π Recurrence formulas of the Euler zeta function ζ E (s) for s N and for s N (n ) s To generalize this rocess, we consider two functions f(x) = x m, f(x) = x m, where < x <. Firstly, let f(x) = x m on < x <. Since f(x) is an even function, we aly the cosine series for f(x) is given by (9) f(x) = a + a n cos nπx, < x <, where a = m+ m+, a n = m ( )k+ m k+ k mp k n k π cosnπ. k From (9), for s ( N) we take the following two formulas: s () ( ) k sp k k π kζ E(k) = s+ s (s+)( s+ ), () () s ( ) k s P k k π kζ E(k) = s (s ) (s )( s ). From () and () we get the formula as below: ζ E (s) = (π) s { s+ s +3 ( ) s sp s s+ (s )(s+) ( ) k (π) kζ E(k) ( ) } sp k s P k. s Therefore from () one has the following theorem. Theorem 4. For s ( N) and ζ E () = π, we get as below: (π) s { s+ s +3 ζ E (s) = ( ) s sp s s+ (s )(s+) ( ) k (π) kζ E(k) ( ) } sp k s P k, s

6 43 H. Y. LEE AND C. S. RYOO where n P r is the ermutation. Note that T. Kim find the Euler zeta function with using the second kind Euler numbers as follows: ζ E (n) = ( )n π n ( 4 n ) (n )!( 4 n ) E n (see [4]). Secondly, let f(x) = x m on < x <, m. The sin series of f(x) on < x < is given by (3) f(x) = b n sin nπx, where b n = xm sin nπx dx. By simle calculations we get the coefficient b n as below: m (4) b n = ( ) n m ( ) k ( m P (k ) nπ ) k. So we aly the sin series to above (4) and we take x =. Then one has the following: m = ( ) n m ( ) k ( ) k sin nπ m P (k ) nπ m (5) = ( ) n m ( ) k ( ) k m P (k ) (n )π m = m ( ) k m P (k ) π k (n ) k. (6) (7) From (5), we get following two relations: s s = ( ) k s P (k ) π k = s ( ) k s 3P (s ) (k ) π k From (6)-(7), one has the following formula: (n ) s = ( )s π s s P (s ) [ 3 4 s ( ) k ( π k s + (n ) k, (n ) k. (n ) k ( s P (k ) s 3 P (k ) ) )].

7 A NOTE ON RECURRENCE FORMULA 43 Theorem 5. For s ( N) and n = π 4, one has as below: (n ) s = ( )s π s s P (s ) [ 3 4 s ( ) k ( π k s + where n P r is the ermutation. (n ) k ( s P (k ) s 3 P (k ) ) )], Note that T. Kim find the relationshi between ( ) k k= and the (k+) (n+) second kind Euler olynomials as follows: k= ( ) k (k +) = ( )ne n( ) (n+) (n)!4 πn+ (see [4]). Acknowledgements. We thank Prof. Kim Tae-kyun of Kwang-woon University whose lectures on theories and careful comments on this aer are greatly areciated. References [] R. Ayoub, Euler and zeta function, Amer. Math. Monthly 8 (974), [] T. Kim, On the analogs of Euler numbers and olynomials associated with -adic q- integral on Z at q =, J. Math. Anal. Al. 33 (7), no., [3], Euler numbers and olynomials associated with zeta functions, Abstr. Al. Anal. 8 (8), Article ID 5858, ages. [4] T. Kim, J. Choi, and Y. H. Kim, A note on values of Euler zeta functions at ositive integers, Adv. Stud. Contem. Math. (), no., [5] D. S. Kim and T. Kim, Euler basis, identities, and their alications, Int. J. Math. Math. Sci. (), Article ID 34398, 5 ages. [6] D. S. Kim, T. Kim, W. J. Kim, and D. V. Dolgy, A note on Eulerian olynomials, Abstr. Al. Anal. (), Article ID 6964, ages. [7] H. Ozden, Y. Simsek, S.-H. Rim, and I. N. Cangul, A note on -adic q-euler measure, Adv. Stud. Contem. Math. 4 (7), no., [8] S. H. Rim and T. Kim, A note on q-euler numbers associated with the basic q-zeta function, Al. Math. Lett. (7), no. 4, [9] D. G. Zill and W. S. Wright, Advanced Engineering Mathematics. 4th, Textbooks, 9. Hui Young Lee Deartment of Mathematics Hannam University Daejeon 36-79, Korea address: normaliz@hnu.kr

8 43 H. Y. LEE AND C. S. RYOO Cheon Seoung Ryoo Deartment of Mathematics Hannam University Daejeon 36-79, Korea address: