Inernaiona Maheaica Foru, Vo 1, 17, no 14, 667-675 HIKARI Ld, www-hikarico hps://doiorg/11988/if177647 On a Cass of wo Diensiona wised -angen Nubers and Poynoias C S Ryoo Deparen of Maheaics, Hanna Universiy, Daejeon 36-791, Korea Copyrigh c 17 C S Ryoo his arice is disribued under he Creaive Coons Aribuion License, which peris unresriced use, disribuion, and reproducion in any ediu, provided he origina work is propery cied Absrac In his paper we inroduce wo diensiona wised -angenubers and poynoias We aso give soe properies, expici foruas, severa ideniies, a connecion wih wo diensiona wised -angen nubers and poynoias, and soe inegra foruas Maheaics Subjec Cassificaion: 11B68, 11S4, 11S8 Keywords: wised angenubers and poynoias, wised -angen nubers and poynoias, wo diensiona wised -angenubers and poynoias 1 Inroducion Many aheaicians have sudied in he area of he Bernoui nubers, Euer nubers, and angenuberssee [1,, 3, 4, 5, 7] In his paper, we sudy soe properies of a new ype of wo diensiona wised -angenubers and poynoias hroughou his paper, we aways ake use of he foowing noaions: N denoes he se of naura nubers, Z + N {}, R denoes he se of rea nubers, and C denoes he se of copex nubers For a rea nuber or copex nuber x, -nuber is defined by [x] 1 x 1 if 1, [x] x if 1
668 C S Ryoo he -binoia coefficiens are defined for posiive ineger n, k as [n]! k [k]![n k]! [n] [n 1] [n k + 1], [k]! where [n]! [n] [n 1] [1], n 1,, 3, and []! 1, which is known as -facoriasee [1] he -anaogue of he funcion x + y n is defined by x + y n x n y, n Z + n For any C wih < 1, he wo for of -exponenia funcions are given by z n e z [n]! and E z n zn [n]! Fro his for we easiy see ha e ze z 1 he -derivaive operaor of a any funcion f is defined by D fx n fx fx, x, 11 1 x and D f f, provided ha f is differeniabe a I happens ceary ha D x n [n] x n 1 Ceary, if he funcion fx is differeniabe on he poin x, he -derivaive in 11 ends o he ordinary derivaive in he cassica anaysis when ends o 1 he -angen poynoias n, x are defined by he generaing funcion: e x e + 1 n n, x n [n]! < π When x, n, n, are caed he -angenuberssee [4] he wo diensiona -angen poynoias n, x, y in x, y are defined by eans of he generaing funcion: e xe y e + 1 he definie -inegra is defined as b n fxd x 1 b n, x, y n [n]! < π j f j b 1 Idenicay, if he funcion fx is Rieann inegrabe on he concerned inervas, he -inegra in 1 ends o he Rieann inegras of fx on he j
wo diensiona wised -angenubers and poynoias 669 corresponding inervas when ends o 1see [1,, 5] In he foowing secion, we inroduce he wo diensiona wised -angenubers and poynoias Afer ha we wi invesigae soe heir properies Finay, we give soe reaionships boh beween hese poynoias and -derivaive operaor and beween hese poynoias and -inegra wised -angenubers and poynoias In his secion, we inroduce he wo diensiona wised -angenubers and poynoias and provide soe of heir reevan properies Le r be a posiive ineger, and e ζ be rh roo of 1 he wised -angen poynoias n,,ζ x are defined by he generaing funcion: e x ζe + 1 n n,,ζ x n [n]! 1 When x, n,,ζ n,,ζ are caed he wised -angenubers Upon seing 1 in 1, we have e x ζe,ζ x n + 1 n!, where n,ζ x are caed faiiar wised angen poynoias Seing ζ 1 in 1, we can obain he corresponding definiions for he -angenubers and poynoias, respeciveysee [4] Nuerous properies of wised angen nubers and poynoias are known More sudies and resus in his subjec we ay see references [3], [4], [5], [6], [7] Abou exensions for he angen nubers can be found in [3, 4, 5, 6, 7] he wo diensiona wised -angen poynoias n,,ζ x, y in x, y are defined by eans of he generaing funcion: ζe + 1 n e xe y n n n,,ζ x, y n [n]! I is obvious ha i 1 n,,ζ x, y n,ζ x + y and n,,ζ x, n,,ζ x By 1, we ge n,,ζ x n [n] n! e x ζe + 1 [ ] 3 n n,,ζ y [n]!
67 C S Ryoo By coparing he coefficiens on boh sides of 3, we have he foowing heore heore 1 For n Z +, we have n,,ζ x n,,ζ x By using Definiion of -derivaive operaor and heore 1, we have he foowing heore heore For n Z +, we have D n,,ζ x [n] n 1,,ζ x By heore and Definiion of he definie -inegra, we have [n] D n 1,,ζ xd x n,,ζ 1 n,,ζ 4 Since n,,ζ n,,ζ, by 4, we have he foowing heore heore 3 For n Z +, we have Using he foowing ideniy: D n 1,,ζ xd x n,,ζ1 n,,ζ [n] ζ ζe + 1 e xe + ζe + 1 e x e x, we have he foowing heore heore 4 For n Z +, we have ζ n,,ζ x, + n,,ζ x x n Subsiuing x in heore 4, we have he foowing coroary Coroary 5 For n Z +, we have ζ n,,ζ n,,ζ
wo diensiona wised -angenubers and poynoias 671 By and he rue of Cauchy produc, we ge n,,ζ x, y n [n]! [ ] n n,,ζ xy [n]! 5 n n By coparing he coefficiens on boh sides of 5, we have he foowing heore heore 6 For n Z +, we have n,,ζ x, y Using he foowing ideniy: we have he foowing heore n,,ζ xy ζ ζe + 1 e + ζe + 1, heore 7 For n Z +, we have ζ n,,ζ + n,,ζ {, if n,, if n By using definiion of -derivaive operaor, we have he foowing heore heore 8 For n Z +, we have D,y n,,ζ x, y [n] n 1,,ζ x, y 3 Soe ideniies invoving wised -angen nubers and poynoias In his secion, we give soe reaionships boh beween hese poynoias and -derivaive operaor and beween hese poynoias and -inegra By 1 and by using Cauchy produc, we ge n n,,ζ x n [n]! e x ζe + 1 e x ζe + e e 1 [ ] n 1 ζ 1, 1,ζ 1xn n [n]! 31 By coparing he coefficiens on boh sides of 31, we have he foowing heore
67 C S Ryoo heore 31 For n Z +, we have n,,ζ x ζ 1 1, 1,ζ 1xn By Definiion of he definie -inegra and heore 1, we ge n,,ζ xd x n,,ζ x d x 1 n,,ζ [ + 1] We aso ge n,,ζ xd x ] [ n 1 ζ 1, 1,ζ 1xn d x 1 ζ 1, 1,ζ 1 1 [n + 1] 3 33 By 3 and 33, we have he foowing heore heore 3 For n Z +, we have n,,ζ ζ 1, 1,ζ 1 [ + 1] [n + 1] Using he foowing ideniy: ζe + 1 e xe y we have n,,ζ x, y n [n] n! n,,ζ x n [n] n! n n k ζe + 1 e x ζe 1 ζe 1e y, ζ n n [n]! 1 ζe 1e y [ ] n ζ k+1,,ζ x, 1 k+1,,ζ x k n+1 B k n k,,ζ y [k + 1] Maching he coefficien of [n]! of boh sides gives he foowing heore [n]!
wo diensiona wised -angenubers and poynoias 673 heore 33 For n Z +, we have ζ k+1,,ζ x, 1 k+1,,ζ x k n+1 n,,ζ x, y B k n k,,ζ y k [k + 1] x n E n,,ζ, y Here B n,,ζ x, y and E n,,ζ x, y denoe he wised -Bernoui and wised -Euer poynoias in x, y which are defined by B n,,ζ x, y ζe 1 e xe y and E n,,ζ x, y By Definiion 1 and by using he foowing ideniy: ζe 1 e xe y we ge ζe ζe + 1 e xe y e + 1 y ζe + 1 ζe 1 e x, B n,,ζ x, y n [n] n! n k k B k k,,ζ x n k + 1 k n B k k,,ζ x n k,,ζ y n k By coparing coefficiens of foowing heore [n]! heore 34 For n Z +, we have B n,,ζ x, y 1 [ ] [ n n k k B n k k,,ζ x k ζ,,ζ y n k 1 k n [n]! [n]! in he above euaion, we arrive a he ζ n k k,,ζ y + k n k,,ζ y By Definiion of he definie -inegra and heore 6, ge y n n,,ζ x, yd y n,,ζ xy +n d y 1 n,,ζ x [n + + 1] 34 ]
674 C S Ryoo By 1, we see ha y n n,,ζ x, yd y y n+1 n,,ζx, y [n + 1] n,,ζx, 1 n+1 [n] n 1,,ζ x, 1 [n + 1] [n + 1] [n + ] + 1 n+1 n+ [n] [n + 1] [n 1] [n + ] Coninuing his process, we obain 1 [n] n+1 y n+1 n 1,,ζx, y [n + 1] d y y n+ n,,ζ x, yd y n 1 y n n,,ζ x, yd y n,,ζx, 1 [n + 1] n+1 n+ [n] [n 1] [n + 1] 1 + n,,ζ x, 1 [n + 1] [n + ] [n + + 1] 1 + 1 n n+1 [n]! [n + 1] [n + ] [n] y n,,ζ x, yd y 35 Hence, by 34 and 35, we have he foowing heore heore 35 For n N, we have 1 n,,ζ x n,,ζx, 1 [n + + 1] [n + 1] n 1 + 1 n+1 n+ [n] [n 1] [n + 1] n,,ζ x, 1 [n + 1] [n + ] [n + + 1] 1 + 1 n n+1 [n]! [n + 1] [n + ] [n] [n + 1] References [1] GE Andrews, R Askey, R Roy, Specia Funcions, Vo 71, Cabridge Press, Cabridge, UK, 1999 hps://doiorg/1117/cbo97811735937 [] YH Ki, HY Jung, CS Ryoo, On he generaized Euer poynoias of he second kind, J App Mah & Inforaics, 31 13, 63-63 hps://doiorg/114317/jai1363 [3] CS Ryoo, A noe on he angenubers and poynoias, Adv Sudies heor Phys, 7 13, no 9, 447-454 hps://doiorg/11988/asp13134
wo diensiona wised -angenubers and poynoias 675 [4] CS Ryoo, Soe properies of wo diensiona -angenubers and poynoias, Goba Journa of Pure and Appied Maheaics, 1 16, 999-37 [5] CS Ryoo, Differenia euaions associaed wih angenubers, J App Mah & Inforaics, 34 16, 487-494 hps://doiorg/114317/jai16487 [6] CS Ryoo, A nuerica invesigaion on he zeros of he angen poynoias, J App Mah & Inforaics, 3 14, 315-3 hps://doiorg/114317/jai14315 [7] H Shin, J Zeng, he -angen and -secanubers via coninued fracions, European J Cobin, 31 1, 1689-175 hps://doiorg/1116/jejc143 Received: June 3, 17; Pubished: June 3, 17