A note on (p, q)-bernstein polynomials and their applications based on (p, q)-calculus

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1 Agyuz ad Acigoz Joural of Iequalities ad Applicatios :81 R E S E A R C H Ope Access A ote o p, q-berstei polyomials ad their applicatios based o p, q-calculus Era Agyuz * ad Mehmet Acigoz * Correspodece: agyuz@gatep.edu.tr Departmet of Mathematics, Faculty of Arts ad Sciece, Uiversity of Gaziatep, Gaziatep, Turey Abstract Nowadays p, q-berstei polyomials have bee studied i may differet fields such as operator theory, CAGD, ad umber theory. I order to obtai the fudametal properties ad results of Berstei polyomials by usig p, q-calculus, we give basic defiitios ad results related to p, q-calculus. The mai purpose of this study is to ivestigate a geeratig fuctio for p, q-berstei polyomials. By usig a approach similar to that of Goldma et al. i SIAM J. Discrete Math. 283: , 2014,we derive some ew idetities,relatios,ad formulas for the p, q-berstei polyomials. Also, we give a plot geeratig fuctio of p, q-berstei polyomials for some selected p ad q values. MSC: Primary 05A15; 05A30; secodary 11B65; 65Q20 Keywords: p, q-calculus; p, q-berstei polyomials; Geeratig fuctio; Fuctioal equatios 1 Itroductio The geeratig fuctios are the ey tool i aalytic umber theory for defiig ad obtaiig properties to special polyomials such as Beroulli, Euler, Hermite, ad Geocchi polyomials. The idea behid the geeratig fuctios are that we ca effectually trasform problems about sequeces ito problems about fuctios. The geeratig fuctios have may advatages. For istace, we ca apply various operatios o geeratig fuctios such as scalig, additio, product, itegratio, ad differetiatio. The Berstei polyomials, which are called by the ame of their creator [2], have may applicatios i mathematics, egieerig, CAGD, ad statistics. The reaso for the emergece of Berstei polyomials is the study of the existece of polyomials that are costatly approachig each fuctio at [0, 1]. May useful ad importat studies have bee carried out i approximatio theory with the defiitio of Berstei polyomials see [3] ad [4]. PhillipsobtaiedBersteipolyomialsthatdepedo q-itegers [5]. Orucad Phillips defied q-berstei polyomialsad Bezier curves ad showed usefulresultsfor these polyomials ad curves [6]. Lewaowicz ad Wozy obtaied some applicatios ad foud iterestig formulae for Bezier represetatios ad coefficiets [7]. Kha ad Lobiyal studied Lupaş type p, q-berstei operators ad Lupaş type p, q-bezier curves ad surfaces [8]. Perez ad Palaci proposed a estimatio for a quatile fuctio by usig theberstei polyomialsassmoothweight fuctios [9]. Esi et al. studied statistical The Authors This article is distributed uder the terms of the Creative Commos Attributio 4.0 Iteratioal Licese which permits urestricted use, distributio, ad reproductio i ay medium, provided you give appropriate credit to the origial authors ad the source, provide a li to the Creative Commos licese, ad idicate if chages were made.

2 Agyuz ad Acigoz Joural of Iequalities ad Applicatios :81 Page 2 of 12 covergece ad derived ew theorems for Berstei polyomials [10]. I recet years, p, q-calculushasbeeprogressedrapidlybymayresearchersi[11 13], ad [14]. I aalytic umber theory, Berstei polyomials are a importat polyomials family because they have very close relatios with special umbers ad polyomials such as Stirlig umbers, Beroulli polyomials, Euler polyomials, ad so o. Obtaiig the geeratig fuctio for the Berstei polyomials [15] has led to a icrease ad acceleratio of studies i this area. For istace, Araci ad Acigoz derived some combiatorial relatios ad properties betwee Berstei ad Frobeius Euler polyomials by usig the cocept of geeratig fuctio ad fermioic p-adic itegral o Z p [16]. Acigoz et al. obtaied a ew formig betwee q-beroulli umbersad polyomialsad q-berstei polyomials see [17]ad[18]. Kim proposed ew q-berstei polyomials ad showed some of their represetatios by usig the fermioic p-adic itegral [19]. Bayad ad Kim foud some theorems ad corollaries related to q-beroulli, q-euler, ad Berstei polyomials [20]. Simse et al. gave some ew results for Berstei ad q-berstei polyomials by usig some special fuctios, polyomials,ad umbers [21, 22], ad [23]. I this article, our mai goal is to obtai a geeratig fuctio of p, q-berstei polyomials by usig the p, q-calculus tools as p, q-derivative operator, p, q-expoetial fuctios, ad p, q-hypergeometric fuctios. After that, we preset some theorems ad idetities by usig the mathematical operators o a geeratig fuctio of p, q-berstei polyomials. 2 Miscellaeous defiitios ad otatios I this part, we give defiitios ad otatios related to p, q-calculus ad use some of them i the subsequet chapters. Let 0 < q < p 1ad Z. The the p, q-aalogue of is defied by [] p q p q. 2.1 Some p, q-itegers are show as follows: [1] 1, [2] p + q, [3] p 2 + pq + q 2, 2.2 ad so o, see [24, 25], ad [26]. The p, q-biomial coefficiets are defied as follows: [ ] []! []![ ]!, 2.3 where 1, 0, []! [] [ 1] [1], 0, cf. [24, 25], ad [26].

3 Agyuz ad Acigoz Joural of Iequalities ad Applicatios :81 Page 3 of 12 The p, q-derivative operator is determied as 2.4 [ ] f px fqx D f x, 2.4 p qx where f : R R ad x R cf. [24, 25], ad [26]. We have two types of p, q-expoetial fuctios as i the q-calculus. These fuctios are defied as follows: e x E x x p 2, p, q; p, q x q 2, p, q; p, q 2.5 where 1 p, q; p, q : p q 1 is called a p, q-aalog of q-shifted operator cf. [24, 25], ad [26]. This operator is described by the followig idetity: 1, a; p, q 1, a; p, q p, aq ;p, q. 2.6 The hypergeometric series cocept has moved to p, q-calculus ad was studied i detail by researchers see [24] ad[25]. The p, q-hypergeometric series are give by the followig equatio: r s a1p, a 1q,...,a rp, a rq, b 1p, b 1q,..., b sp, b sq,p, q, z a 1p, a 1q,...,a rp, a rq, p, q p, q, b 1p, b 1q,...,b sp, b sq, p, q p, q, p, q q 2 1+s r 1, 2.7 p where a1p, a 1q,...,a rp, a rq, p, q r a1p, a 1q, p, q. j1 By usig the defiitio of 2.6, we obtai p, q-biomial coefficiets by meas of p, q- hypergeometric series as follows: r s a, b,, p, q, z p, bz; p, q p, az; p, q. 2.8

4 Agyuz ad Acigoz Joural of Iequalities ad Applicatios :81 Page 4 of 12 Substitutig a, b 1, 0 ad 0, 1 ito 2.7, we have p, q-expoetial fuctios by aid of p, q-hypergeometric series i the followig theorem: e x r s 1, 0,, p, q, z 2.9 ad E x r s 0, 1,, p, q, z Now, we give a p, q-expoetial fuctio by usig p, q-factorial i the followig idetity: ε x x p 2 []! We have some results betwee 2.5ad2.11 as follows: ε xe p qx, 0<q < p 1, ε q xe q 2.12 p qx, p p <1. Remar 1cf.[27] q- adp, q-itegers are a good bit similar. However, they have a importat differece. Oe may obtai a equality by usig the defiitio of p, q-iteger as follows: [] p 1 [] q p If we tae p 1i2.13, [] reduces to [] q,buttheoppositeisottrue.thatis,we caot obtai [] by usig [] q. Therefore, we ca describe that q-itegers are a special case of p, q-itegers. 3 Mai results I this sectio, we start with defiitio for p, q-berstei polyomials by meas of p, q- aalogof q-shifted operator. Thas to this defiitio, we give the geeratig fuctio for p, q-berstei polyomials. A importat applicatio of this fuctio is a method for represetig p, q-berstei polyomials i differet ways. We begi by givig a defiitio of p, q-berstei polyomials i the followig equatio. Defiitio 1 cf. [28] Let ad be arbitrary positive itegers. The p, q-berstei polyomial of degree is give by [ ] B, x; p, qp 2 2 where 1 2 is show by 2. x 1 x,, 3.1

5 Agyuz ad Acigoz Joural of Iequalities ad Applicatios :81 Page 5 of 12 We ote that whe p 1, Defiitio 1 reduces to the results of Goldma et al. [1] ad Oruc ad Philips [6]. These polyomials are give as follows: B, x; p, q p 2 2 1, x; p, q p, q +1 ; p, q x p, q; p, q Geeratig fuctio of p, q-berstei polyomials So far we have metioed p, q-calculus ad Berstei polyomials. We tur our attetio ow to the cocept of the geeratig fuctio of these polyomials based o p, q- calculus. By usig a approach similar to that of Goldma et al. [1], we arrive at the followig theorem. Theorem 1 For 0<q < p 1, Ϝ x, 2 2 xt tp []! ε t ε xt B, x; p, q []!. 3.3 Proof The proof of this theorem is similar to those of Goldma et al. [1] ad[22]. Let Ϝ x, t adf x, t, c be geeratig fuctios of p, q-berstei polyomials. So we have the followig formal defiitios, respectively: Ϝ x, t B, x; p, q []! 3.4 ad F x, t, c B, x; p, q , c; p, q By substitutig the right-had side of 3.2 ito3.5, we have F x, t, c p 2 2 1, x; p, q p, q +1 ; p, q x p, q; p, q x t p 2 1, c; p, q 1, c; p, q p 2 1, x; p, q p, q +1 ; p, q p, q; p, q p, cq ; p, q 2 1 1, x, p +1, q +1 ; p, cq,p, q, t x t p 2 2 From 3.4ad3.5, we have the followig equality: 1, c; p, q. 3.6 Ϝ x, t B, x; p, q []!

6 Agyuz ad Acigoz Joural of Iequalities ad Applicatios :81 Page 6 of 12 p q B, x; p, q p, q; p, q F 1, x, p qt,p, q. 3.7 From the right-had side of 3.7, we get F 1, x, p qt,p, q p 2 2 1, x; p, q p, q +1 ; p, q x p 2 xp qt p, q; p, q p, q; p, q p, q; p, q p q p, q; p, q 1, x; p, q p, q +1 ; p, q p, q; p, q p +1, q +1 ; p, q 1 0 1, x, ; p, q, p qt p 2 2 xp qt p, q; p, q by usig the defiitio of p, q-hypergeometric fuctios p 2 2 xp qt p, q; p, q p, xp qt, p, q p,p qt, p, q p 2 2 xt p, xp qt, p, q []! p,p qt, p, q p 2 2 xt e p qt []! e p qxt. From 2.12, we obtai that Ϝ x, 2 2 xt tp []! ε t ε xt. Therefore, we arrive at the desired result. Remar 2 The above defied fuctio ca be called a geeratig fuctio of p, q- Berstei polyomials because whe p 1, the geeratig fuctio reduces to the geeratig fuctio of q-berstei polyomial i [1]. However, the p, q-berstei polyomials are related to ε x. We say that ε x coverges by usig the ratio test, ad also this fuctio is well defied for all x < 1 1 q p if q p < The graph of a geeratig fuctio of p, q-berstei polyomials I this sectio, by usig a similar method to that of the wor of Kucuoglu ad Simse [29], i which a simulatio for the extesio of uificatio of Berstei type basis fuctios is provided, we provide a simulatio for a geeratig fuctio of the p, q-berstei polyomials with their graph for some special values i Fig Some properties for the geeratig fuctio of p, q-berstei polyomials I this sectio, by usig Theorem 1 i Sect. 3.1, wegivesomeresultsforthegeeratig fuctio. Havig obtaied these results, we give some ew idetities for the p, q- Berstei polyomials from arisig p, q-calculus.

7 Agyuz ad Acigoz Joural of Iequalities ad Applicatios :81 Page 7 of 12 Figure 1 The geeratig fuctio of p, q-berstei polyomials Thus we give the followig theorem. Theorem 2 For 0 < q < p 1 ad N {0}, x, tz 0 ε xtzε t p 2. ε xt 3.8 Proof By usig a similar method to that of Goldma et al. [1], we ca prove this theorem. By usig the series expasio for 3.3 depedig o ad z, we have the followig equality: x, tz 0 0 xtz ε t p2 p 2 []! ε xt ε xtzε t p 2. ε xt Corollary 1 B, x; p, q Proof If we tae z 1 i 3.8, we obtai the followig idetity: x, t p 2 ε t. 0 If we expad both sides of 3.3 to a series depedig o, we have x, t 0 0 t. B, x; p, q []! 3.10

8 Agyuz ad Acigoz Joural of Iequalities ad Applicatios :81 Page 8 of 12 By applyig the Cauchy product to the above equatio, we get []! B, x; p, q 0 []! By equatig the coefficiets of []! i 3.11, the desired result is obtaied. Remar 3Whep 1, Corollary 1 is reduced to Corollary 4.2 i [1]. Corollary 2 [ ] l [ ] x l p l 2 B l x; p, qp 2 Proof We give a alterative form of the geeratig fuctio for p, q-berstei polyomialsasfollows: ε xtϝ x. x, 2 2 xt ε t tp []! By usig the defiitio of a geeratig fuctio ad Taylor series for p, q-expoetial fuctios, we obtai p 2 xt []! B, x; p, q []! p 2 2 xt []! By applyig the Cauchy product to the above equatio, we get [ ] l x l p l 2 B l x; p, q t []! t p 2 []!. [ x p 2 ] []! Comparig the coefficiets of []! i 3.13, we arrive at the desired result. Remar 4Whep 1, Corollary 2 is reduced to Corollary 4.3 i [1]. 3.4 Recurrece relatios ad derivative of B, x; p, q I this sectio, we obtai ot oly the derivative operator to a geeratig fuctio of B, x; p, q but also some relatios ad properties for Ϝ x, t. Corollary 3 Let 0<q < p 1. The we have [ ] D,x Ϝ x, t p 2 2 t ε t px 1 ε pxt+ []! ε qxt [] x 1, 3.14 where max{ q/p, x, y } <1 ad t <1/ 1 q/p. Proof By usig the same method as that of Theorem 4.7 i the wor of Goldma et al. [1], we ca prove this corollary. By usig the derivative of the geeratig fuctio for

9 Agyuz ad Acigoz Joural of Iequalities ad Applicatios :81 Page 9 of 12 B, x; p, qwithrespecttox,wehave D,x Ϝ x, t D,x p 2 2 xt []! p 2 2 t ε t []! ε t ε xt x D,x ε xt From the divisio property of p, q-derivative, we obtai D,x Ϝ x, t p 2 2 t ε tx [ ] 1 px 1 D,x + []! ε xt ε qxt [] x 1. After some basic calculatios based o p, q-calculus i the above equatio, we arrive at the desired result. Corollary 4 D,t Ϝ. x, t p 1 ε ptx Ϝ 1 ε qt x, ε pt qt+q ε qt Ϝ x, qt ε xpt ε xqt Ϝ x, qt Proof Multiplyig both sides of the geeratig fuctio for B, x; p, qi3.3byε xt, we have ε xtϝ x, 2 2 xt tp []! ε t By differetiatig the equatio i 3.16with respect to t,we obtai D,t ε xtϝ x, t D,t p 2 2 xt []! ε t p 2 x 2 []! D,t t ε t Byapplyigtheproductruleof p, q-derivative i 3.17, weget Ϝ x, ptε xpt+ε xqtd,t Ϝ x, t p 2 2 x []! ε pt[] t 1 +qt ε pt Applyig the defiitio of Ϝ x, ti3.18, we complete the proof of the corollary. Corollary 5 Let 0<q < p 1. The we have xy, 2 2 s 2 l [l]![s]! tp x l s y s t l s Ϝ l x, tϝ s x, yt, 3.19 []! Ϝ where max{ q/p, x, y } <1 ad t <1/ 1 q/p.

10 Agyuz ad Acigoz Joural of Iequalities ad Applicatios :81 Page 10 of 12 Proof Now, we re-cosider Ϝ xy, t i the followig equatio: Ϝ xy, 2 2 xyt ε t tp []! ε xyt p 2 2 xyt ε t ε xt []! ε xt ε yxt p 2 2 s 2 l [l]![s]! x l s y s t l s []! p l2 2 xtl ε t p s2 2 xts [l]! ε xt [s]! p 2 s 2 l 2 [l]![s]! []! ε t ε xt x l s y s t l s Ϝ l x, tϝ s x, yt From the above equatio, we arrive at the desired result. Corollary 6 B, xy; p, q p j 2 B j, x; p, qb, j y; p, q j0 Proof By usig 3.3for 0, we obtai the followig equality: 0 x, t B 0, x; p, q []! Ϝ p 2 1 p l xq l By usig 3.3 agai i the followig equatios, we have []! Ϝ Ϝ xy, t B, xy; p, q []!, y, xt B, y; p, q xt []! By substitutig 3.22 ad3.23 ito3.19, after some calculatios, we arrive at the desired result. Corollary 7 p 2 1 p l xtq l j 1 p j 2+ j 2 B j, x; p, q p l tq l j0 Proof Usig 3.21ad3.1for 0ady t,weget B 0, xt; p, q p j 2 B j, x; p, qb 0, j t; p, q 3.25 j0

11 Agyuz ad Acigoz Joural of Iequalities ad Applicatios :81 Page 11 of 12 ad B 0, x; p, qp 2 1 p j xq j. j0 After replacig j by j i 3.25 ad some basic calculatios, the proof of the corollary is completed. 3.5 p, q-aalogue of Marsde s idetity Ithis sectio, wecostruct p, q-marsde s idetity by usig p, q-berstei polyomials. Corollary 8 p 2 1 tp l xq l p j 2+ j 2 1 j q j 2 Bj,x; p, qb j, t; p [ ] q, j0 j p q p q >1. Proof By usig the same method as that of Corollary 4.15 i Goldma et al. [1] othe iterval [0, 1], we ca prove this corollary. By replacig t by 1/t i 3.24, we have p 2 1 p l xt 1 q l j 1 p j 2+ j 2 B j, x; p, q p l t 1 q l, 3.26 j0 ad the multiplyig both sides of 3.26by,weget p 2 1 tp l xq l j 1 p j 2+ j 2 B j, x; p, q tp l q l j j0 By usig 3.1, we obtai j 1 j tp l q l j 1 1 j q j 2 t j q l 1 t p 1 j q j 2 B j,t; p q [ ] j p q By combiig 3.27 ad3.28, we derive a p, q-aalogue of Marsde s idetity o the [0, 1]. Thus, the proof of the corollary is completed. 4 Coclusio I this study, we have first itroduced a geeratig fuctio of p, q -Berstei polyomials. By meas of this ew fuctio, we have costructed may ew results ad idetities which are geeralizatios of q-berstei basis polyomials earlier origially itroduced by Goldma et al. i [1].

12 Agyuz ad Acigoz Joural of Iequalities ad Applicatios :81 Page 12 of 12 Acowledgemets The first author is thaful to the Scietific ad Techological Research Coucil of Turey TUBITAK for their support with the Ph.D. scholarship BIDEB Competig iterests The authors declare that they have o competig iterests. Authors cotributios The authors declare that the study was realized i collaboratio with the same resposibility. All authors read ad approved the fial mauscript. Publisher s Note Spriger Nature remais eutral with regard to jurisdictioal claims i published maps ad istitutioal affiliatios. Received: 12 February 2018 Accepted: 3 April 2018 Refereces 1. Goldma, R., Simeoov, P., Simse, Y.: Geeratig fuctios for the q-berstei bases. SIAM J. Discrete Math. 283, Berstei, S.: Demostratio du théoréme de Weierstrass fodeé sur la calcul des probabilités. Comm. Math. Soc. Charow. 132, Ostrovsa, S.: q-berstei polyomials ad their iterates. J. Approx. Theory 123, Ostrovsa, S.: The covergece of q-berstei polyomials. Math. Nachr. 2822, Phillips, G.M.: Berstei polyomials based o the q-itegers. A. Numer. Math. 4, Oruc, H., Phillips, G.M.: q-berstei polyomials ad Bézier curves. J. Comput. Appl. Math. 1511, Lewaowicz, S., Wozy, P.: Bezier represetatio of the costraied dual Berstei polyomials. Appl. Math. Comput. 2188, Kha, K., Lobiyal, D.K.: Bézier curves based o Lupaşp, q-aalogue of Berstei polyomials i CAGD. J. Comput. Appl. Math. 317, Perez, J.M., Palaci, A.F.: Estimatig the quatile fuctio by Berstei polyomials. Comput. Stat. Data Aal. 5, Esi, A., Araci, S., Acigoz, M.: Statistical covergece of Berstei operators. Appl. Math. If. Sci. 6, Mursalee, M., Asari, J.A., Kha, A.: Some approximatio results by p, q-aalogue of Berstei Stacu operators. Appl. Math. Comput. 264, Mursalee, M., Nasiuzzama, N., Nurgali, A.: Some approximatio results o Berstei-Schurer operators defied by p, q-itegers.j.iequal.appl.2015, Mursalee, M., Al-Abied, A.A.H., Alotaibi, A.: O p, q-szász Miraya operators ad their approximatio properties. J. Iequal. Appl. 2017, Kaat, K., Sofyalıoglu, M.: Some approximatio results for Stacu type Lupaş Schurer operators based o p, q-itegers. Appl. Math. Comput , Acigoz, M., Araci, S.: O the geeratig fuctio of the Berstei polyomials. Num. Aal. Appl. Math. Amer. Ist. Phys. Cof. Proc. CP1281, Araci, S., Acigoz, M.: A ote o the Frobeius Euler umbers ad polyomials associated with Berstei polyomials. Adv. Stud. Cotemp. Math. 223, Acigoz, M., Erdal, D., Araci, S.: A ew approach to q-beroulli umbers ad q-beroulli polyomials related to q-berstei polyomials. Adv. Differ. Equ. 2010, Article ID Araci, S., Acigoz, M.: A ote o the values of the weighted q-berstei polyomials ad weighted q-geocchi umbers. Adv. Differ. Equ. 2015, Kim, T., Jag, L.C., Yi, H.: A ote o the modified q-berstei polyomials. Discrete Dy. Nat. Soc. 2010, Article ID Bayad, A., Kim, T.: Idetities ivolvig values of Berstei, q-beroulli, ad q-eulerpolyomials. Russ. J. Math. Phys. 18, Simse, Y., Acigoz, M.: A ew geeratig fuctio of q-berstei-type polyomials ad their iterpolatio fuctio. Abstr. Appl. Aal. 2010,Article ID Simse, Y.: Fuctioal equatios from geeratig fuctios: a ovel approach to derivig idetities for the Berstei basis fuctios. Fixed Poit Theory Appl. 2013, Simse, Y.: O parametrizatio of the q-berstei basis fuctios ad their applicatios. J. Iequal. Spec. Fuct. 81, Charabarti, R., Jagaatha, R.: A p, q-oscillator realizatio of two-parameter quatum algebras. J. Phys. A, Math. Ge. 24, L Jagaatha, R., Rao, K.S.: Two-parameter quatum algebras, twi-basic umbers ad associated geeralized hypergeometric series. arxiv:math/ [math.qa] 26. Sadjag, P.N.: O the fudametal theorem of p, q-calculus ad some p, q-taylorformulas. arxiv: [math.qa] 27. Gupta, V., Aral, A.: Berstei Durrmeyer operators based o two parameters. Facta Uiv. 311, Mursalee, M., Asari, K.J., Kha, A.: O p, q-aalogue of Berstei operators. Appl. Math. Comput. 266, Kucuoglu, I., Simse, Y.: A ote o geeratig fuctios for the uificatio of the Berstei type basis fuctios. Filomat 304,