Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 5 Issue 3 (2013), Pages 18-26
HYPERGEOMETRIC REPRESENTATION FOR
BASKAKOV-DURRMEYER-STANCU TYPE OPERATORS
(COMMUNICATED BY HÜSEYIN BOR)
VISHNU NARAYAN MISHRA, HUZOOR H. KHAN, KEJAL KHATRI AND LAKSHMI
NARAYAN MISHRA
Abstract. In the present paper, we introduce and study the mixed summationintegral type operators having Baskakov and Beta basis functions in summation and integration, respectively. First, we estimate moments of these operators using hypergeometric series. Next, we obtain an error estimation in
simultaneous approximation for Baskakov-Durrmeyer-Stancu operators.
1. Introduction
Khan [4] and Mishra [5] have proved some results dealing with the degree of approximation of functions in Lp - spaces using different types of operators. BaskakovDurrmeyer operators were first considered by Sahai-Prasad [8] in 1985 . Sinha et
al. [9] improved and corrected the results of [8]. In 2005, Finta [1], introduced a
new type of Baskakov-Durrmeyer operator by taking the weight function of Beta
operators on L[0, ∞) as
Dn (f, x) =
∞
X
k=1
pn,k (x)
Z
∞
bn,k (t)f (t)dt + pn,0 (x)f (0),
(1)
0
k
k
(n+1)(n+2)k
x
t
k
where pn,k (x) = (n)
k! (1+x)n+k and bn,k (t) =
k!
(1+t)n+k+2 .
Wafi and Khatoon [11] have proved inverse theorem for generalized Baskakov operators. Recently, Gupta and Yadav [3] introduced the Baskakov -Beta- Stancu
operator and invetigated like asymptotic formula, moments of these operators using hypergeometric series and errors estimation in simultaneous approximation. we
0
2000 Mathematics Subject Classification: 41A25, 41A35.
Keywords and phrases. Hypergeometric series, Baskakov-Durrmeyer-Stancu operators, Estimation
of errors.
c 2013 Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted December 19, 2012. Published June 28, 2013.
18
HYPERGEOMETRIC REPRESENTATION FOR BASKAKOV-DURRMEYER-STANCU TYPE OPERATORS
19
write the operators (1) as
Dn (f, x) =
=
(n + 1)
(n + 1)
∞
X
(n)k
k=1
Z ∞
0
xk
k! (1 + x)n+k
Z
∞
0
tk
f (0)
(n + 2)k
f (t)dt +
k!
(1 + t)n+k+2
(1 + x)n
∞
X
(xt)k
f (0)
(n)k (n + 2)k
f (t)(1 + x)
dt +
.
[(1 + x)(1 + t)]n+2
(k!)2
[(1 + x)(1 + t)]k
(1 + x)n
2
k=1
P∞
(a)k (b)k k
By hypergeometric series 2 F1 (a, b; c; x) =
k=0 (c)k k! x and the Pochhammer
symbol (n)k is
(n)k = n(n + 1)(n + 2)(n + 3)....(n + k − 1), using the equality (1)k = k!, we can
write
Z ∞
f (0)
xt
f (t)(1 + x)2
− 1 dt +
.
Dn (f, x) = (n + 1)
2 F1 n, n + 2; 1;
n+2
[(1 + x)(1 + t)]
(1 + x)(1 + t)
(1 + x)n
0
Now using 2 F1 (a, b; c; x) = 2 F1 (b, a; c; x) and applying Pfaff- Kummer transformation
x
−a
2 F1 (a, b; c; x) = (1 − x)
2 F1 a, c − b; c;
x−1
we have
Dn (f, x) = (n+1)
Z
0
∞

 2 F1
f (t)(1 + x)2 

−xt
n + 2, 1 − n; 1; 1+x+t
(1 + x +
t)n+2
−


1
 dt+ f (0) .

n+2
[(1 + x)(1 + t)]
(1 + x)n
(2)
This is the form of the operators (1) in terms of hypergeometric functions.
Verma et al. [10] considered Baskakov -Durrmeyer- Stancu operators and studied
some approximation properties of these operators. Very recently, Mishra and Patel
[6] introduced a simple Stancu generalization of q-analogue of well known Durrmeyer operators. We first estimate moments of q− Durrmeyer-Stancu operators.
They also established the rate of convergence as well as Voronovskaja type asymptotic formula for q− Durrmeyer-Stancu operators. Here, we introduce Baskakov
-Durrmeyer- Stancu operators in terms of hypergeometric functions, for 0 ≤ α ≤ β
as
−xt
2 F1 n + 2, 1 − n; 1; 1+x+t
Z ∞ 1
nt + α
(1 + x)2
dt
−
Dn,α,β (f, x) = (n + 1)
f
n+β
(1 + x + t)n+2
[(1 + x)(1 + t)]n+2
0
+
f (0)
.
(1 + x)n
(3)
For α = β = 0 the operators (3) reduces to the operators (1).
we know that
Z ∞
Z ∞
∞
∞
X
X
1
bn,k (t) = n + 1,
bn,k (t)dt = 1.
,
pn,k (x) = 1,
pn,k (x)dx =
n−1
0
0
k=0
k=1
We take
Cν [0, ∞) = {f ∈ C[0, ∞) : f (t) = O(t)ν , ν > 0}.
20
VISHNU NARAYAN MISHRA, HUZOOR H. KHAN, KEJAL KHATRI AND LAKSHMI NARAYAN MISHRA
The operators Dn,α,β (f, x) are well defined for f ∈ C[0, ∞). The behavior of these
operators is very similar to the operators recently introduced in [7] by Mishra et al.
In the present note, first, we estimate moments of Baskakov -Durrmeyer- Stancu
operators with the help of hypergeometric series. Next, we give an error estimation
in simultaneous approximation for the operators (3)
2. Auxiliary results
In the sequel we shall need several lemmas.
Lemma 2.1. For n > 0 and s > −1, we have
x
Γ(n − s + 1)Γ(s + 1)
(1 + x)s 2 F1 1 − n, −s; 1;
− (1 + x)−n .
Dn (ts , x) =
Γ(n + 1)
1+x
(4)
Moreover,
Dn (ts , x) =
(n + s − 1)!(n − s)! s s(s − 1)(n + s − 2)!(n − s)! s−1
x +
x
+ O(n−2 ).
n!(n − 1)!
n!(n − 1)!
(5)
Proof. Taking f (t) = ts , t = (1 + x)u and using Pfaff-Kummer transformation the
right-hand side of (2), we get
Z ∞
∞
X
(1 + x)s+3 us
(1 − n)k (n + 2)k (−x(1 + x)u)k
s
du
Dn (t , x) = (n + 1)
[(1 + x)(1 + u)]n+2
(k!)2
[(1 + x)(1 + u)]k
0
k=0
Γ(n − s + 1)Γ(s + 1)
(1 + x)−n = Q1 + Q2 (say).
+
Γ(n + 1)
Q1
=
=
(n + 1)
(n + 1)
∞
X
(n + 2)k (1 − n)k
k=0
∞
X
k=0
=
(n + 1)
(k!)2
k
s−n+1
(−x) (1 + x)
Z
0
∞
us+k
du
(1 + u)n+k+2
(n + 2)k (1 − n)k
(−x)k (1 + x)s−n+1 B(s + k + 1, n − s + 1)
(k!)2
∞
X
(n + 2)k (1 − n)k
k=0
(k!)2
(−x)k (1 + x)s−n+1
Γ(s + k + 1)Γ(n − s + 1)
.
Γ(n + k + 2)
Using Γ(n + k + 2) = Γ(n + 2)(n + 2)k , we have
Q1
= (n + 1)
∞
X
(n + 2)k (1 − n)k
k=0
= (1 + x)s−n+1
(k!)2
(−x)k (1 + x)s−n+1
Γ(s + 1)(s + 1)k Γ(n − s + 1)
Γ(n + 2)(n + 2)k
∞
Γ(s + 1)Γ(n − s + 1) X (s + 1)k (1 − n)k
(−x)k
Γ(n + 1)
(k!)2
k=0
Γ(s + 1)Γ(n − s + 1)
= (1 + x)s−n+1
2 F1 (1 − n, 1 + s; 1; −x).
Γ(n + 1)
x
Using 2 F1 (a, b; c; x) = (1 − x)−a 2 F1 a, c − b; c; x−1
, we have
x
Γ(n − s + 1)Γ(s + 1)
s
.
(1 + x) 2 F1 1 − n, −s; 1;
Q1 =
Γ(n + 1)
1+x
HYPERGEOMETRIC REPRESENTATION FOR BASKAKOV-DURRMEYER-STANCU TYPE OPERATORS
21
Combining Q1 and Q2 , we get
x
Γ(n − s + 1)Γ(s + 1)
s
−n
.
(1 + x) 2 F1 1 − n, −s; 1;
− (1 + x)
Dn (t , x) =
Γ(n + 1)
1+x
s
The other consequence (5) follows from the above equation by writting the expansion of hypergeometric series.
Lemma 2.2. For 0 ≤ α ≤ β and m > 0, we have
Dn,α,β (ts , x) =
+
+
+
ns
(n + s − 1)!(n − s)!
s
(n + β)
n!(n − 1)!
(n + s − 2)!(n − s)!
ns
ns−1 (n + s − 2)!(n − s + 1)!
s−1
s(s − 1)
+ sα
x
(n + β)s
n!(n − 1)!
(n + β)s
n!(n − 1)!
s−1
n
(n
+
s
−
3)!(n
−
s
+
1)!
xs−2 s(s − 1)(s − 2)α
(n + β)s
n!(n − 1)!
s(s − 1) 2 ns−2 (n + s − 3)!(n − s + 2)!
+ O(n−m ).
α
2
(n + β)s
n!(n − 1)!
xs
Proof. Using binomial theorem, the relation between operators (2) and (3) can be
defined as
s
s
Z ∞
∞
X
α
nt + α
Dn,α,β (ts , x) =
pn,k (x)
bn,k (t)
dt + (1 + x)−n
n+β
n+β
0
k=1
s
Z
∞
∞
∞
X
X
s (nt)j αs−j
α
−n
=
pn,k (x)
bn,k (t)
dt + (1 + x)
j (n + β)s
n+β
0
j=0
k=1
s
∞ j s−j X
α
s n α
j
−n
−n
D
(t
,
x)
−
(1
+
x)
0
+
(1
+
x)
.
=
n
n+β
j (n + β)s
j=0
Using (5), we get Lemma (2.2).
S
Lemma 2.3. [2] For m ∈ N {0}, if
Un,m (x) =
∞
X
k=0
pn,k (x)
m
k
−x
,
n
then Un,0 (x) = 1, Un,1 (x) = 0 and we have the recurrence relation:
′
nUn,m+1 (x) = x(1 + x) Un,m
(x) + mUn,m−1 (x) .
Consequently, Un,m (x) = O n−[(m+1)/2] , where [m] is integral part of m.
S
Lemma 2.4. [10] For m ∈ N {0}, if
µn,m (x)
=
=
Dn,α,β ((t − x)m , x)
m
m
Z ∞
∞
X
nt + α
α
bn,k (t)
pn,k (x)
−x
−x
dt + pn,0 (x)
n+β
n+β
0
k=1
then
µn,0 (x) = 1, µn,1 (x) =
α − βx
n+β
22
VISHNU NARAYAN MISHRA, HUZOOR H. KHAN, KEJAL KHATRI AND LAKSHMI NARAYAN MISHRA
and for n>m we have recurrence relation:
n+β
µn,m+1 (x) = x(1 + x) µ′n,m (x) + mµn,m−1 (x)
(n − m)
n
n+β
α
+ (m + nx) +
− x (n − 2m) µn,m (x)
n
n+β
α
n+β
α
− 1 mµn,m−1 (x).
−x
−x
−
n+β
n+β
n
From the recurrence relation, it easily verified that for all x ∈ [0, ∞), we have
µn,m (x) = O(n−[(m+1)/2] ).
Lemma 2.5. [2] There exist the polynomials qi,j,s (x) on [0, ∞), independent of n
and k such that
X
ds
xs (1 + x)s s pn,k (x) =
ni (k − nx)j qi,j,s (x)pn,k (x).
dx
2i+j≤s
i,j≥0
3. Main result
(s)
In this section, we give an estimate of the degree of approximation by Dn,α,β (f (t), x)
for smooth functions.
Theorem 3.1. Let f ∈ Cν [0, ∞) for some ν > 0, m > 0 and s ≤ q ≤ s + 2. If f (q)
exists and is continuous on (a − η, b + η) ⊂ (0, ∞), η > 0, then for sufficiently large
n
(s)
||Dn,α,β (f, x)−f (s) (x)||C[a,b] ≤ C1 n−1
q
X
||f i ||C[a,b] +C2 n1/2 ω(f (q) , n1/2 )+O(n−m ),
i=s
(6)
where C1 , C2 are constants independent of f and n, ω(f, δ) is the modulus of
continuity of f on (a − η, b + η) and ||.||C[a,b] denotes the sup-norm on [a, b].
Proof. By the Taylor’s, expansion, we have
f (t) =
q
X
f (i) (x)
i=0
i!
(t − x)i +
f (q) (x) − f (q) (ξ)
(t − x)q χ(t) + h(t, x)(1 − χ(t))
q!
where ξ lies between t and x, and χ(t) is the characteristic function on interval
(a − η, b + η).
For t ∈ (a − η, b + η) and x ∈ [a, b], we have
f (t) =
q
X
f (i) (x)
i=0
i!
(t − x)i +
f (q) (x) − f (q) (ξ)
(t − ξ)q .
q!
For t ∈ [0, ∞)\(a − η, b + η) and x ∈ [a, b], we define
h(t, x) = f (t) −
q
X
f (i) (x)
i=0
i!
(t − x)i .
HYPERGEOMETRIC REPRESENTATION FOR BASKAKOV-DURRMEYER-STANCU TYPE OPERATORS
23
Now,
(s)
Dn,α,β (f, x)
−f
(s)
(x) =
X
q
i=0
(q)
f (x) − f (q) (ξ)
f (i) (x) (s)
(s)
i
(s)
Dn,α,β ((t − x) , x) − f (x) + Dn,α,β
i!
q!
(s)
(t − x)q χ(t), x + Dn,α,β (h(t, x)(1 − χ(t)), x) = F1 + F2 + F3 .
Using Lemma 2.2, we get
F1
=
+
+
+
q
i X
f (i) (x) X i
ds
nj
(n + j − 1)!(n − j)!
(−x)i−j s xj
j
i! j=0 j
dx
(n + β)
n!(n − 1)!
i=0
nj
(n + j − 2)!(n − j)!
nj−1 (n + j − 2)!(n − j + 1)!
xj−1 j(j − 1)
+
jα
(n + β)j
n!(n − 1)!
(n + β)j
n!(n − 1)!
j−1
(n + j − 3)!(n − j + 1)!
n
xj−2 j(j − 1)(j − 2)α
(n + β)j
n!(n − 1)!
j−2
j(j − 1) 2 n
(n + j − 3)!(n − j + 2)!
−m
+ O(n ) − f (s) (x).
α
2
(n + β)j
n!(n − 1)!
Hence
||F1 ||C[a,b] ≤ C1 n−1
q
X
||f i ||C[a,b] + O(n−m ), unif ormly on [a, b].
i=s
Next, we estimate F2 as
|F2 | ≤
+
≤
∞
X
q
(q)
f (x) − f (q) (ξ) nt + α
χ(t) dt
bn,k (t) −
x
n + β
q!
0
k=1
q
α
(n + s − 1)!
(1 + x)−n−s − x χ(t)
(n − 1)!
n+β


nt+α
q
Z
−
x
∞
∞
(s)
n+β
 nt + α
X

ω(f , δ)
(s)

dt
bn,k (t) 
|pn,k (x)|
1
+
−
x


q!
δ
n+β
0
(s)
|pn,k (x)|
Z
∞
k=1
+
≤
+

α
q −
x
n+β
 α

(n + s − 1)!
−n−s 

(1 + x)
 n + β − x
1 +
(n − 1)!
δ

q
q+1 ∞
Z ∞
nt + α
nt + α
ω(f (s) , δ) X (s)
dt
bn,k (t) − x + δ −1 − x
|pn,k (x)|
q!
n+β
n+β
0
k=1
q
q+1 (n + s − 1)!
−1 α
−n−s α
(1 + x)
− x + δ − x
(n − 1)!
n+β
n+β
24
VISHNU NARAYAN MISHRA, HUZOOR H. KHAN, KEJAL KHATRI AND LAKSHMI NARAYAN MISHRA
Now, on application of Schwarz inequality for integration and then for summation,
we get
q
Z ∞
12
Z ∞
∞
∞
X
X
nt + α
j
j
− x ≤
pn,k (x)|k − nx|
bn,k (t)dt
pn,k (x)|k − nx|
bn,k (t)
n+β
0
0
k=1
k=1
Z ∞
2q 12
nt + α
bn,k (t)
−x
dt
n+β
0
X
12 X
∞
∞
≤
pn,k (x)(k − nx)2j
pn,k (x)
k=1
k=1
Z
∞
0
2s 21
nt + α
bn,k (t)
−x
dt .
n+β
Using Lemma 2.3, we get
∞
X
2j
pn,k (x)(k − nx)
= n
k=1
2j
X
∞
2j
−n
pn,k (x)(k/n − x) ) − (1 + x)
2j
(−x)
k=0
= n2j O(n−j ) + O(n−r ) (f or any r > 0).
= O(nj ).
(7)
Similarly, using Lemma 2.4, we get
2q
Z ∞
∞
X
nt + α
−x
dt
pn,k (x)
bn,k (t)
n+β
0
=
O(nq ) − (1 + x)−n (−x)2q
=
O(n−q ) + O(n−r ) (f or any r > 0).
k=1
= O(n−s ).
Hence
Z
∞
X
j
pn,k (x)|k − nx|
k=1
∞
0
(8)
q
nt + α
bn,k (t)
− x = O(nj/2 )O(n−q/2 ) = O(n(j−q)/2 ), unif ormly on [a, b]. (9)
n+β
Therefore, by Lemma 2.5 and (9), we get
q
Z ∞
Z ∞
∞
∞
X
X
X ni qi,j,s (x)
nt + α
(s)
j
p
(x)|k
−
nx|
|pn,k (x)|
bn,k (t)
− x ≤
bn,k (t)
n,k
n+β
xs (1 + x)s
0
0
2i+j≤s
k=1
k=1
i,j≥0
q
nt + α
− x dt
n+β
q !
Z ∞
∞
X
nt
+
α
pn,k (x)|k − nx|j
bn,k (t)
− x dt
n+β
0
≤K
X
2i+j≤s
i,j≥0
=K
X
2i+j≤s
i,j≥0
ni
k=1
ni O(n(j−q)/2 ) = O(n(s−q)/2 ), unif ormly on [a, b], (10)
HYPERGEOMETRIC REPRESENTATION FOR BASKAKOV-DURRMEYER-STANCU TYPE OPERATORS
25
q
(x)
−1/2
where K = sup 2i+j≤s supx∈[a,b] xsi,j,s
and making use of
(1+x)s . Choosing δ = n
i,j≥0
(10), we get for any m > 0,
||F2 ||C[a,b] ≤
ω(f (q) , n−1/2 )
[O(n(s−q)/2 ) + n1/2 O(n(s−q−1)/2 ) + O(n−m )] ≤ C2 (n−(q−s)/2 )ω(f (q) , n−1/2 ).
q!
For t ∈ [0, ∞)\(a − η, b + η), we can choose δ such that |t − x| ≥ δ for all x ∈ [a, b].
Thus by Lemma 2.5, we get
Z
∞
X ni qi,j,s (x) X
j
pn,k (x)|k − nx|
bn,k (t)|h(t, x)|dt
|F3 | ≤
xs (1 + x)s
|t−x|≥δ
2i+j≤s
i,j≥0
k=1
(n + s − 1)!
α
−n−s +
(1 + x)
, x .
h
(n − 1)!
n+β
We can find a constant M1 such that
β
nt + α
− x f or |t − x| ≥ δ,
|h(t, x)| ≤ M1 n+β
where β ≥ (ν, q). Hence applying Schwarz inequality, (7) and (8), it is easy to see
that F3 = O(n−r ) for any r > 0 uniformly on [a, b]. Combining the estimates of
F1 , F2 and F3 , the required result follows.
Acknowledgement. Authors are grateful to the anonymous learned referees
for their valuable suggestions and comments for improvement of this paper. Authors mention their sense of gratitude to Professor Hüseyin Bor, editorial board of
BMAA, Professor Faton Z. Merovci, secretary of BMAA for supporting this work.
Kejal Khatri, one of the author, is thankful to the financial support of ”Ministry
of Human Resource Development”, New Delhi, Govt. of India, for encouragement
to this work.
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Indian Institute of Technology, Roorkee, Uttarakhand, India, (2007).
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26
VISHNU NARAYAN MISHRA, HUZOOR H. KHAN, KEJAL KHATRI AND LAKSHMI NARAYAN MISHRA
Vishnu Narayan Mishra and Kejal Khatri
Department of Applied Mathematics and Humanities, Sardar Vallabhbhai National Institute of Technology, Ichchhanath Mahadev Road
Surat (Gujarat), India.
E-mail address: vishnunarayanmishra@gmail.com and kejal0909@gmil.com
Huzoor H. Khan
Department of Mathematics, Aligarh Muslim University
Aligarh, India.
E-mail address: huzoorkhan@yahoo.com
Lakshmi Narayan Mishra
L. 1627 Awadh Puri Colony Beniganj, Phase -III, Opp. I.T.I., Ayodhya Main Road
Faizabad, (Uttar Pradesh).
E-mail address: lakshminarayanmishra04@gmail.com