On a new sequence of q -Baskakov-Szasz-Stancu operators

ISSN: 2347-2529

Available online atwww.ijaamm.com

International Journal of Advances in

Applied Mathematics and Mechanics

On a new sequence of

q

-Baskakov-Szasz-Stancu operators

Research Article

Sangeeta Garg

1,∗

1_{(Research Scholar) Department of Mathematics, Mewar University, Chittorgarh, Rajasthan,India}

Received 02 July 2014; accepted (in revised version) 25 August 2014

Abstract:

This paper deals with Stancu type generalization ofq-analogue of Baskakov-Szasz operators introducing a new sequence of positiveq-integral operators. We show that it is a weighted approximation process in the polynomial space of continuous functions defined on[0,∞). An estimate for the rate of convergence and weighted approximation properties are also obtained for these operators.

MSC:

41A25• 41A35

Keywords:

Stancu type generalization• Continuity of functions•q-integrals• q-Baskakov-Szasz operators•Rate of Con-vergence•Weighted approximation.

c

2014 IJAAMM all rights reserved.

1.

Introduction

In approximation theory,q-calculus makes our research very interesting. In the year 1987, firstlyq-analogue of classical Bernstein polynomials was given by A. Lupas[11]. In 1997, the most importantq-analogue of the Bernstein polynomials was introduced by Phillips[13]. After that many researchers worked in this direction and proposed manyq-operators and motivated their different properties related to some special functions, number theory and convergence behaviour. Gupta et al. [6]established the generating functions of someq-basis functions. In the theory of approximation, the convergence is important. In this context we mention some results for the convergence ofq-discrete operators due to[2],[7]etc.

In the year 2012, Gupta-Kim-Lee[6]proposed theq-analogue of a new sequence of linear positive operators and modified the well known Baskakov-Szasz operators by Agrawal-Mohammad[1], which is given as

Dq_n(f;x) = [n]_q ∞ X

v=1 pq_n,v(x)

Z q/(1−qn)

0

q−vsq_n,v−1(t)f(t q−v)dqt+p q

n,0(x)f(0), (1)

where

pnq,v(x) = 

n+v−1

v ‹

qv(v−1)/2 xv (1+x)−q(n+v) snq,v(t) =Eq(−[n]qt)

([n]_qt)v [v]q!

In caseq=1, we get the original Baskakov-Szasz operators. It is observed that the above operators reproduce con-stant as well as linear functions.

Now we discuss Stancu type generalization[15]of the aboveq-Baskakov-Szasz operators. Very recently some results on Szasz-Mirakyan- Stancu operators are obtained by Maheshwari-Garg[12]. The Stancu variant is based on two

∗ Corresponding author.

E-mail address:sangeetavipin@rediffmail.com

parametersα,βsatisfying 0≤α≤β. So for 0<q<1 andx∈[0,∞), we proposeq-Baskakov-Szasz-Stancu operators as

M_nq_,α,β(f;x) = [n]_q ∞ X

v=1

p_nq_,v(x)

Z q/(1−qn₎

0

q−vs_nq_,v−1(t)f

q−v _[

n]_qt+α [n]_q+β

dqt+pnq,0(x)f

α [n]_q+β

, (2)

where Baskakov operatorspnq,vand Szasz operatorssnq,vare defined as above.

The literature ofq-calculus is described in different branches of Mathematics and Physics. Recently theq-analogues of the Baskakov operators and their Kantorovich and Durrmeyer variants have been studied by Aral-Gupta[2],[3]

and Gupta-Radu[8]respectively. The history ofq-calculus and its use to obtain some results can be seen in Ernst[4]

and Kac-Cheung[9]. Some of the notations and concepts inq-calculus are recalled here. Hence for a real number

q_∈(0, 1)andn_∈N

(x;q)_n= (1+x)n_q =:(1+x)(1+q x)...(1+qn−1x), n=1, 2, ...

=: 1 n=0.

Theq-binomial coefficients are given by

• n v ˜

q

= [n]q!

[v]_q![n−v]_q!, 0≤v≤n.

Theq-derivative of a functionf that isDqf is given by

(Dqf)(x) =

f(x)₋f(q x)

(1−q)x , x6=0

and(Dqf)(0) =f′(0)providedf′(0)exists.

Theq-Gamma integral is defined by Koornwinder[10]as

Γq(t) = Z ₁₋1_q

0

xl−1Eq(−q x)dqx, l>0,

where

Eq(x) = ∞ X

n=0

qn(n2−1) x

n

[n]_q!; Γq(l+1) = [l]qΓq(l), Γq(1) =1.

Furthermore,q-Beta function defined by Solo-Kac[14]is

Bq(l,m) =K(A,l) Z ∞/A

0

xl−1

(1+x)l_q+mdqx,

whereK(x,l) =_x1₊₁xl ₁₊1

x l

q(1+x)

1−l

q . AlsoK(x,l)is aq-constant, that is,K(q x,l) =K(x,l)and in case ofl to be

an integer, it is independent ofx. In particular for any integern∈N, we have

K(x,n) =qn(n2−1), _K(x, 0_{) =}1, _B

q(l,m) =

Γq(l)Γq(m)

Γq(l+m)

.

For details onq-Beta functions, we refer the readers to Sole-Kac[14].

Now forf ∈C[0,∞),q>0 andn∈N Aral-Gupta[2]introducedq-Baskakov operators are defined as

Bn,q(f;x) = ∞ X

v=0

•

n+v−1

v ˜

qv(v2−1) x

v

(1+x)n_q+vf

[v]_q

qv−1_[n]

q

=:

∞ X

v=0

p_nq_,v(x)f _[

v]_q qv−1_[n]

q

. (3)

2.

Moment estimations

In this section, we give moments and higher order moments for operators(3)and(1). Also we estimate the corre-sponding moments for new operators(2). Also we give higher order moments.

Lemma 2.1.

For the operators(3), the following equalities hold:

Bn,q(1;x) =1, Bn,q(t;x) =x,

Bn,q(t2;x) =x2+ x [n]_q



1+x q ‹

.

Proves are along the lines of Aral-Gupta[2].

Lemma 2.2.

For the operators(1), the following equalities hold:

D_nq(1;x) =1, D_nq(t;x) =x,

D_nq(t2;x) =x2+ x [n]_q



1+q+x q ‹

.

Proves can be seen in Gupta et al.[6].

Lemma 2.3.

For q∈(0, 1)and x∈[0,∞),we have

Dq_n((t₋x)2;x) =x(x+q[2]q) q[n]_q .

Lemma 2.4.

For q∈(0,_∞),n∈N and0< α < β,the equalities hold

M_nq_,α,β(1,x) =1, M_nq_,α,β(t,x) =[n]qx+α [n]_q+β ,

M_nq_,α,β(t2,x) =

[n]_q+q[n]2_q

q([n]_q+β)2x

2₊(1+q+2α)[n]q ([n]_q+β)2 x+

α [n]_q+β

2

.

Proof. ObviouslyM_nq_,α,β(1,x) =1. Now we proceed as

M_nq_,α,β(t,x) = [n]_q ∞ X

v=1

p_nq_,v(x)

Zq/(1−qn₎

0

q−vs_nq_,v−1(t)q−v _[

n]_qt+α [n]_q+β

dqt+p

q n,0(x)

 _α

n+β

‹

= _[

n]_q [n]_q+β

D_nq(t;x) + α [n]_q+βD

q n(1;x)

=[n]qx+α [n]_q+β ,

using Lemma2.2. In the similar way, we have

M_nq_,α,β(t2,x) = _[

n]_q [n]_q+β

2

D_nq(t2;x) + 2α[n]q ([n]_q+β)2D

q n(t;x) +

α [n]_q+β

2

= _[

n]_q [n]_q+β

2

x2+ x

[n]_q(1+q+ x q)

+ 2α[n]qx ([n]_q+β)2+

α [n]_q+β

2

=

[n]_q+q[n]2_q

q([n]_q+β)2x

2₊(1+q+2α)[n]q ([n]_q+β)2 x+

α [n]_q+β

2

.

Lemma 2.5.

For0< α < βand0<q<1,the central moments for our operators(2)are

T_nq_,m,α,β(x) =M_nq_,α,β((t−x)m,x)

= [n]_q ∞ X

v=1

p_nq_,v(x)

Z q/(1−qn₎

0

q−vs_nq_,v−1(t)q−m v

[n]_qt+α

[n]_q+β −x m

dqt+p q n,0(x)

 _α

n+β−x

‹m

.

Then we have the first three moments as

T_nq_,0,α,β(x) =1 (4)

T_nq_,1,α,β(x) = α−βx

[n]_q+β (5)

T_nq_,2,α,β(x) =

α [n]_q+β

2 +

–

(1+q+2α) [n]q ([n]_q+β)2−

2α [n]_q+β

™

x+

– q[n]2

q+ [n]q ([n]_q+β)2 −

2[n]_q [n]_q+β+1

™

x2. (6)

Proof. From Lemma2.2it is obvious thatT_nq_,0,α,β(x) =1. Further we get

T_nq_,1,α,β(x) =M_nq_,α,β((t−x),x) =M_nq_,α,β(t,x)−x M_nq_,α,β(1,x)

= α−βx [n]_q+β,

T_nq_,2,α,β(x) =M_nq_,α,β((t−x)2,x)

=M_nq_,α,β(t2,x)−2x M_nq_,α,β(t,x) +x2M_nq_,α,β(1,x)

=

α [n]_q+β

2 +

–

(1+q+2α) [n]q ([n]_q+β)2−

2α [n]_q+β

™

x+

–

q[n]2_q+ [n]_q

([n]_q+β)2 −

2[n]_q [n]_q+β +1

™ x2.

Remark 2.1.

For 0< α < βand 0<q<1, it can be proved easily from above lemma that

M_nq_,α,β((t−x)2,x)≤x(x+q[2]q) q([n]_q+β) .

Higher Order MomentsNow we consider for higher order moments for the operators(2).

Lemma 2.6 ([5]).

For0<q<1,we have for operators(3)

Bn,q(t3;x) =

1

[n]_qx+

1+2q q2

[n+1]_q

[n]2_q x

2₊ 1

q3

[n+1]_q[n+2]_q

[n]2_q x

3

Bn,q(t4;x) =

1

[n]3_qx+

1+3q+3q2 q3

[n+1]_q

[n]3_q x

2₊(1+3q+5q2+3q3)

q5_[2]

q

[n+1]_q[n+2]_q

[n]3_q x

3

+(1+3q+5q

2_+6q3_+5q4_+3q5_+q6₎

q6_[2]

q[3]q[4]q

[n+1]_q[n+2]_q[n+3]_q

[n]3_q x

Lemma 2.7 ([6]).

Let0<q<1,then for operators(1)we have

D_nq(t3;x) =[n+1]q[n+2]q q3_[n]2

q

x3+

1+2q q2

[n+1]_q

[n]2_q +

2+3q+q2 [n]_q

x2+1+2q+2q

2_+q3

[n]2_q x

D_nq(t4;x) =(1+3q+5q

2_+6q3_+5q4_+3q5_+q6₎

q6_[2]

q[3]q[4]q

[n+1]_q[n+2]_q[n+3]_q

[n]3_q x

4

+

1+3q+5q2+3q3

q5[2]_q +q(3+2q+q

2₎

_[

n+1]_q[n+2]_q [n]3_q x

3 +    

1+3q+3q2 q3

[n+1]_q

[n]_q +

(1+2q)(3+2q+q2) q

[n+1]_q

[n]3_q

+q

2_(3+4q+3q2_+q3₎

[n]2_q +

q(3+4q+3q2+q3) [n]3_q

    x2 + 1

[n]_q +

q(3+5q+6q2+5q3+3q4+q5) [n]3q

x.

Lemma 2.8.

For q∈(0, 1)and0< α < β,we have

M_nq_,α,β(t3;x) =

α [n]_q+β

3 +

_[ n]_q [n]_q+β

3_[

n+1]_q[n+2]_q q3_[n]2_q x

3₊

_[ n]_q [n]_q+β

3

×

2+3q+q2 [n]_q +

1+2q q2

[n+1]_q

[n]_q +

(q[n]_q+1)3α

q[n]2q

x2

+ _[

n]_q [n]_q+β

3

(1+2q+2q2+q3) +3(1+q)α [n]2_q +

3α2 [n]2_q

x

and

M_nq_,α,β(t4;x) =

(1+3q+5q2+6q3+5q4+3q5+q6) q6[2]_q[3]_q[4]_q

[n]_q[n+1]_q[n+2]_q[n+3]_q

([n]_q+β)4

x4+

₁

+3q+q2(5+4α[2]_q) +3q3+q6(3+2q+q2)[2]_q

q5_[2]

q

[n]_q[n+1]_q[n+2]_q

([n]_q+β)4

x3

+{1+3q+3q

2

q3

[n+1]_q[n]3_q

([n]_q+β)4 +

(1+2q)(4α+3q+2q2+q3) q2

[n+1]_q[n]_q

([n]_q+β)4

+ [n]2_q

([n]_q+β)4+

[n]_q

([n]_q+β)4}x

2₊• [n] 3

q

([n]_q+β)4+2α{2+3α+2α

2_{+ (4+3α)q}

+4q2+2q3} [n]2_q

([n]_q+β)4+q(3+5q+6q

2_+5q3_+3q4_+q5₎ [n]q ([n]_q+β)4

˜

x+

α [n]_q+β

4

.

Proof.

M_nq_,α,β(t3;x) = [n]_q ∞ X

v=1

p_nq_,v(x)

Z q/(1−qn₎

0

q−vs_nq_,v−1(t)q−3v

[n]_qt+α

[n]_q+β 3

dqt+p q n,0(x)

α [n]_q+β

3

= _[

n]_q [n]_q+β

3

Dq_n(t3;x) + 3α [n]2_q

([n]_q+β)3D

q n(t

2_{;x) +} 3α2[n]q ([n]_q+β)3D

q n(t;x) +

α [n]_q+β

3

=

[n]_q

[n]_q+β 3

([n+1]q[n+2]q q3_[n]2

q

x3+

1+2q q2

[n+1]_q

[n]2_q +

2+3q+q2 [n]_q

x2

+1+2q+2q

2_+q3

[n]2_q x) +

3α[n]2_q

([n]_q+β)3(x

2₊ x

[n]_q 

1+q+x q ‹

+ 3α

2_[n]

q ([n]_q+β)3x

+

α [n]_q+β

3

using Lemma2.2and Lemma2.6. Collecting the coefficients ofx,x2,x3we have the required result. Similarly we find that

M_nq_,α,β(t4;x) = [n]_q ∞ X

v=1

p_nq_,v(x)

Z q/(1−qn₎

0

q−vs_nq_,v−1(t)q−4v

[n]_qt+α

[n]_q+β 4

dqt+pnq,0(x)

α [n]_q+β

4

=

[n]_q

[n]_q+β 4

Dq_n(t4;x) + 4α [n]3_q

([n]_q+β)4D

q n(t

3_{;x) +} 6α 2_[n]2

q ([n]_q+β)4D

q n(t

2_;x)

+ 4α

3_[n]

q ([n]_q+β)4D

q n(t;x) +

α [n]_q+β

4

=

[n]_q

[n]_q+β 4_

(1+3q+5q2+6q3+5q4+3q5+q6) q6_[2]

q[3]q[4]q

[n+1]_q[n+2]_q[n+3]_q

[n]3_q x

4

+

1+3q+5q2+3q3

q5[2]_q +q(3+2q+q

2₎

_[

n+1]_q[n+2]_q [n]3_q x

3₊

•_1+3q+3q2

q3

[n+1]_q

[n]_q + (3+4q+3q

2_+q3₎

q2 [n]2_q +

q [n]3_q

+[n+1]q [n]3_q

×3+8q+5q2+2q3 q

˜ x2+

1

[n]_q +

q(3+5q+6q2+5q3+3q4+q5) [n]3_q

x

‹

+ 4α [n]3

q ([n]_q+β)4

•_[n+1_]

q[n+2]q q3_[n]2

q

x3+

1+2q q2

[n+1]_q

[n]2_q +

2+3q+q2 [n]_q

x2

+1+2q+2q

2_+q3

[n]2_q x ˜

+ 6α

2_[n]2

q ([n]_q+β)4

x2+ x

[n]_q 

1+q+x q

‹ + 4α

3_[n]

q ([n]_q+β)4x

+

α [n]_q+β

4

using Lemmas2.2and Lemma2.6. Rearranging the coefficients ofx,x2,x3,x4we get the required lemma.

Definition 2.1 (Peetre’sK-functional).

Let us consider the spaceCB[0,∞)of all the continuous and bounded functionsf that isf ∈CB[0,∞)and endowed

with the normkfk={|f(x)|:x∈[0,∞)}, then theK-functional

K2(f,δ) = inf

g∈W2 ∞

{kf −gk+δkg′′k},

whereδ >0 andW_∞2 ={g ∈CB[0,∞):g′,g′′∈CB[0,∞)}. Also there exists an absolute constantC >0 such that K2(f,δ)≤Cω2(f,

p

δ), where

ω2(f,

p

δ) = sup 0<h<pδ

sup

x∈[0,∞)|

f(x+2h)−2f(x+h) +f(x)|

is the second order modulus of smoothness off ∈CB[0,∞).

Also, forf ∈CB[0,∞)a usual modulus of continuity is given by

ω(f,δ) = sup 0<h<δ

sup

x∈[0,∞)|

f(x+h)−f(x)|.

Definition 2.2 (Rate of convergence).

LetBx2[0,∞)be the set of all functions f ∈[0,∞)satisfying the condition|f(x)| ≤M_f(1+x2),M_f is a constant

depending onf. We denote the subspace of all continuos functions byCx2[0,∞)belonging toB_x2[0,∞). Again, we

supposeC_x∗2[0,∞)be the subspace of all the functionsf ∈Cx2[0,∞), for which limx→∞1f+(xx)2is finite. The norm on C_x∗2[0,∞)is defined askfkx2=sup_x→∞|f

(x)_|

1+x2. We denote the usual modulus of continuity off on the closed interval

[0,a]fora>0, by

ωa(f,δ) = sup |t−x|≤δ

sup

x,t∈[0,a]|

f(t)−f(x)|.

Definition 2.3.

Here we define some classes of functions

If there exists some constantMf >0 corresponding to functionf, then we have

Cm[0,∞) =:

f ∈C[0,_∞):_|f(x)_|<Mf(1+xm);kfkm:= sup x∈[0,∞)

|f(x)|

1+xm

,

C_m∗[0,_∞) =:

§

f ∈Cm[0,∞): lim x→∞

|f(x)|

1+xm <∞,∀m∈N ª

,

3.

Direct estimates

In this section we give some direct theorems and asymptotic formula using our operators(2).

Theorem 3.1.

Let f ∈CB[0,∞)and0<q<1.Then for all x∈[0,∞)and n∈N,there exists an absolute constant C>0such that

|M_nq_,α,β(f;x)₋f(x)_{| ≤}Cω2 f,

v u

tx(x+q[2]q q([n]_q+β)

!

.

Proof. Letg∈W_∞2 andx,t∈∈[0,_∞), then by Taylor’s expansion

g(t) =g(x) + (t−x)g′(x) + Z t

x

(t−w)g′′(w)d w.

From Lemma2.5and Remark2.1

M_nq_,α,β(g(t);x)−g(x) =g′(x)M_nq_,α,β(t−x;x) +M_nq_,α,β Z t

x

(t−w)g′′(w)d w;x

.

UsingRt

x(t−w)g′′(w)d w≤(t−x)

2_kg′′_{k, we have}

|M_nq_,α,β(g(t);x)−g(x)|=M_nq_,α,β(t−x;x)g′(x) +M_nq_,α,β((t−x)2;x)kg′′k

≤x(x+q[2]q) q([n]_q+β)kg

′′_k.

Also from(2)

|M_nq_,α,β(f;x)|= [n]_q ∞ X

v=1

p_nq_,v(x)

Z q/(1−qn)

0

q−vs_nq_,v−1(t) f

q−v

_[

n]_qt+α [n]_q+β

dqt

+p_nq_,0(x) f

α [n]_q+β

≤ kfk. Hence we can have

|M_nq_,α,β(f(t);x)₋f(x)_{| ≤ |}M_nq_,α,β(f −g;x)₋(f −g)(x)_|+_|M_nq_,α,β(g;x)₋g(x)_|

≤ kf −gk+x(x+q[2]q) q([n]_q+β)kg

′′_k.

Taking infimum overallg∈W_∞2 and then from Peetre’sK-function, we get

|M_nq_,α,β(f(t);x)−f(x)| ≤Cω2 f,

v u

tx(x+q[2]q q([n]_q+β)

!

.

Hence the required theorem.

Theorem 3.2.

Let f ∈Cx2,q∈(0, 1)andω(a+1(f,δ))be its modulus of continuity on the finite interval[0,a+1]⊂[0,∞),∀a>0.Then for every n>2,we have

kM_nq_,α,β(f)−f)k ≤6Mfa(1+a

2_)(a+2)

q([n]_q+β) +2ω(f, v u

ta(a+q[2]q) q([n]_q+β).

Proof. Forx∈[0,a]andt>a+1, ast−x>1, we have

|f(t)−f(x)| ≤Mf(2+x2+t2) ≤Mf(2+3x2+ (t−x)2)

≤6Mf(1+a2)(t−x)2. (7)

Forx∈[0,a]andt≤a+1, we have

|f(t)₋f(x)_{| ≤}ωa+1(f,|t−x|)≤



1+|t−x| δ

‹

ωa+1(f,δ), δ >0 (8)

From(7)and(8), forx∈[0,a]andt≥0, we have

|f(t)−f(x)| ≤6Mf(1+a2)(t−x)2+ 

1+|t−x| δ

‹

ωa+1(f,δ). (9)

Hence

|M_nq_,α,β(f;x)−f(x)| ≤M_nq_,α,β(|f(t)−f(x)|;x)

≤6Mf(1+a2)M q

n,α,β((t−x)2;x) +ωa+1(f,δ)



1+1 δM

q

n,α,β((t−x)2;x)1 /2‹_.

Therefor by using Schwarz inequality and Remark 1,

|M_nq_,α,β(f;x)−f(x)| ≤6Mf(1+a

2_)(x(x+q[2]

q)

q([n]_q+β) +ωa+1(f,δ) 1+

1

δ v u

tx(x+q[2]q) q([n]_q+β)

!

≤6Mfa(1+a

2_)(a+2)

q([n]_q+β) +ωa+1(f,δ) 1+

1

δ v u

ta(a+q[2]q) q([n]_q+β)

!

.

Takingδ= r

a(a+q[2]_q)

q([n]q+β), we get the required assertion.

Theorem 3.3.

Let qn ∈(0, 1),then the sequence Mnqn,α,β(f)converges to f uniformly on[0,A],for each f ∈C2∗[0,∞)if and only if limn→∞qn=1.

Its proof is obvious as q,qn∈(0, 1).

Theorem 3.4.

If qn∈(0, 1),qn →1and qnn →1as n→ ∞for any f ∈C2∗[0,∞)such that f′,f′′∈C2∗[0,∞),the following equality

holds

lim

n→∞[n]qn(M

qn

n,α,β(f;x)−f(x)) = (α−βx)f′(x) +

1

2x(x+2)f

′′_(x).

Proof. Assume thatf,f′,f′′∈C₂∗[0,_∞)for allx∈[0,_∞). Therefore by Taylor’s formula

f(t) =f(x) +f′(x)(t−x) +1

2f

′′_(x)(t−x)2_{+r(t;x)(t−x)}2_{, (10)}

wherer(t,x)is the Peano form of the remainder andr(t,x)_∈C₂∗[0,_∞). Also limt→xr(t,x) =0. Now applying the

operatorsMqn

n,α,βin(10), we get [n]_qn(Mqn

n,α,β(f;x)−f(x)) =f′(x)[n]qnM

qn

n,α,β(t−x;x) +

1 2f

′′_(x)[n] qnM

qn

n,α,β((t−x)

2_;x)

+[n]_qnMqn

n,α,β(r(t;x)(t−x)

2_;x).

By the Cauchy-Schwarz inequality, we have

Mqn

n,α,β(r(t;x)(t−x)

2_;x)≤r_Mqn

n,α,β(r2(t;x);x) r

Mqn

n,α,β(((t−x)2;x)) (11)

Asr2(x;x) =0 andr2(.;x)∈C₂∗[0,_∞), it follows that_∀x∈[0,A]

l i mn→∞Mnqn,α,β(r

2_{(t;x);x) =r}2_{(x;x) =0 (12)}

uniformly and so RHS of(11)becomes zero. Therefore we get

l i mn→∞[n]qn(M

qn

n,α,β(f;x)−f(x)) =l i mn→∞f′(x)[n]qnM

qn

n,α,β(t−x;x) +

1 2f

′′_(x)[n] qnM

qn

n,α,β((t−x)

2_;x)

+[n]_qnMqn

n,α,β(r(t;x)(t−x)

2_;x)

= (α−βx)f′(x) +1

2x(x+2)f

′′_(x)

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