(p, q)-type BETA FUNCTIONS OF SECOND KIND

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1 Adv. Oper. Theory 6, o., ISSN: 538-5X electroic p, q-type BETA FUNCTIONS OF SECOND KIND ALI ARAL ad VIJAY GUPTA Commuicated by A. Kamisa Abstract. I the preset article, we propose the p, q-variat of beta fuctio of secod id ad establish a relatio betwee the geeralized beta ad gamma fuctios usig some idetities of the post-quatum calculus. As a applicatio, we also propose the p, q-basaov Durrmeyer operators, estimate momets ad establish some direct results.. Itroductio The quatum calculus q-calculus i the field of approimatio theory was discussed widely i the last two decades. Several geeralizatios to the q variats were recetly preseted i the boo [3. Further there is possibility of etesio of q-calculus to post-quatum calculus, amely the p, q-calculus. Actually such etesio of quatum calculus ca ot be obtaied directly by substitutio of q by q/p i q-calculus. But there is a li betwee q-calculus ad p, q-calculus. The q calculus may be obtaied by substitutig p i p, q-calculus. We metioed some previous results i this directio. Recetly, Gupta [8 itroduced p, q geuie Berstei Durrmeyer operators ad established some direct results. p, q geeralizatio of Szász Miraya operators was defied i [. Also authors ivestigated a Durrmeyer type modificatios of the Berstei operators i [9. We ca also metio other papers as Berstei operators [, Berstei Stacu operators [. Bleima Butzer Hah operators ad Szász Miraya Katorovich Copyright 6 by the Tusi Mathematical Research Group. Date: Received: Oct. 7, 6; Accepted: Nov. 9, 6. Correspodig author. Mathematics Subject Classificatio. Primary 4A5; Secodary 4A39. Key words ad phrases. p, q-beta fuctio of secod id, p, q-gamma fuctio, Basaov operator, Durrmeyer variat. 34

2 p, q-type BETA FUNCTIONS OF SECOND KIND 35 operators. Besides this, we also refer to some recet related wor o this topic: e.g. [5, [ ad [3. Some basic otatios of p, q-calculus are metioed below: The p, q-umbers are defied as [ p q p q. Obviously, it may be see that [ p [ q/p. I The p, q-factorial is defied by [! [,, [!. The p, q-biomial coefficiet is give by [ [! [! [!,. For details see [5 ad [6. Defiitio.. The p, q-power basis is defied below ad it also has a li with q-power basis as a ap qap q a p q a. a ap qap q a p q a. Defiitio.. The p, q-derivative of the fuctio f is defied as D f f p f q, p q ad D f f, provided that f is differetiable at. Note also that for p, the p, q-derivative reduces to the q derivative. The p, q-derivative fulfils the followig product rules D fg fpd g gqd f D fg gpd f fqd g. The followig assertios hold true: ad D a. D a [ p a, D a [ a q,, Defiitio.3. [4Let is a oegative iteger, we defie the p, q-gamma fuctio as Γ p q p q [!, < q < p.

3 36 A. ARAL ad V. GUPTA Propositio.4. The formula of p, q-itegratio by part is give by b a f p D g d f b g b f a q a b a g q D f d. I the preset paper, we propose the p, q-basaov Durrmeyer operators ad estimate some approimatio properties, which iclude asymptotic formula ad covergece i terms of modulus of cotiuity.. p, q-beta Fuctio of Secod Kid Let m, N, we defie p, q-beta fuctio of secod id as B m, m p m d Theorem.. Let m, N. We have the followig relatio betwee p, q-beta ad p, q-gamma fuctio: Proof. We ow that B m, q [ mm / p mm/ Γ m Γ. Γ m D p [ p If we choose f m ad g p[m m B, B m, m p m d p m p [m p m p [m p m [m p [m p m [m q m [m p [m pq m [m p m D D m m ad use. we have m q m q m q m m p [m pq m [m B m,, p d p [ d d q m m d d d D d p [

4 p, q-type BETA FUNCTIONS OF SECOND KIND 37 ad B m, p [m pq m [m B m, p [m pq m [m p [m pq m [m p [m pq m [m qp m Γ m Γ pq m m/ Γ m p [m pq m [m B m, p [m p pq m B, [m pq [ p [m pq m [m p q pq [ pq [ 3. p, q-basaov Durrmeyer Operators ad Momets The p, q-aalogue of Basaov operators for [, ad < q < p is defied as p B, f; b, f [, 3. q [ where b, [ p / q / I case p, we get the q-basaov operators [. If p q, we get at oce the well ow Basaov operators. Remar 3.. Startig with the followig relatios betwee p, q-calculus ad q- calculus: [ [ p ad a p / a q/p ad usig momets of q-basaov operators see [, [3, it ca easily be verified by simple computatio that B, ;, B, t;, B, t ; p pq [. q/p.

5 38 A. ARAL ad V. GUPTA Defiitio 3.. Usig p, q-beta fuctio of secod id, we propose below for [,, < q < p the p, q aalogue of Basaov Durrmeyer operators D f; [ [ b, q[ / p / t pt fp td t 3. where b, is as defied i 3.. Lemma 3.3. For [,, < q < p, we have D ; D t; 3 D t ; qp [ [ p q [ p [ p q 3 [ [ 3 5 qqp[3 p 6 q 4 [ [ 3 p5 qqp p 3 q[ q pqp p 9 q [ [3 p q 3 [ 3 p [3 q[ [3 p q 6 [ [ 3. p 7 Proof. Usig 3. ad Remar 3., we have D ; [ [ [ [ b, q[ / p / t pt d t b, q[ / p / B, ;. B,

6 p, q-type BETA FUNCTIONS OF SECOND KIND 39 Net usig the idetity [ q p[ ad applyig Remar 3., we have D t; [ [ b, q[ / p / q [ / p / b, [ p q b,. [ [ [ p q b, q p[ [ qp B [, ; B p q, t; [ [. qp [ p q [ [ t p d pt t p B, Further usig the idetity [ q pq p [ ad by Remar 3., we get D t ; [ [ [ b, q[ / p / t p pt d t q [ / p / b, [ p.b 3, 3 q 3 p 5 b,. [ [ [ [ 3 b, p 5.q 3 p 3 [ q p[ p [ q [ [ [ 3 [ p b, [ [ 3 q [ p 7 q 5 p p 3 [ p q [ q 3 [ q 4

7 4 A. ARAL ad V. GUPTA q 3 [ B, ; p 3 [ p [ B [ [ 3 q 4, t; [ [ 3 p 7 q 5 [ B, t ; [ [ 3 [ [ p [ q 3 [ [ 3 p 3 q 4 [ [ 3 [ p pq p 7 q 5 [ [ 3 [ [ [ p [ q 3 [ [ 3 p 3 q 4 [ [ 3 [ p pq p 7 q 5 [ [ 3 [. [ [ p [ q 3 [ [ 3 p 3 q 4 [ [ 3 [ [ p 7 q 5 [ [ 3 p 8 q 5 [ [ 3 [. p 7 q 6 [ [ 3 [ [ p [ q 3 [ [ 3 p 3 q 4 [ [ 3 p 7 [ p 9 q [ [3 p q 3 [ 3 [ [3 p q 5 [ [ 3 [ [ p 8 q 5 [ [ 3 p 7 q 6 [ [ 3 [ p 5 q q p [3 p5 q q p q 3 [ [ 3 p 6 q 4 [ [ 3 p 3 q[ q pq p [3 p 9 q [ p q 3 [ 3 p [3 q[ [3 p q 6 [ [ 3 p Weighted approimatio We cosider the followig class of fuctios:

8 p, q-type BETA FUNCTIONS OF SECOND KIND 4 Let H [, be the set of all fuctios f defied o [, satisfyig the coditio f M f, where M f is a costat depedig oly o f. By C [,, we deote the subspace of all cotiuous fuctios belogig to H [,. Also, let C [, be the subspace of all fuctios f C [,, f for which lim is fiite. The orm o C f [, is f [,. Now we shall discuss the weighted approimatio theorem, where the approimatio formula holds true o the iterval [,. Theorem 4.. Let p p ad q q satisfies < q < p ad for sufficietly large p, q ad q ad p. For each f C [,, we have lim Dp,q f f. Proof. Usig the Theorem i [7 we see that it is sufficiet to verify the followig three coditios lim Dp,q t ν, ν, ν,,. 4. Sice, the first coditio of 4. is fulfilled for ν. We ca write for > 3 ad t, q p [ p,q [ p,q p q[ p,q p [, D p,q t, q p q p p 9 q [ p,q [, p 7 [, p p [3 p,q q [ p,q [3 p,q q[ 3 3 p q[ 6 p,q [ 3 p,q [3 p,q p 5 q q p [3 p,q p5 q q p p 6 q[ 4 p,q [ 3 p,q p 3 q [ p,q [, [ p,q q[ 3 p,q [ 3 p,q which implies that lim Dp,q t,

9 4 A. ARAL ad V. GUPTA ad lim Thus the proof is completed. D p,q t,. We give the followig theorem to approimate all fuctios i C [,. Theorem 4.. Let p p ad q q satisfies < q < p ad for sufficietly large p, q ad q ad p. For each f C [,, we have lim [, Proof. For ay fied >, D p,q f, f [, α D p,q f, f α. f, f α D p,q f f C[, f f α. D p,q f, f α t, α The first term of the above iequality teds to zero from well ow Korovi s p,q D theorem. By Lemma 3.3 for ay fied > it is easily see that t, α teds to zero as. We ca choose > so large that the last part of above iequality ca be made small eough. Remar 4.3. For q, ad p q, it is see that lim [ /q p. I order to cosider covergece of p, q Basaov operators we assume p p ad q q such that < q < p ad for sufficietly large p, q ad p ad q. 5. Quatitative approimatio Let C B [, deote the space of all real valued cotiuous ad bouded fuctios o [,. I this space we cosider the orm f CB f. [, Now we give the first ad secod order modulus of cotiuity of fuctio f C B see [4, [6 The first modulus of cotiuity is defied as ω f; δ f u f v,u,v u v δ

10 p, q-type BETA FUNCTIONS OF SECOND KIND 43 ad the secod order modulus of cotiuity is defied ω f; δ f u f u v f v, δ.,u,v u v δ We will use the Stelov mea fuctio for f C B f h 4 h h Sice f h C B we ca write f h f 4 h h It is obvious that ad h h [f u v f u v dudv. 5. [f u v f u v f dudv. f h f ω f; h If f is cotiuous, the f h C B ad Thus we have Similarly f h C B ad f h f CB ω f; h. 5. f h 4 h [ f v h f v dv h h f h v f v dv f h CB 5 h ω f; h. 5.3 f h CB 9 h ω f; h Theorem 5.. Let q, ad p q,. The operator D ito C B ad maps space C B D f CB f CB.

11 44 A. ARAL ad V. GUPTA Proof. Let q, ad p q,. From Lemma 3.3. we have D f; [ b, q[ / p / [, [ f [ [ t pt t fp t d t b, q[ / p / pt f D ; f CB. [, d t We are goig to study the degree of approimatio i terms of ω f; δ ad ω f; δ, first ad secod order modulus of cotiuity. Theorem 5.. Let q, ad p q,. If f C B, the f; f 5ω f; [ D [ p q [ p [ qp [ p 9 ω 7 p 7 f; [ q pq p p 8 qp 7 p 9 q p 7 [ [3 [ p [3 q[ [3 p q 3 [ 3 p q 6 [ 3 p 5 q q p [3 p5 q q p [ p 6 q 4 [ 3 p 3 q qp q 3 [ 3 Proof. We use the Stielov fuctio f h defied by 5.. For ad N, we have D f; f D f f h ; D f h f h ; f h f. By 5. we ca write D f f h ; D f f h CB f f h CB ω f; h.

12 Sice D D p, q-type BETA FUNCTIONS OF SECOND KIND 45 is a liear positive operator we get f h f h ; f h D t ; By Lemma 3.3, 5.3 ad 5.4 we have D where D f h f h ; 5 h ω f; h 9 h ω f; h D t ; p 7 p 7 p 7 [3 qp [ t ;, f D CB t ;. [ p q [ p q pq p p 8 qp 7 p 9 q [ p [3 q[ [3 p q 3 [ 3 p q 6 [ [ 3 p 5 q q p [3 p5 q q p p 6 q 4 [ [ 3 p 3 q[ qp [ [ q 3 [ [ 3 for, h >. Settig h [, we have desired result. Remar 5.3. From Theorem 5. we ca say that that the order of approimatio of D f; to f is at least as good as the order of approimatio to f by classical Basaov Durrmeyer operators for ay [, as a depedig o selectio of q ad p. If we choose p ad q as i Remar 4.3, we have a approimatio process with the aid of operator 3.. Refereces. T. Acar, p, q-geeralizatio of Szász Miraya operators, Math. Methods i the Appl. Sci. 39 6, o., A. Aral ad V. Gupta, Geeralized q Basaov operators, Math. Slovaca 6, o. 4, A. Aral, V. Gupta, ad R. P. Agarwal, Applicatios of q-calculus i Operator Theory, Spriger M. Becer, Global approimatio theorems for Szasz - Miraya ad Basaov operators i polyomial weight spaces, Idiaa Uiv. Math. J , o., Q. B. Cai ad G. Zhou, O p, q-aalogue of Katorovich type Berstei-Stacu-Schurer operators, Appl. Math. Comput. 76 6, 6. Z. Ditzia ad V. Toti, Moduli of smoothess, Spriger, New Yor A. D. Gadžhiev, Theorems of the type of P. P. Korovi type theoremsrussia, Preseted at the Iteratioal Coferece o the Theory of Approimatio of Fuctios Kaluga, 975. Mat. Zameti 976, o. 5, V. Gupta, p, q-geuie Berstei Durrmeyer operators, Boll. Uioe Mat. Ital. 9 6, o. 3,

13 46 A. ARAL ad V. GUPTA 9. V. Gupta ad A. Aral, Berstei Durrmeyer operators based o two parameters, Facta Uiv Ser. Math. Ifor. 3 6, o., M. Mursalee, K. J. Asari, ad A. Kha, O p, q-aalogue of Berstei Operators, Appl. Math. Comput [Erratum: Appl. Math. Comput. 78 6, M. Mursalee, K. J. Asari, ad A. Kha, Some Approimatio Results by p, q-aalogue of Berstei-Stacu operators, Appl. Math. Comput. 64 5, 39 4 [Corrigedum: Appl. Math. Comput 69 5, M. Mursalee, Md. Nasiruzzama, A. Kha ad K. J. Asari, Some approimatio results o Bleima-Butzer-Hah operators defied by p, q-itegers, Filomat 3 6, o. 3, M. Mursalee, A. Alotaibi, ad K. J. Asari, O a Katorovich variat of p, q-szász Miraja operators, J. Fuct. Spaces 6, Art. ID 3553, 9 pp. 4. P. N. Sadjag, O the p, q-gamma ad the p, q-beta fuctios, arxiv v. 5. P. N. Sadjag, O the fudametal theorem of p, q-calculus ad some p, q-taylor formulas, arxiv: [math.qa. 6. V. Sahai ad S. Yadav, Represetatios of two parameter quatum algebras ad p, q- special fuctios, J. Math. Aal. Appl , Departmet of Mathematics, Kiriale Uiversity, TR-745, Yahsiha Kiriale, Turey. address: aliaral73@yahoo.com Departmet of Mathematics, Netaji Subhas Istitute of Techology, Sector 3 Dwara, New Delhi-78, Idia. address: vijaygupta@hotmail.com