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6 1 0 2 r a M 7 (p,q)-Beta Functions and Applications 1 in Approximation ] A C Gradimir V. Milovanović, Vijay Gupta and Neha Malik . h t a m Abstract. In the present paper, we consider (p,q)-analogue of theBeta operatorsand usingit,weproposetheintegralmodification ofthegen- [ eralizedBernsteinpolynomials.Weestimatesomedirectresultsonlocal 2 and global approximation. Also, we illustrate some graphs for the con- v vergence of (p,q)-Bernstein-Durrmeyeroperators for different values of 7 theparameters p and q using Mathematicapackage. 0 Mathematics Subject Classification (2010). Primary 33B15; Secondary 3 41A25. 6 0 Keywords.(p,q)-Betafunction,(p,q)-Gammafunction,Bernstein poly- . nomial, Durrmeyervariant,direct results. 2 0 6 1 : v 1. Introduction i X In the last decade, the generalizations of several operators to quantum vari- r a ant have been introduced and their approximation behavior have been discussed [see for instance [1], [5], [8], [12], [10] etc]. The further generalization ofquantumcalculusisthepost-quantumcalculus,denotedby(p,q)-calculus. Recently, some researchers started working in this direction (cf. [2], [6], [13], [16]). Some basic definitions and theorems, which are mentioned below may be found in these papers and references therein. pn qn [n]p,q :=pn−1+pn−2q+pn−3q2+···+pqn−2+qn−1 = p−q · − n The (p,q)-factorial is given by [n] ! = [k] , n 1, [0] ! = 1. The p,q p,q ≥ p,q k=1 (p,q)-binomial coefficient satisfies Q n [n] ! p,q = , 0 k n. hkip,q [n−k]p,q![k]p,q! ≤ ≤ ThispaperwassupportedbytheSerbianMinistryofEducation,ScienceandTechnological Development(No.#OI174015). 2 G.V. Milovanović,Vijay Gupta and Neha Malik The (p,q)-power basis is defined as (x a)n = (x a)(px qa)(p2x q2a) (pn 1x qn 1a). ⊖ p,q − − − ··· − − − The (p,q)-derivative of the function f is defined as f(px) f(qx) D f(x)= − , x=0. p,q (p q)x 6 − Asaspecialcasewhenp=1,the(p,q)-derivativereducestotheq-derivative. The (p,q)-derivative fulfils the following product rules D (f(x)g(x)) = f(px)D g(x)+g(qx)D f(x), p,q p,q p,q D (f(x)g(x)) = g(px)D f(x)+f(qx)D g(x). p,q p,q p,q Obviously D (x a)0 =0 and for n 1, we have p,q ⊖ p,q ≥ Dp,q(x⊖a)np,q = [n]p,q(px⊖a)pn,−q1, D (a x)n = [n] (a qx)n 1. p,q ⊖ p,q − p,q ⊖ p,−q Letf be anarbitraryfunctionanda R. The (p,q)-integraloff(x) on ∈ [0,a] (see [15]) is defined as a ∞ pk pk p f(x)d x=(q p)a f a if <1 p,q − qk+1 qk+1 q Z0 Xk=0 (cid:18) (cid:19) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12) (cid:12) and (cid:12) (cid:12) a ∞ qk qk q f(x)d x=(p q)a f a if <1. p,q − pk+1 pk+1 p Z0 Xk=0 (cid:18) (cid:19) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12) (cid:12) The formula of (p,q)-integration by part is given by (cid:12) (cid:12) b f(px)D g(x)d x = f(b)g(b) f(a)g(a) p,q p,q − Z a b g(qx)D f(x)d x (1.1) p,q p,q − Z a Very recently Gupta and Aral [7] proposed the (p,q) analogue of usual Durrmeyer operators by considering some other form of (p,q) Beta functions, which is not commutative. In the present article, we define different (p,q)-variantof Beta function of first kind and find an identity relationwith (p,q)-Gamma functions. It is observed that (p,q)-Beta functions may satisfy the commutative property, by multiplying the appropriate factor while choosing (p,q) Beta function. As far as the approximation is concerned, order is important in post-quantum calculus. We propose a generalization of Durrmeyer type operators and establish some direct results. (p,q)-Beta Functions and Applications 3 2. (p,q)-Gamma and (p,q)-Beta Functions Definition 2.1 ([14]). Let n is a nonnegative integer, we define the (p,q)- Gamma function as (p q)n Γ (n+1)= ⊖ p,q =[n] !, 0<q <p. p,q (p q)n p,q − Definition 2.2. Let m,n N, we define (p,q)-Beta integral as ∈ 1 Bp,q(m,n)= xm−1(1⊖qx)pn,−q1dp,qx. (2.1) Z 0 Theorem2.3. The (p,q)-Gamma and (p,q)-Beta functions fulfil thefollowing fundamental relation Γ (m)Γ (n) B (m,n)=p(n 1)(2m+n 2)/2 p,q p,q , (2.2) p,q − − Γ (m+n) p,q where m,n N. ∈ Proof. For any m,n N since ∈ 1 Bp,q(m,n)= xm−1(1⊖qx)pn,−q1dp,qx, Z 0 using (1.1) for f(x) = (x/p)m−1 and g(x) = (1 x)n /[n] with the − ⊖ p,q p,q equality Dp,q(1⊖x)n =−[n]p,q(1⊖qx)n−1 we have 1 [m 1] B (m,n) = − p,q xm 2(1 qx)n d x p,q pm 1[n] − ⊖ p,q p,q − p,q Z 0 [m 1] = − p,q B (m 1,n+1). (2.3) pm 1[n] p,q − − p,q Also we can write for positive integer n 1 B (m,n+1) = xm 1(1 qx)n d x p,q − ⊖ p,q p,q Z 0 1 = xm−1(1⊖qx)pn,−q1 pn−1−qnx dp,qx Z 0 (cid:0) (cid:1) 1 1 = pn−1 xm−1(1⊖qx)pn,−q1dp,qx−qn xm(1⊖qx)pn,−q1dp,qx Z Z 0 0 = pn 1B (m,n) qnB (m+1,n). − p,q p,q − 4 G.V. Milovanović,Vijay Gupta and Neha Malik Using (2.3), we have [m] B (m,n+1)=pn 1B (m,n) qn p,q B (m,n+1), p,q − p,q − pm[n] p,q p,q which implies that pn qn Bp,q(m,n+1)=pn+m−1pn+m−qn+mBp,q(m,n). − Further, by definition of (p,q) integration 1 1 B (m,1) = xm 1d x= p,q − p,q [m] · Z p,q 0 We immediately have B (m,n) = pn+m−2 pn−1−qn−1 B (m,n−1) p,q pn+m−1−qn+m−1 p,q = pn+m−2 pn−1−qn−1 pn+m−3 pn−2−qn−2 B (m,n−2) pn+m−1−qn+m−1 pn+m−2−qn+m−2 p,q = pn+m−2 pn−1−qn−1 pn+m−3 pn−2−qn−2 ··· pn+m−1−qn+m−1 pn+m−2−qn+m−2 p−q ×pm B (m,1) pm+1−qm+1 p,q = pn+m−2 pn−1−qn−1 pn+m−3 pn−2−qn−2 ··· pn+m−1−qn+m−1 pn+m−2−qn+m−2 p−q 1 ×pm pm+1−qm+1[m] p,q (p⊖q)n−1 = pm+(m+1)+···+(m+n−2) p,q (p−q) (pm⊖qm)n p,q (p⊖q)n−1 = p(n−1)(2m+n−2)/2 p,q (p−q). (2.4) (pm⊖qm)n p,q Following [14], we have (a b)n+m =(a b)n (apn bqn)m thus (2.4) ⊖ p,q ⊖ p,q ⊖ p,q leads to Bp,q(m,n) = p(n−1)(22m+n−2)(p(mp⊖⊖qq)mnp,−)qn1 (p−q) p,q = p(n−1)(22m+n−2)((pp−⊖qq))nnp,−−q11 · ((pp⊖−qq))mpm,q−−11 · (p(p⊖−q)qm)m−−11(p(mp−⊖qq)mn−)n1 (p−q) p,q p,q = p(n−1)(22m+n−2)((pp−⊖qq))nnp,−−q11 · ((pp⊖−qq))mpm,q−−11 · ((pp−⊖qq))mm++nn−−11 p,q = p(n−1)(22m+n−2)Γp,q(m)Γp,q(n)· Γ (m+n) p,q This completes the proof of the theorem. (cid:3) (p,q)-Beta Functions and Applications 5 Remark 2.4. Thefollowingobservationshavebeenmadefor(p,q)Betafunc- tions: For m,n N, we have • ∈ B (m,n+1)=pn 1B (m,n) qnB (m+1,n). p,q − p,q p,q − The (p,q)-Beta integral defined by (2.1) is not commutative. In order • to make commutative, we may consider the following form 1 Bp,q(m,n)= pm(m−1)/2xm−1(1⊖qx)pn,−q1dp,qx. Z 0 e For this form, (p,q)-Gamma and (p,q)-Beta functions fulfill the following fundamental relation Γ (m)Γ (n) B (m,n)=p(2mn+m2+n2 3m 3n+2)/2 p,q p,q , (2.5) p,q − − Γ (m+n) p,q whereem,n N.Obviouslyforform(2.5),wegetB (m,n)=B (n,m). p,q p,q ∈ e e 3. (p,q) Bernstein-Type Operators and Moments For n N and k 0, we have the following identity, which can be easily ∈ ≥ verified using the principle of mathematical induction: n n pk(k 1)/2xk(1 x)n k =pn(n 1)/2. (3.1) k p,q − ⊖ p,−q − Xk=0h i Using the above identity, we consider the (p,q)-analogueof Bernstein operators for x [0,1] and 0<q <p 1 as ∈ ≤ n pn k[k] B (f,x)= bp,q(1,x)f − p,q , (3.2) n,p,q n,k [n] k=0 (cid:18) p,q (cid:19) X where the (p,q)-Bernstein basis is defined as n bp,q(1,x)= p[k(k 1) n(n 1)]/2xk(1 x)n k. n,k k p,q − − − ⊖ p,−q h i Remark 3.1. Other form of the (p,q)-analogue of Bernstein polynomials has been recently considered by Mursaleen et al. [13]. Remark 3.2. Using the identity (3.1) and the following recurrence relation (for m 1, Up,q (x)=B (e ,x)=B (tm,x)): ≥ n,m n,p,q m n,p,q [n] Up,q (px)=pnx(1 px)D [Up,q (x)]+[n] pxUp,q (px), p,q n,m+1 − p,q n,m p,q n,m the (p,q)-Bernstein polynomial satisfy pn 1x(1 x) B (e ,x)=1, B (e ,x)=x, B (e ,x)=x2+ − − , n,p,q 0 n,p,q 1 n,p,q 2 [n] p,q where e =ti, i=0,1,2. i 6 G.V. Milovanović,Vijay Gupta and Neha Malik Recently, Gupta and Wang (see [9]) discussed the q-variant of certain Bernstein-Durrmeyer type operators. We now extend these studies and propose the following(p,q)-Bernstein-Durrmeyeroperatorsbasedon(p,q)-Beta function. For x [0,1] and 0 < q < p 1, the (p,q)-analogue of Bernstein- ∈ ≤ Durrmeyer operators is defined as n Dp,q(f,x) = [n+1] p (n k+1)(n+k)/2bp,q(1,x) n p,q − − n,k k=1 X 1 bp,q (t)f(t)d t+bp,q(1,x)f(0), (3.3) × n,k 1 p,q n,0 − Z 0 where bp,q(1,x) is defined by (3.2) and n,k n bpn,,qk(t)= k p,qtk(1⊖qt)pn,−qk. h i It maybe remarkedhere that for p=1, these operatorswill notreduce to the q-Durrmeyer operators; but for p = q = 1, these will reduce to the Durrmeyer operators. Lemma3.3. Let e =tm, m N 0 , then for x [0,1] and 0<q <p 1, m ∈ ∪{ } ∈ ≤ we have p[n] x Dp,q(e ,x)=1, Dp,q(e ,x)= p,q , n 0 n 1 [n+2] p,q (p+q)pn+1[n] x ([n] pn 1)p2q[n] x2 Dp,q(e ,x)= p,q + p,q− − p,q n 2 [n+2] [n+3] [n+2] [n+3] · p,q p,q p,q p,q Proof. Using (2.2) and (2.1) and Remark 3.2, we have n Dp,q(e ,x) = [n+1] p [(n+1 k)(n+k)/2]bp,q(1,x) n 0 p,q − − n,k k=1 X 1 n × k 1 tk−1(1⊖qt)np,+q1−kdp,qt+bpn,,q0(1,x) Z0 (cid:20) − (cid:21)p,q n n = [n+1] p [(n+1 k)(n+k)/2]bp,q(1,x) p,q − − n,k k 1 k=1 (cid:20) − (cid:21) X B (k,n k+2)+bp,q(1,x) × p,q − n,0 n [n] ![k 1] ! = [n+1]p,q p−[(n+1−k)(n+k)/2]bpn,,qk(1,x) [np+,q1 −k] p,q! p,q k=1 − X [k 1] ![n k+1] ! × p[(n+1−k)(n+k)/2] − p[,nq+1−] ! p,q +bpn,,q0(1,x) p,q = B (1,x)=1. n,p,q (p,q)-Beta Functions and Applications 7 Next, applying Remark 3.2, we have n Dnp,q(e1,x) = [n+1]p,q p−[(n+1−k)(n+k)/2]bpn,,qk(1,x) k=1 X 1 n tk(1 qt)n+1 kd t × k 1 ⊖ p,q − p,q Z (cid:20) − (cid:21)p,q 0 n = [n+1] p [(n+1 k)(n+k)/2]bp,q(1,x) p,q − − n,k k=1 X n B (k+1,n k+2) p,q × k 1 − (cid:20) − (cid:21)p,q n n = [n+1] p [(n+1 k)(n+k)/2] bp,q(1,x) p,q − − n,k k 1 k=1 (cid:20) − (cid:21)p,q X [k] ![n k+1] ! p(n+1−k)(n+k+2)/2 p,q − p,q × [n+2] ! p,q n [k] = pn k+1bp,q(1,x) p,q − n,k [n+2] p,q k=1 X p[n] n pn k[k] p[n] x = p,q bp,q(1,x) − p,q = p,q [n+2] n,k [n] [n+2] · p,q p,q p,q k=1 X Further, using the identity [k+1] =pk+q[k] and by Remark 3.2, p,q p,q we get n Dp,q(e ,x) = [n+1] p [(n+1 k)(n+k)/2] bp,q(1,x) n 2 p,q − − n,k k=1 X 1 n × k 1 tk+1(1⊖qt)np,+q1−kdp,qt Z0 (cid:20) − (cid:21)p,q n = [n+1] p [(n+1 k)(n+k)/2] bp,q(1,x) p,q − − n,k k=1 X n B (k+2,n k+2) p,q × k 1 − (cid:20) − (cid:21)p,q n n = [n+1] p [(n+1 k)(n+k)/2] bp,q(1,x) p,q − − n,k k 1 k=1 (cid:20) − (cid:21)p,q X [k+1] ![n k+1] ! p(n+1−k)(n+k+4)/2 p,q − p,q × [n+3] ! p,q n [k] [k+1] = p2(n k+1) bp,q(1,x) p,q p,q − n,k [n+2] [n+3] p,q p,q k=1 X 8 G.V. Milovanović,Vijay Gupta and Neha Malik i.e., Dp,q(e ,x) = n p2(n−k+1) bp,q(1,x) [k]p,q(pk+q[k]p,q) n 2 n,k [n+2] [n+3] Xk=1 p,q p,q pn+2[n] n pn−k[k] = p,q bp,q(1,x) p,q [n+2] [n+3] n,k [n] p,q p,q Xk=1 p,q p2q[n]2 n pn−k[k] 2 + p,q bp,q(1,x) p,q [n+2] [n+3] n,k (cid:18) [n] (cid:19) p,q p,q Xk=1 p,q pn+2[n] x p2q[n]2 pn−1x(1−x) = p,q + p,q x2+ [n+2] [n+3] [n+2] [n+3] (cid:18) [n] (cid:19) p,q p,q p,q p,q p,q pn+2[n] x p2q[n]2 x2 pn+1q[n] x(1−x) = p,q + p,q + p,q [n+2] [n+3] [n+2] [n+3] [n+2] [n+3] p,q p,q p,q p,q p,q p,q (p+q)pn+1[n] x ([n] −pn−1)p2q[n] x2 = p,q + p,q p,q · [n+2] [n+3] [n+2] [n+3] p,q p,q p,q p,q (cid:3) Remark 3.4. Using above lemma, we can obtain the following central moments: (p[n] [n+2] )x 1◦ Dnp,q((t−x),x)= p,[qn−+2] p,q p,q (p+q)pn+1[n] x 2◦ Dnp,q (t−x)2,x = [n+2] [n+3p],q p,q p,q (cid:0) (cid:1) [([n] pn 1)p2q[n] 2p[n] [n+3] +[n+2] [n+3] ]x2 p,q − p,q p,q p,q p,q p,q + − − . [n+2] [n+3] p,q p,q Lemma 3.5. Let n be a given natural number, then 6 1 Dp,q((t x)2,x) ϕ2(x)+ , n − ≤ [n+2] [n+2] p,q (cid:18) p,q(cid:19) where ϕ2(x)=x(1 x), x [0,1]. − ∈ Proof. In view of Lemma 3.3, we obtain (p+q)pn+1[n] x Dp,q((t x)2,x)= p,q n − [n+2] [n+3] p,q p,q [([n] pn 1)p2q[n] 2p[n] [n+3] +[n+2] [n+3] ]x2 p,q − p,q p,q p,q p,q p,q + − − . [n+2] [n+3] p,q p,q By direct computations, using the definition of the (p,q)-numbers, we get (p+q)pn+1[n]p,q =pn+1(p+q)(pn−1+pn−2q+pn−3q2+ +pqn−2+qn−1)>0 ··· for every q (q ,1). 0 ∈ Furthermore, the expression (p+q)pn+1[n] +([n] pn 1)p2q[n] 2p[n] [n+3] +[n+2] [n+3] p,q p,q − p,q p,q p,q p,q p,q − − (p,q)-Beta Functions and Applications 9 is equal to (p+q)pn+1[n] +p2q[n]2 pn+1q[n] p,q p,q − p,q 2p[n] (pn+2+qpn+1+q2pn+q3[n] ) p,q p,q − +(pn+1+qpn+q2[n] )(pn+2+qpn+1+q2pn+q3[n] ) 6. p,q p,q ≤ In conclusion, for x [0,1], we have ∈ 6 Dp,q((t x)2,x) δ2(x), (3.4) n − ≤ [n+2] n p,q which was to be proved. (cid:3) 4. Local and Global Estimates Inthissection,weestimatesomedirectresults,viz.,localandglobalapprox- imation in terms of modulus of continuity. Our first main result is a local theorem. Forthis,wedenoteW2 = g C[0,1]:g C[0,1] ,forδ >0,K functional ′′ ∈ ∈ − is defined as (cid:8) (cid:9) K2(f,δ)=inf f g +η g′′ :g W2 , k − k k k ∈ where norm- . denotes the u(cid:8)niform norm on C[0,1]. F(cid:9)ollowing the well kk known inequality due to DeVore and Lorentz [3], there exists an absolute constant C >0 such that K (f,δ) Cω (f,√δ), (4.1) 2 2 ≤ where the second order modulus of smoothness for f C[0,1] is defined as ∈ ω (f,√δ)= sup sup f(x+h) f(x). 2 | − | 0<h √δx,x+h [0,1] ≤ ∈ The usual modulus of continuity for f C[0,1] is defined as ∈ ω(f,δ)= sup sup f(x+h) f(x). | − | 0<h δx,x+h [0,1] ≤ ∈ Theorem 4.1. Let n > 3 be a natural number and let 0 < q < p 1, q = 0 ≤ q (n) (0,p) be defined as in Lemma 3.5. Then, there exists an absolute 0 ∈ constant C >0 such that 2x |Dnp,q(f,x)−f(x)|≤Cω2 f,[n+2]p−,1q/2δn(x) +ω f,[n+2] , (cid:16) (cid:17) (cid:18) p,q(cid:19) where f C[0,1], δ2(x) =ϕ2(x)+ 1 , ϕ2(x)=x(1 x), x [0,1] and ∈ n [n+2]p,q − ∈ q (q ,1). 0 ∈ Proof. For f C[0,1] we define ∈ p[n] x Dp,q(f,x) = Dp,q(f,x)+f(x) f p,q . n n − [n+2] (cid:18) p,q(cid:19) e 10 G.V. Milovanović,Vijay Gupta and Neha Malik Then, by Lemma 3.3, we immediately observe that D˜p,q(1,x)=Dp,q(1,x)=1 n n and p[n] x D˜p,q(t,x)=Dp,q(t,x)+x p,q =x. n n − [n+2] p,q By applying Taylor’s formula t g(t)=g(x)+(t x)g (x)+ (t u)g (u)du, ′ ′′ − − Z x we get t Dp,q(g,x) = g(x)+Dp,q (t u)g (u)du,x n n  − ′′  Z x e e  t  = g(x)+Dnp,q (t−u)g′′(u)du,x Z x   p[n]p,qx [n+2]p,q p[n] x p,q u g′′(u)du. − [n+2] − Z (cid:18) p,q (cid:19) x Thus, Dp,q(g,x) g(x) | n − | p[n]p,qx e t [n+2]p,q p[n] x Dp,q t u g (u) du ,x + p,q u g (u)du ≤ n (cid:12)(cid:12)Zx | − |·| ′′ | (cid:12)(cid:12)  (cid:12)(cid:12) Zx (cid:12)(cid:12) [n+2]p,q − (cid:12)(cid:12)| ′′ | (cid:12)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)  (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) p[n](cid:12) x (cid:12) 2 (cid:12) (cid:12) (cid:12) ≤ Dnp,q((t−x)2,x)kg′′k+ [n+p2,]q −x kg′′k. (4.2) (cid:18) p,q (cid:19) Also, we have p[n] x 2 Dp,q((t x)2,x)+ p,q x n − [n+2] − (cid:18) p,q (cid:19) 2 6 1 (p[n] [n+2] )x ϕ2(x)+ + p,q− p,q ≤ [n+2] [n+2] [n+2] p,q (cid:18) p,q(cid:19) (cid:18) p,q (cid:19) 10 1 ϕ2(x)+ . (4.3) ≤ [n+2] [n+2] p,q (cid:18) p,q(cid:19) Hence, by (4.2) and with the condition n>3 and x [0,1], we have ∈ 10 |Dnp,q(g,x)−g(x)|≤ [n+2] δn2(x) kg′′k. (4.4) p,q e