New Generalization of Eulerian Polynomials and their Applications

Transcription

1 J. Aa. Num. Theor. 2, No. 2, Joural of Aalysis & Number Theory A Iteratioal Joural New Geeralizatio of Euleria Polyomials ad their Applicatios Sera Araci 1,, Mehmet Acigoz 1, ad Erdoğa Şe 2, 1 Departmet of Mathematics, Faculty of Sciece ad Arts, Uiversity of Gaziatep, Gaziatep, Turey 2 Departmet of Mathematics, Faculty of Sciece ad Letters, Namı Kemal Uiversity, Teirdağ, Turey Received: 14 Feb. 2014, Revised: 20 Apr. 2014, Accepted: 22 Apr Published olie: 1 Jul Abstract: I the preset paper, we itroduce Euleria polyomials with parameters a ad b ad give the defiitio of them. By usig the defiitio of geeratig fuctio for our polyomials, we derive some ew idetities i Aalytic Numbers Theory. Also, we give relatios betwee Euleria polyomials with parameters a ad b, Berstei polyomials, Poly-logarithm fuctios, Beroulli ad Euler umbers. Moreover, we see that our polyomials at a 1 are related to Euler-Zeta fuctio at egative ietegers. Fially, we get Witt s formula for ew geeralizatio of Euleria polyomials which we express i this paper. Keywords: Euleria polyomials, Poly-logarithm fuctios, Stirlig umbers of the secod id, Berstei polyomials, Beroulli umbers, Euler umbers ad Euler-Zeta fuctio, p-adic fermioic itegral oz p MATHEMATICS SUBJECT CLASSIFICATION. Primary 05A10, 11B65; Secodary 11B68, 11B73. 1 Itroductio The Beroulli umbers ad polyomials, Euler umbers ad polyomials, Geocchi umbers ad polyomials, Stirlig umbers of the secod id, Berstei polyomials ad Euleria polyomials possess may iterestig properties ot oly i complex aalysis, ad aalytic umbers theory but also i mathematical physics related to ot theory ad ζ-fuctio, ad p-adic aalysis. These polyomials have bee studied by may mathematicias for a log time for details, see [1-30. Euleria polyomial sequece{a x} 0 is give by the followig summatio: l0 l x l A x 1 x +1, x <1. 1 It is well-ow that the Euleria polyomial, A x, of degree ca be itroduced as A x 1 A,x, A 0 z1 2 where A, are called the Euleria umbers that ca be computed by usig A, +1 j 1 j j, 1, 3 where A,01. Euleria polyomials, A x, are also give by meas of the followig expoetial geeratig fuctio: e Axt 0 A x t! 1 x e t1 x x where A x : A x, symbolically. Euleria polyomials ca be foud via the followig recurrece relatio: { A t+t 1 1 t, if 0 ta t 5 0, if 0, for details, see [5, [6, [25, [9 ad [10. Now also, we give the defiitio of Euleria fractio, α x, ca be expressed as 4 α x : A x x Correspodig author mtsr@hotmail.com, acigoz@gatep.edu.tr, erdoga.math@gmail.com Natural Scieces Publishig Cor.

2 60 S. Araci et. al. : New Geeralizatio of Euleria Polyomials... We wat to ote that Euleria fractio is very useful i the study of the Euleria umbers, Euleria polyomials, Euler fuctio ad its geeralizatio, Jorda fuctio i Number Theory for details, see [16. Firstly, Acigoz ad Araci itroduced the geeratig fuctio of Berstei polyomials as follows: B, x t! tx e t1 x, t C, 7! where B, x are called Berstei polyomials, which are defied by B, x x 1 x, 0 x 1, 8 for details o this subject, see [8. The Poly-logarithms ca be defied by the series: ew theoretical properties for them. Also, we show that our polyomials are related to poly-logarithm fuctio, the Berstei polyomials, Beroulli umbers, Euler umbers, Geocchi umbers, Euler-Zeta fuctio ad Stirlig umbers of the secod id. Fially, we get Witt s formula for ew geeralizatio of Euleria polyomials which seems to be iterestig for further wor i p-adic aalysis. 2 O the ew geeralizatio of Euleria polyomials I this sectio, we start by givig the followig defiitio of ew geeralizatio of Euleria polyomials. Defiitio 1.Let b R + positive real umbers ad a Cfield of complex umbers, the we defie the followig: Li z 1 z 9 e taa,b 0 A a,b t! 1 a b t1 a a 14 for 0, ad z <1. We easily see that if 0 Li 0 z z 1 z. Also, Poly-logarithms ca be give by the itegral represetatio, as follows: z Li z 0 Li z dz z i C\[1,. We ote that Li 1 z log1 z is the usual logarithm see [27. I [28, [29, Luo et al. defied the geeralizatio of the Beroulli ad Euler polyomials with parameters a,b,c as follows: t log b tc xt b t a t 2c xt b t + a t 0 0 B x;a,b,c t,! E x;a,b,c t,! So that, obviously, a, < 2π 10 t log b a. < π 11 B x;1,e,e : B x ad E x;1,e,e : E x. 12 Here B x ad E x are the classical Beroulli polyomials ad the classical Euler polyomials, respectively. Next, for the classical Beroulli umbers, B ad the classical Euler umbers, E we have B 0 : B ad E 0 : E. 13 By the same motivatio of all the above geeralizatios, we cosider, i this paper, the geeralizatio of Euleria polyomials ad derive some where A a,b are called the geeralizatio of Euleria polyomials or Euleria polyomials with parameters a ad b. Also, A a,b : A a,b, symbolically. So that, obviously, A x,e : A x. By 14, we have the followig recurrece relatio for the Euleria polyomials with parameters a ad b: e taa,b 1 a e t1 alb a. By applyig combiatorial techiques to the above equality, the we easily derive the followig theorem: Theorem 1.The followig recurrece relatio holds: [A a,b+1 alb aa a,b1 aδ,0 15 where δ,0 is the Kroecer s symbol. We ow cosider for > 0 i 15, becomes A a,b 1 1 A a,b1 a lb We wat to ote that taig a x ad b e i 16 reduces to A x 1 1 x 1 A x1 x 17 0 see [5 ad [25. We see that 17 is proportioal with Berstei polyomials which we state i the followig theorem: Natural Scieces Publishig Cor.

3 J. Aa. Num. Theor. 2, No. 2, / 61 Theorem 2.The followig idetity A x 1 0 A xb, x x +1 x Let us ow cosider lim t 0 d dt i 14, the we readily arrive at the followig theorem. Theorem 3.Let b R + ad a C, the we have [ d 1 a A a,blim t 0 dt b t1 a. 18 a By 18, we easily coclude the followig corollary. Corollary 1.The followig Cauchy-type itegral holds true: 1 1 a A a,b! t 1 2πi C b t1 a a dt where C is a loop which starts at, ecircles the origi oce i the positive directio, ad the returs. 0 By 14, we discover the followig: A a 2,b 2 t! [ 1 a b t1+a1 a a [ 1+a b t1 a1+a a [ 1+a A a,b t 1 a 0![ A a,b t 0! By usig Cauchy product o the above equality, the we get the followig theorem. Theorem 4.The followig equality A a 2,b a A a,ba a,b1 a 19 After the basic operatios i 19, we discover the followig corollary. Corollary 2.The followig property holds: A a 2,b B, aa a,ba a,b. a 0 Now also, we cosider geometric series i 14, the we compute as follows: A a,b t 0! 1 a e t1 a lb a 1 1 a 1 a 1 1 a 1 e t1 alb a j jt1 a lb e 1 1 a j a j 1 a lb t 0! [ a a j j 1 a lb t!. By comparig the coefficiets of t! o the above equatio, the we readily derive the followig theorem.. Theorem 5.The followig 1 lb A a,b[ a lb j1 a j j The above theorem is related to Poly-logarithm fuctio, as follows: 1 lb A a,b[ lb Li a 1. a 20 I [27, it is well-ow that Li x x dx d x 1 x 0!S+1,+1 x +1 1 x 21 where S, are the Stirlig umbers of the secod id. By 20 ad 21, we have the followig iterestig theorem. Theorem 6.The followig holds true: aa a,b lb 0 3 Further Remars 0 1!S+1,+1. Now, we cosider 14 for evaluatig at a 1, as follows: A 1,b t! 2 b 2t where A 1,b are called Euleria polyomials with parameter b. By 22, we derive the followig equality i complex plae: i A 1,b t 0! 2 b 2it e 2it lb + 1. From this, we discover the followig: i A 1,b t 0! E 2 i lb t 0! 23 where E are -th Euler umbers which are defied by the followig expoetial geeratig fuctio: 0 t E! 2 e t, t <π By 23 ad 24, we have the followig theorem. Theorem 7.Let N field of atural umbers ad b C, the we get A 1,b 2 E lb. Natural Scieces Publishig Cor.

4 62 S. Araci et. al. : New Geeralizatio of Euleria Polyomials... We ow give the defiitio of Beroulli umbers for sequel of this paper via the followig expoetial geeratig fuctio: 0 t B! t e t, t <2π By usig 22 ad 25, we see that A 1,b t 0! 2 e 2t lb [ t So from above 2t e 2t lb 1 4t e 4t lb. 1 A 1,b t 0! [2 lb B 4 lb B t 1 0!. By comparig the coefficiets of t o the above equatio, the we ca state the followig theorem. Theorem 8.The followig idetity holds true. A 1,b 2+1 lb B By 22, we obtai the followig: 0 A 1,b t! 1 t 0 2 lb G t! 0 2 lb G t 1!. That is, we reach the followig theorem. Theorem 9.The followig holds true: A 1,b 2+1 lb +1 G where G are the familiar Geocchi umbers which is defied by 0! 0 G t! 2t e t + 1. We recosider 22 ad usig defiitio of geometric series, the we compute as follows: t2 A 1,b 2 1 j jt lb e 0 2lb 1 j j Therefore, we obtai the followig theorem Theorem 10.For >0, the we have t!. A 1,b 2 +1 lb 1 j j. 26 j1 As is well ow, Euler-zeta fuctio is defied by ζ E s2 j1 1 j j s, s C see[3. 27 From 26 ad 27, we obtai the iterpolatio fuctio of ew geeralizatio of Euleria polyomials at a 1, as follow: A 1,b2 lb ζ E. 28 Equatio 28 seems to be iterpolatio fuctio at egative itegers for Euleria polyomials with parameter b. Let us ow cosider Witt s formula for our polyomials at a 1, so we eed the followig otatios: Imagie that p be a fixed odd prime umber. Throughout this paper, we use the followig otatios. By Z p, we deote the rig of p-adic ratioal itegers, Q deotes the field of ratioal umbers,q p deotes the field of p-adic ratioal umbers, ad C p deotes the completio of algebraic closure of Q p. Let N be the set of atural umbers adn N {0}. The ormalized p-adic absolute value is defied by p p 1 p. Let q be a idetermiate with q 1 p < 1. Let UDZ p be the space of uiformly differetiable fuctios o Z p. For a positive iteger d with d, p 1, let X X d lim Z/d p Z d p 1 a+d pz p a0 with a+d p Z p {x X x amodd p } where a Z satisfies the coditio 0 a < d p ad let σ : X Z p be the trasformatio itroduced by the iverse limit of the atural trasformatio Z/d p Z Z/p Z. If f is a fuctio o Z p, the we will utilize the same otatio to idicate the fuctio f σ. For a cotiuous fuctio f : X C p, the p-adic fermioic itegral o Z p is defied by T. Kim i [2 ad [3, as follows: I 1 f X f υdµ 1υ Z p f υdµ 1 υlim p 1 υ0 1υ f υ. 29 By 29, it is well-ow that I 1 f 1 +I 1 f2 f 0 30 Natural Scieces Publishig Cor.

5 J. Aa. Num. Theor. 2, No. 2, / 63 where f 1 υ : f υ+ 1. Substitutig f υ b 2υt ito 30, we get the followig: X e 2tυ lb dµ 1 υ 2 b 2t + 1 A 1,b t 0!. 31 By 31 ad usig Taylor expasio of e 2tυ lb, we obtai Witt s formula for our polyomials at a 1, as follows: Theorem 11.The followig holds true: A 1,blb 2 υ dµ 1 υ. 32 Equatio 32 seems to be iterestig for our further wors i the cocept of p-adic itegrals. Refereces [1 T. Kim, Idetities ivolvig Frobeius-Euler polyomials arisig from o-liear differetial equatios, Joural of Number Theory, 132, [2 T. Kim, Some idetities o the q-euler polyomials of higher order ad q-stirlig umbers by the fermioic p-adic itegral oz p, Russia J. Math. Phys., 16, [3 T. Kim, Euler umbers ad polyomials associated with zeta fuctios, Abstract ad Applied Aalysis, vol. 2008, Article ID , 11 pages, [4 T. Kim, Some idetities for the Beroulli, the Euler ad the Geocchi umbers ad polyomials, Adv Stud Cotemp Math., 20, [5 D. S. Kim, T. Kim, W. J. Kim ad D. V. Dolgy, A ote o Euleria polyomials, Abstract ad Applied Aalysis, Volume , Article ID , 10 pages. [6 D. S. Kim, T. Kim, Y. H. Kim, ad D. V. Dolgy, A ote o Euleria polyomials associated with Beroulli ad Euler umbers ad polyomials, Advaced Studies i Cotemporary Mathematics, 22, [7 M. Acigoz ad Y. Simse, O multiple iterpolatio fuctios of the Nörlud-type q-euler polyomials, Abstract ad Applied Aalysis, 2009, Article ID , 14 pages. [8 M. Acigoz ad S. Araci, O the geeratig fuctios for Berstei polyomials, Numerical Aalysis ad Applied Mathematics, Amer. Ist. Phys. Cof. Proc. CP1281, [9 S. Araci, M. Acigoz ad D. Gao, O the Dirichlet s type of Euleria polyomials, arxiv: [math.nt [10 S. Araci ad M. Acigoz, Dirichlet s type of twisted Euleria polyomials i coectio with twisted Dirichlet s type-l-fuctio, arxiv: [math.nt [11 S. Araci, D. Erdal ad J. J. Seo, A study o the fermioic p-adic q-itegral represetatio o Z p associated with weighted q-berstei ad q-geocchi polyomials, Abstract ad Applied Aalysis, 2011, Article ID , 10 pages. [12 S. Araci, M. Acigoz, ad J. J. Seo, Explicit formulas ivolvig q-euler umbers ad polyomials, Abstract ad Applied Aalysis, 2012, Article ID , 11 pages. X [13 E. Ceti, M. Acigoz, I. N. Cagul, ad S. Araci, A ote o the h, q-zeta-type fuctio with weight α, Joural of Iequalities ad Applicatios, 2013, 2013:100. [14 S. Araci, M. Acigoz, ad A. Kilicma, Exteded p-adic q-ivariat itegrals o Z p associated with applicatios of umbral calculus, Advaces i Differece Equatios 2013, 2013:96. [15 S. Araci, M. Acigoz, ad F. Qi, O the q-geocchi umbers ad polyomials with weight zero ad their iterpolatio fuctios, Noliear Fuctioal Aalysis ad Applicatios, 18, [16 G. Birhoff, C. de Boor, Piecewise polyomial iterpolatio ad approximatio, Proc. Sympos. Geeral Motors Res. Lab.,, Elsevier Publ. Co., Amsterdam, 1965, [17 I. N. Cagul, H. Ozde, ad Y. Simse, Geeratig fuctios of the h, q extesio of twisted Euler polyomials ad umbers, Acta Mathematica Hugarica, 120, [18 L. Carlitz, Euleria umbers ad polyomials, Mathematics Magazie, 32, [19 L. Carlitz, q-beroulli ad Euleria umbers, Trasactios of the America Mathematical Society, 76, [20 L. Carlitz, A combiatorial property of q-euleria umbers, Amer. Math. Mothly, 82, [21 F. Hirzebruch, Euleria polyomials, Müster J. of Math., 1, [22 L. C. Jag, V. Kurt, Y. Simse, ad S. H. Rim, q-aalogue of the p-adic twisted l-fuctio, Joural of Cocrete ad Applicable Mathematics, 6, , [23 H. Jolay, R. E. Alielaye ad S. S. Mohamad, Some results o the geeralizatio of Beroulli, Euler ad Geocchi polyomials, Acta Uiversitatis Apulesis, [24 H. Jolay ad H. Sharifi, Some results for the Apostol- Geocchi polyomials of higher order, Bull. Malays. Math. Sci. Soc., 36, [25 D. Foata, Euleria polyomials: from Euler s time to the preset, The legacy of Alladi Ramarisha i the mathematical scieces, , Spriger, New Yor, [26 J. Choi ad H. M. Srivastava, The multiple Hurwitz Zeta fuctio ad the multiple Hurwitz-Euler zeta fuctio, Taiwaese Joural of Mathematics, 15, [27 L. Lewi, Polylogarithms ad associated fuctios, North Hollad, [28 Q. M. Luo, F. Qi, ad L. Debath, Geeralizatios of Euler umbers ad polyomials, IJMMS. 2003, [29 Q. M. Luo, B. N. Guo, F. Qi, ad L. Debath, Geeralizatio of Beroulli umbers ad polyomials, IJMMS, 2003, [30 H. M. Srivastava ad J. Choi, Series Associated with the Zeta ad Related Fuctios, Kluwer Academic Publishers, Dordrecht, Bosto ad Lodo, Natural Scieces Publishig Cor.