SOME IDENTITIES FOR THE GENERALIZED POLY-GENOCCHI POLYNOMIALS WITH THE PARAMETERS A, B AND C

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1 Joural of Mathatical Aalysis ISSN: , URL: Volu 8 Issu , Pags SOME IDENTITIES FOR THE GENERALIZED POLY-GENOCCHI POLYNOMIALS WITH THE PARAMETERS A, B AND C BURAK KURT Abstract. I this work, w dfi th gralizd poly-gocchi polyoials with th paratr a, b ad c. W prov so proprtis for ths polyoials. Also, w giv closd forula ad sytry proprtis for ths polyoials. 1. Itroductio Th classical Broulli polyoials ad classical Gocchi polyoials ar dfid by as of th followig gratig fuctios rspctivly, t F x, t t 1 xt B x t, t < 2π, Q x, t 2t t + 1 xt G x t, t < π. 1.1 I th cas x 0 i Eq. 1.1, th classical Broulli ubrs ad classical Gocchi ubrs ar dfid th followig gratig fuctios rspctivly [1], F t Q t t t 1 t B, t < 2π, 2t t + 1 t G, t < π. Lt k Z, k > 1. Th k th polylogarith fuctios is dfid by z L ik z k 1.2 [4]-[7], [11]-[14], [18]-[24], [26]. This fuctio is covrgt for z < 1. If k < 0, polylogarith fuctio is ratioal fuctio [6] Mathatics Subjct Classificatio. 05A10, 11B73, 11A07. Ky words ad phrass. Poly-Broulli polyoials ad ubrs, Poly-Gocchi polyoials ad ubrs, polylogarith, poly-broulli ubrs, ultipl-polylogarith. c 2017 Ilirias Rsarch Istitut, Prishtië, Kosovë. Th prst ivstigatio was supportd by th Scitific Rsarch Projct Adiistratio of Akdiz Uivrsity. Subittd Dcbr 20, Publishd Jauary 10, Couicatd by H.M. Srivastava. 156

2 GENERALIZED POLY-GENOCCHI POLYNOMIALS 157 I th cas k 1 i Eq. 1.2, L i1 z log1 z. Th poly-broulli polyoials ad ubrs ar dfid by followig gratig fuctios rspctivly: F 1 x, t B k x t L i k 1 t t xt, 1 G 1 x, t B k t L i k 1 t t [4], [10], [11]. I th cas k 1 i Eq. 1.3, B 1 B, B 1 x B x. Th poly-gocchi polyoials ad ubrs ar dfid by followig gratig fuctios rspctivly: Q 1 x, t G k x t 2L i k 1 t t xt, + 1 [21]. I th cas k 1 i Eq. 1.4, G k t 2L i k 1 t t + 1 G 1 G, G 1 x G x. Ki t al. [21] dfid th odifid poly-gocchi polyoials as follows: G k t,2 x 2L i k 1 2t t xt By this otivatio, w dfi th odifid poly-broulli polyoials ad th aalogu of th odifid poly-gocchi polyoials as follows: F 2 x, t B k t,2 x L i k 1 2t t xt, ad Q 2 x, t G k t,2 2x 2L i k 1 2t t 2xt Fro 1.3, 1.4, 1.5 ad 1.6, w writ th followig qualitis; t tf 1 t 2 + Gt Q 1 t, 1.7 F 2 t 2F 1 2t 1 2 Q 2 t, 1.8 2t 2Qt Q 1 t Q 1, t Q 1 t, 1.9 F 1 x, t F 1 y, t F 1 t F 1 x + y, t 1.10

3 158 BURAK KURT ad F 1 x + 1 2, t + F 1 x, t Q 2 x, t Th Stirlig ubrs of th scod kid is also dfid by: S 2, t t 1! 1.12 [2]-[5], [9]-[11]. Firstly, Kako dfid poly-broulli ubrs [19]. Bayad t al. [6], [7] itroducd ad ivstigatd th poly-broulli polyoials ad gav so rlatios for ths polyoials. Haahata i [11] dfid poly-eulr polyoials ad provd th closd forula for th poly-eulr polyoials. Haahata t al. [12], [13] gav th rcurrc rlatios for ulti-poly-broulli ubrs ad spcial ultipoly-broulli ubrs. Jolay t al. i [14], [18] itroducd ad ivstigatd th gralizd poly-broulli polyoials with th paratrs a, b ad c. Ki t al. [21], [22] dfid poly-gocchi polyoials ad gav th so proprtis for th ultipl-poly-broulli ubrs ad ultipl-zta valus. Prgrio i [23], [24] provd th closd forula for th poly-broulli ubrs. Srivastava ad Srivastava t al. [16], [17] gav so basic proprtis ad thors for th Broulli, Eulr ad Gocchi polyoials. Araci [2], [3] itroducd ad ivstigatd q-gocchi ubrs ad polyoials. I this work, w giv so rlatios btw poly-broulli ubrs ad poly- Gocchi ubrs. Also, w dfi odifid poly-broulli polyoials ad prov so rlatios for ths polyoials. W itroduc ad ivstigat th poly- Gocchi polyoials with th paratrs a, b ad c. W prov th closd forula for th poly-gocchi polyoials with th paratrs a, b ad c. 2. Mai Rsults I this sctio, w prov so idtitis btw th poly-gocchi ubrs ad poly-broulli ubrs. Also, w giv a rlatio btw for th odifid poly-gocchi polyoials ad poly-broulli polyoials. Propositio 2.1. Thr ar th followig rlatios o th poly-broulli ubrs ad poly-gocchi ubrs which ar btw ths ubrs ad polyoials: B k Gk 1 + B G k, ad 2G k B k,2 2+1 B k 1 2 Gk,2, 2.2 G G k B k x B k y p0 0 p G k,2 2x 2+1 B k x G 1 G k, 2.3 B p k B k p x + y 2.4 B k x. 2.5

4 GENERALIZED POLY-GENOCCHI POLYNOMIALS 159 Proof. Th proofs of ca b obtai asily fro 1.7, 1.8, 1.9, 1.10 ad 1.11 rspctivly. Thor 2.2. Thr is th followig rlatio btw th poly-gocchi ubrs, Eulr ubr ad th Stirlig ubrs of th scod kid: { } G k 1 + r r k E r p!s 2 r, p. 2.6 r0 1 Proof. Fro 1.4 ad 1.12: G k t 2 1 t 1 t + 1 k k t + 1 p! S 2 l, p 1 l t l l! 1 l0 1 t k E p! S 2 l, p 1 l t l l!. 1 By usig Cauchy product, coparig th cofficit of t, w hav 2.6. Dfiitio 2.3. W dfi th gralizd poly-gocchi polyoials with th paratrs a, b ad c as: G k x; a, b, c t 2L ik 1 ab t a t + b t c xt. 2.7 ad If w put a 1, b c i 2.7, w hav th dfiitio of Ki s t al. [21] G k x; 1,, t 2L i k 1 t t xt. + 1 Fro 2.7, w ca obtai asily G k x; a, b, c G k x + 1; a, b, c G k 0 l0 x; ac, bc, c G k 0; a, b x l c +. By th otivatio of th dfiitio of odifid poly-gocchi polyoials [21]. W dfi gralizd odifid-poly-gocchi polyoials with th paratrs a, b ad c as: 1 ab 2t 2L G k t,2 x; a, b, c ik a t + b t c xt. 2.8 Thor 2.4. Thr is th followig rlatio btw th gralizd odifid poly-gocchi polyoials ad poly-broulli polyoials: { } G k x l c + x l c,2 x; a, b, c B k B k

5 160 BURAK KURT Proof. G k t,2 x; a, b, c 2L ik 1 ab 2t a t + b t 2L ik 1 ab 2t a 2t b 2t 2L ik 1 ab 2t 1 ab 2t t t xt l c 2t c xt a t b t c xt 2L ik 1 ab 2t 2t lab 1 t +x l c+2 t+x l c+2 l a 2L i k 1 2tb t+x l c+2 2L i k 1 2tb 1 + 2t lab 1 + 2t lab 2L i k 1 2tb 2t lab[ +x l c+2 2 b ] 2L ik 1 2tb 1 + 2t lab 1 + 2t lab + x l c B k 2 [ + ] t 2 + x l c + B k 2 [ + ] t 2 +. Coparig th cofficits of both sids, w hav 2.9. tx l c+ 2t lab[ x l c+ 2 b ] Corollary 2.5. If w put i 2.9 a 1 ad b c, w hav th rsult of Ki t. al. [21]. 3. Gralizd Poly-Gocchi Polyoials with No-positiv Idx I this sctio, w itroduc two-variabl polyoials with o-positiv idx-k to coct G k x; a, b, c, G k x; a, b, c. Dfiitio 3.1. For, > 0, w dfi C x, y; a, b G k x 1 + k k0 k0 0 G k t u k k! + ; a, b y t t t u k0 >0 2 t + 1 >0 k uk k! k t u+t t + 1 u + t. 3.2 u+t

6 GENERALIZED POLY-GENOCCHI POLYNOMIALS 161 Thor 3.2. Thr is th followig rlatio C x, y; a, b t u k k! 2 1 t t t 1 u 1 y+ 0 Proof. Fro 3.1; 0 0 k0 C x, y; a, b t 0 k0 u k k! G k x 1 + k put th k l G k x 1 + y 1+ +u k0 G k y 1+ +u y 1+ + ; a, b y 1 + +u x+ +t. 3.3 k t u +!, l + ; a, b t y t x ; a, b u k k! x 1+ +t G k t k0 +u x t 2 1 t t + 1 u + t Corollary 3.3. For a b, C 1 t u+t t + 1 u + t u+t y+ +u x+ +t. u+t x, y; a, b C x, y; a, b. u k k! u u l! l! Thor 3.4. Th gralizd poly-gocchi polyoials with th paratrs a, b ad c satisfy th followig closd forula: C x, y; a, b { { 1 i j! 2 l l j i0 r0 j0 { r r j r0 { r r j l0 } y + } r x + i + + } x + i l + r } l0 l { l j } l y Proof. Fro 1.12 ad 3.3; C x, y; a, b t u k k! 0 y+ 2 +u x+ +t 1 i it 1 t t 1 j u 1 j i0 j0

7 162 BURAK KURT 1 i i0 1 i i0 j0 { u 1 j y+ +u t 1 j x+i+ j0 { u 1 j y+ +u t 1 j x+i 1+ } +t } +t { { } 1 i l u u j! y + j +! i0 j0 0 { } t t j! x + i + j +! 0 { } u u j! y + j +! 0 { } t } t j! x + i 1 +. j +! 0 By usig Cauchy product, coparig th cofficits of t u!, w hav 3.4. Corollary 3.5. Spcial cass, for a 1 ad b ; { C x, y; 1, 1 i j! 2 l l j i0 j0 l0 } y l r0 r { r j Ackowldgts. Th prst ivstigatio was supportd by th Scitific Rsarch Projct Adiistratio of Akdiz Uivrsity. Rfrcs [1] M. Abraowitz ad I. A. Stgu, Hadbook of athatical fuctios with forulas, graphs ad athatical to 1972 Natioal Burau of Stadards App. Math. Sris 55. [2] S. Araci, Novl idtitis for q-gocchi ubrs ad polyoials, Joural of Fuctio Spacs ad Applicatios, Volu 2012, 2012, Articl ID , 13 pag. [3] S. Araci, Novl idtitis ivolvig Gocchi ubrs ad polyoials arisig fro applicatios of ubral calculus, Applid Mathatics ad Coputatio , [4] T. Arakawa ad M. Kako, O poly-broulli ubrs, Cotarıı Mathatic Uiv. Sact. Pauli [5] T. Arakawa, T. Ibukiyaa ad M. Kako, Broulli ubrs ad zta fuctios, Sprigr Moographs i Mathatics [6] A. Bayad ad Y. Haahata, Polylogariths ad poly-broulli polyoials, Kyushu. J. Math [7] A. Bayad ad Y. Haahata, Multipl polylogariths ad ulti-poly-broulli polyoials, Fuctios t App [8] C. H. Chag ad C. W. Ha, O rcurrc rlatio for Broulli ad Eulr ubrs, Bull. Austral Math. Soc [9] G. S. Cho, A ot o th Broulli ad Eulr polyoials, App. Math. Lttrs [10] B. S. Dsouky, Multiparatr poly-cauchyad poly-broulli ubrs ad polyoials, It. J. of Math. Aalysis [11] Y. Haahata, Poly-Eulr polyoials ad Arakawa-Kako typ fuctios, Fuctios t Approxiatio Cotari Math [12] Y. Haahata ad H. Masubuchi, Rcurrc forula for ulti-poly-broulli ubrs, Itgr lctroic J. of Cobiotorial Nubr Thory #A46. } [ x + i r x + i 1 r].

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