Applications of Fourier Series and Zeta Functions to Genocchi Polynomials

Transcription

1 Appl. Math. Inf. Sci., No. 5, (8) 95 Applied Mathematics & Information Sciences An International Journal Applications of Fourier Series Zeta Functions to Genocchi Polynomials Serkan Araci, Mehmet Acikgoz Department of Economics, Faculty of Economics, Administrative Social Science, Hasan Kalyoncu University, TR-74 Gaziantep, Turkey Department of Mathematics, Faculty of Science Arts, Gaziantep University, TR-73 Gaziantep, Turkey Received: 5 Jun. 8, Revised: Jul. 8, Accepted: 9 Jul. 8 Published online: Sep. 8 Abstract: In this paper, we firstly consider the properties of Genocchi polynomials, Fourier series Zeta functions. In the special cases, we see that the Fourier series yield Zeta functions. From here, we show that zeta functions for some special values can be computed by Genocchi polynomials. Secondly, we consider the Fourier series of periodic Genocchi functions. For odd indees of Genocchi functions, we construct good links between Genocchi functions Zeta function. Finally, since Genocchi functions reduce to Genocchi polynomials over the interval [, ), we see that Zeta functions have integral representations in terms of Genocchi polynomials. Keywords: Zeta Functions, Genocchi polynomials, Fourier series. Mathematics Subject Classification. Primary B68; S8; Secondary 5A9; 4B5. Introduction Fourier series plays an important role in the principal methods of analysis for mathematical physics, engineering, signal processing. While the original theory of Fourier series applies to periodic functions occurring in wave motion, such as with light sound, its generalizations often relate to wider settings, such as the time-frequency analysis underlying the recent theories of wavelet analysis local trigonometric analysis. Hurwitz found the Fourier epansion of the Bernoulli polynomials over a century ago. In general, Fourier analysis can be fruitfully employed to obtain properties of the Bernoulli polynomials related functions in a simple manner. Very recently, the Fourier series epansions of some special polynomials have been studied in details, see []-[8]. The Riemann zeta function is useful in number theory for investigating properties of prime numbers including special function of mathematics physics that arises in definite integration given by ζ(s)= Γ (s) t s e t e t dt where Γ (s) is Gamma function for Re(s)> with s C known as Γ (s)= t s e t dt (see []). Srivastava [3, 4, 8] developed the family of rapidly converging series representations for the Riemann Zeta function ζ(s) at s = n +. We also note that the Bernoulli numbers are interpolated by the Riemann zeta function at negative integers, which plays an important role in analytic number theory has applications in physics, probability theory applied statistics, as follows: ζ( n)= B n+ n+ where B n sts for Bernoulli numbers defined by means of the following generating function: ζ(s) := n= n s (s C; Re(s)>) () Corresponding author mtsrkn@hotmail.com t n B n n= n! = t e t ( t <π). c 8 NSP

2 95 S. Araci, M Acikgoz: Applications of Fourier series Zeta... A few Bernoulli numbers are listed below: B =, B =, B = 6, B 4 = 3, B 6 = 4, B n+ = for n (see [5],[6],[], [8], [] for details). Euler also gave the following interesting equality ζ(n)= (π)n ( ) n+ B n, n=,,,. From here one may see that Riemann zeta functions at even natural numbers can be computed by using this equality (see [4],[5],[],[3],[4],[8]), for eample, ζ()= π π4, ζ()=, ζ(4)= 6 9,. λ β functions are also defined, respectively, by λ(s) := m= β(s)= (m+) s = s s ζ(s) (s C;Re(s)>) m= () ( ) m (m+) s (s C;Re(s)>) (3) (see [4],[5]). A further generalization of Riemann zeta function is Hurwitz zeta function given by ζ(s,a)= n= (n+a) s (s C;Re(s)>;a,,, ). Obviously that ζ(s, ) := ζ(s) (see []). Digamma function is also known as Ψ(s)= d ds logγ (s)= Γ (s) Γ(s) (s C;Re(s)>) which has integral series representations, respectively, as follows: ( e ) Ψ(s)= + e s e d Ψ(s)= γ s + k= ( k ) k+s where γ :=, is Euler-Mascheroni constant (see []). Apostol Frobenius-Euler polynomials are known as H n (,u,λ) tn n= n! = u λ e t u et (4) (u, λ, u λ) (see [9],[],[6],[7],[9],[],[]). In the special cases, H n (,,) := E n ()= G n+() n+ where E n () G n () are called, respectively, Euler polynomials Genocchi polynomials. These polynomials in the value = reduce to Euler numbers Genocchi numbers denoted by E n () := E n ve G n () := G n, cf. [8],[]. Recently, Araci Acikgoz have derived Fourier epansion of Apostol Frobenius-Euler polynomials in the Laurent series form, as follows: where H n (,u,λ)= u ( u ) u n! λ a k,n (λ,u)z k (5) k Z z=e πi a k,n (λ,u)= ( ( )) n+ (see [8]). πik log λu In the special cases of the parameters u λ, one can easily derive that Fourier epansion of Euler polynomials ([3],[8]): H n (,,) := E n ()= n! (πi) n+ k Z e πi(k+ ) ( k+ ) n+. Fourier epansion of Genocchi polynomials ([],[8]): nh n (,,) := G n ()= n! (πi) n e πi(k+ ) ( ) k Z k+ n. (6) In this paper, our main focus is going to be Genocchi numbers polynomials whose history can be traced back to Italian mathematician Angelo Genocchi (87889). From Genocchi s time to the present, Genocchi numbers polynomials have been etensively studied in many different contet in such branches of Mathematics as, for instance, elementary number theory, comple analytic number theory, Homotopy theory (stable Homotopy groups of spheres), differential topology (differential structures on spheres), theory of modular forms (Eisenstein series), p-adic analytic number theory (p-adic L-functions), quantum physics (quantum groups). The works of Genocchi numbers their combinatorial relations have received much attention [,3,7,,5]. For showing the value of this type of numbers polynomials, we list some properties known in the literature: c 8 NSP

3 Appl. Math. Inf. Sci., No. 5, (8) / Series representations of Genocchi polynomials arising from its Fourier epansion (see []): G n () = 4( )n (n)! π n m= G n+ () = 4( )n (n+)! π n+ cos((m+)π) (m+) n, (7) m= sin((m+)π) (m+) n+.(8) Generating function of Genocchi polynomials (see [7], [5]): n= G n () tn n! = t e t + et. A few Genocchi polynomials (see [7],[5]): G ()=, G ()=, G ()=, G 3 ()=3 3,. The symmetric property for Genocchi polynomials (see [7],[5]): G n ( )=( ) n G n (). A recurrence relation for Genocchi polynomials (see [7],[5]): ( ) n+ G n ( )=G n () n n. The n th periodic Genocchi function may be introduced in the following way: G n () := G n (),( <) G n (+)= G n () ( R) (see []). Note that the period of G n is since G n (+)= G n (+)= G n (). Theorem.The following equality holds ζ(n)= 4 n+ 4 (π) n ( ) n (n)! G n which seems to be a similar formula given by Euler as follows ζ(n)= (π)n ( ) n+ B n. Proof.Substituting = into Eq. (8) yields π n ( ) n 4(n)! G n = m= = λ(n). n (m+) By using well-known equality G n = ( 4 n )B n in [4], λ(n)= πn ( ) n ( 4 n )B n. (9) Thus, from ([5]) (9), we complete the proof. Theorem.The following equality holds true β(n+)= ( )n π n+ ( ) 4(n+)! G n+. Proof.It is proved by using (7) for the value =. So we omit the proof. Theorem shows that β functions at odd positive integers can be computed by Genocchi polynomials at =, for eample, β()= π π3, β(3)= 4 3,. Recall from Eq. (8) that for R G n+ ()=4( ) n (n+)! m= sin[(m+)π] ((m+)π) n+. () When > we can derive uniformly convergent series for G n+ () from () integration from zero to infinity using the well-known formula in comple analysis sin(a) d= π (a>). () Main Results Now we are in a position to state prove the zeta functions the uniform integral representations for Genocchi polynomials as follows. Theorem 3.For n N, ζ(n+)= 4 n ( ) n π n (n+)!( n+ ) G n+ () d. () c 8 NSP

4 954 S. Araci, M Acikgoz: Applications of Fourier series Zeta... Proof.By () (), we see that G n+ () d = 4( ) n (n+)! m= ((m+) π) n+ = 4( )n (n+)! π n+ m= = ( )n (n+)! π n λ(n+). π (m+) n+ which, from (), gives the following relation: ζ(n+)= Thus our assertion is proven. 4 n ( ) n π n (n+)!( n+ ) Theorem 4.The following equality holds true G n+ () d= G n+()( k= G n+ () d = = sin[(m+) π] d G n+ () d. (+k)(+k ) ) d. (3) Proof.From the definition of periodic Genocchi function, G n+ () G n+ () d+ d G n+ () k+ d+ k= k G n+ () d. Integrating by substitution =t+k in the second integral on the right h side of the above equality yields G n+ () d= G n+ () d+ k= Now it is sufficient to analyse the integral follows: G n+ (t) t+ k dt = = = = Thus, the proof is completed. Theorem 5.For n N, G n+ (t) t+ k dt. (4) G n+ (t) t+k dt as G n+ (t) t+ k dt+ G n+ (t) t+ k dt G n+ (u ) u +k du+ G n+ (t) t+ k dt G n+ (u) u +k du+ G n+ (t) t+ k dt G n+ () (+k)(+k ) d. ζ(n+)= 4n ( ) n π n (n+)!( n+ ) G n+() ( +Ψ( ) ( Ψ )) d. Proof.From (4), we derive ( ) ( Ψ Ψ = ) + k= +k +k = k=(+k)(+k ). One may see that this equality is closely related to Eq. (3). So we derive that ( ) ( Ψ Ψ + ) = k=(+k)(+k ). Thus, our assertion follows from Eq.(3) last equality on the above. 3 Perspective We have derived new identities related to zeta functions. From these identities, computed some values of zeta functions using Genocchi polynomials. After that, considered the Fourier series of periodic Genocchi functions. For odd indees of Genocchi functions, constructed some new relations between Genocchi functions Zeta functions. Since Genocchi functions reduce to Genocchi polynomials over the interval [,), we see that Zeta functions for odd positive integers have integral representations in terms of Genocchi polynomials. References [] Luo, Q-M., Fourier epansions integral representations for Genocchi polynomials, J. Integer Seq., Vol., Article (9) [] Luo, Q-M., Fourier epansions integral representations for the Apostol-Bernoulli Apostol-Euler polynomials, Math. Comp., Vol. 78, No. 68, 93-8 (9). [3] Bayad, A., Fourier epansions for Apostol-Bernoulli, Apostol-Euler Apostol-Genocchi polynomials, Math. Comp., Vol. 8, No. 76, 9- (). [4] Scheufens, E.E., Euler polynomials, Fourier series zeta numbers, Int. J. Pure Appl. Math., Vol. 78, (). [5] Scheufens, E.E., Bernoulli polynomials, Fourier series Zeta numbers, Int. J. Pure Appl. Math., Vol. 88, No., (3). [6] Ryoo, C.S., Kwon, H.I., Yoon, J., Jang, Y.S., Fourier series of the periodic Bernoulli Euler functions, Abstr. Appl. Anal., Article ID 85649, 4 pages (4). [7] Kim, T., Kim, D.S., Jang, G-W., Kwon, J., Fourier series of sums of products of Genocchi functions their applications, J. Nonlinear Sci. Appl., Vol., (7). [8] Araci, S, Acikgoz, M., Construction of Fourier epansion of Apostol Frobenius-Euler polynomials its applications, Adv. Difference Equ., 8:67. [9] Simsek, Y., Generating functions for q-apostol type Frobenius-Euler numbers polynomials, Aioms, Vol., (). [] Kim, T., Identities involving Frobenius-Euler polynomials arising from non-linear differential equations, J. Number Theory, Vol. 3, (). c 8 NSP

5 Appl. Math. Inf. Sci., No. 5, (8) / [] Srivastava, HM, Choi, J., Zeta q-zeta functions associated series integrals, Elsevier Science Publishers, Amsterdam, London New York (). [] Kim, T. Rim, S.-H, Simsek, Y, Kim, D., On the analogs of Bernoulli Euler numbers, related identities Zeta L-functions, J. Korean Math. Soc., Vol. 45, No., pp (8). [3] Srivastava, H.M., Further series representations for ζ(n + ), Appl. Math. Comput., Vol. 97, -5 (998). [4] Srivastava, H.M., Some rapidly converging series for ζ(n+), Proc. Amer. Math. Soc., Vol. 7, No., (999). [5] Araci, S, Şen, E, Acikgoz, M., Theorems on Genocchi polynomials of higher order arising from Genocchi basis, Taiwanese J. Math., Vol. 8, No,, (4). [6] Araci, S., Duran, U., Acikgoz, M., Abdel-Aty, M., Theorems on q-frobenius-euler polynomials under sym 4, Utilitas Mathematica, Vol., pp.9-37 (6). [7] Araci, S., Acikgoz, M., On the von Staudt-Clausen s theorem related to q-frobenius-euler numbers, J. Number Theory, Vol. 59, (6). [8] Srivastava, H.M., Some simple algorithms for the evaluations representations of the Riemann zeta function at positive integer arguments, J. Math. Anal. Appl., Vol. 46, (). [9] Duran, U., Acikgoz, M., Araci, S., On higher order (p, q)-frobenius-euler polynomials, TWMS J. Pure Appl. Math., Vol. 8 No,, 98-8, 7. [] Araci, S., Acikgoz, M., Diagana, T., Srivastava, H. M., A novel approach for obtaining new identities for the λ etension of q-euler polynomials arising from the q-umbral calculus, J. Nonlinear Sci. Appl., Vol, (7). [] Araci, S, Acikgoz, M., A note on the Frobenius-Euler numbers polynomials associated with Bernstein polynomials, Adv. Stud. Contemp. Math., Vol., No. 3, (). [] Araci, S, Acikgoz, M., Computation of Nevanlinna characteristic functions derived from generating functions of some special numbers, J. Inequal. Appl. 8:8. Serkan Araci was born in Hatay, Turkey, on October, 988. He has published over than papers in reputed international journals. His areas of specialization include p-adic Analysis, Theory of Analytic Numbers, q-series q-polynomials, Theory of Umbral calculus, Fourier series Nevanlinna theory. Currently, he works as a lecturer at Hasan Kalyoncu University, Gaziantep, Turkey. On the other h, Araci is an editor a referee for several international journals. For details, one can visit the link: Mehmet Acikgoz received M. Sc. And Ph. D. From Cukurova University, Turkey. He is currently Full Professor at University of Gaziantep. His research interests are approimation theory, functional analysis, p-adic analysis analytic numbers theory. c 8 NSP