EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 11, No. 3, 2018, 575-579
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

A General Family of the Srivastava-Gupta Operators
Preserving Linear Functions

Vijay Gupta1, H. M. Srivastava2,3,∗

1 Department of Mathematics, Netaji Subhas Institute of Technology, Sector 3, Dwarka,
New Delhi 110078, India
2 Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia
V8W 3R4, Canada
3 Department of Medical Research, China Medical University Hospital,
China Medical University, Taichung 40402, Taiwan, Republic of China

Abstract. The general sequence of positive linear operators containing some well-known operators
as special cases were introduced in the earlier work by Srivastava and Gupta [9], which reproduce
only the constant functions. In the present sequel, we provide a general sequence of operators
which preserve not only the constant functions, but also linear functions.

2010 Mathematics Subject Classifications: Primary 41A35, 41A36; Secondary 33C05, 33C15.
Key Words and Phrases: Srivastava-Gupta operators; Phillips operators; Central moments;
Generalized hypergeometric function; Kummer’s confluent hypergeometric function; Gauss hyper-
geometric function.

1. Introduction and Preliminaries

In the year 2003, Srivastava and Gupta [9] introduced a general sequence of positive
linear operators defined by

Vn,c(f, x) = n
∞∑
k=1

pn,k(x, c)

∫ ∞
0

pn+c,k−1(t, c)f(t)dt

+ pn,0(x, c)f(0) (n ∈ N := {1, 2, 3, · · · }), (1)

where

pn,k(x, c) =
(−x)k

k!
φ(k)n,c(x).

The following special cases of the operator defined by (1) are worthy of mention here:

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v11i3.3314

Email addresses: harimsri@math.uvic.ca (H. M. Srivastava),
vijaygupta2001@hotmail.com (V. Gupta)

http://www.ejpam.com 575 c© 2018 EJPAM All rights reserved.



V. Gupta, H. M. Srivastava / Eur. J. Pure Appl. Math, 11 (3) (2018), 575-579 576

• If c = 0 and φn,c(x) = e−nx, then we get

pn,k(x, 0) = e−nx
(nx)k

k!
;

• If c ∈ N and φn,c(x) = (1 + cx)−
n
c , then we obtain

pn,k(x, c) =

(
n
c

)
k

k!

(cx)k

(1 + cx)
n
c
+k
,

where, and in what follows, (λ)n denotes the Pochhammer symbol (or the shifted
factorial) defined, for λ ∈ C, by

(λ)0 = 1 and (λ)n = λ(λ+ 1) · · · (λ+ n− 1) (n ∈ N);

• If c = −1 and φn,c(x) = (1− x)n, then

pn,k(x,−1) =

(
n

k

)
xk(1− x)n−k.

Here, for the last case when c = −1, we have x ∈ [0, 1] whereas, for c ∈ N ∪ {0}, we
have x ∈ [0,∞). These operators were further discussed by Ispir and Yüksel [6], Atakuta
and Büyükyazici [2], Deo [3], Gupta and Tachev [5], Kumar [7] and Yadav [13], and other
authors (see also the closely-related works by Acar et al. [1], Maheshwari [8], Srivastava
and Gupta [10], Srivastava and Zeng [11], and Verma and Agrawal [12]).

The moments of the above-defined operator Vn,c(f, x) of order r (r ∈ N) are given, in
terms of the Gauss hypergeometric function 2F1 and Kummer’s confluent hypergeometric
function 1F1, as follows:

Vn,c(er, x) =



r! · (nx)

(n− c)(n− 2c) · · · (n− rc)
· 2F1

(n
c

+ 1, 1− r; 2;−cx
)

(c ∈ N ∪ {−1})

r! · (nx)

nr
1F1(1− r; 2;−nx) (c = 0),

(2)

where `Fm

(
α1, · · · , α`;β1, · · · , βm;x

)
denotes the generalized hypergeometric function

with ` numerators parameters α1, · · · , α` and m denominator parameters β1, · · · , βm de-
fined by

`Fm

(
α1, · · · , α`;β1, · · · , βm;x

)
=

∞∑
n=0

∏̀
j=1

(αj)n

m∏
j=1

(βj)n

xn

n!
(3)



V. Gupta, H. M. Srivastava / Eur. J. Pure Appl. Math, 11 (3) (2018), 575-579 577(
`,m ∈ N0 := N ∪ {0}; ` 5 m+ 1; ` 5 m and |x| <∞;

` = m+ 1 and |x| < 1; ` = m+ 1, |x| = 1 and <(ω) > 0
)
,

where

ω :=

m∑
j=1

βj −
∑̀
j=1

αj

(
αj ∈ C (j = 1, · · · , `); βj ∈ C \ Z−0 (j = 1, · · · , m)

)
.

The operators Vn,c(f, x) are known to preserve the constant functions only except for
the case c = 0. We observe that in (1) the suffix n in the basis function pn,k(x, c) has a
difference of c under summation and integration. But, if we have a difference of 2c in place
of c under summation and integration, we may get the modified operators which preserve
the constant functions as well as linear functions.

2. Modified Operators Preserving Linear Functions

Here, in this section, we introduce a modification of the operators Vn,c(f, x) which do
preserve linear functions as well. For m an integer and c ∈ N0 := N ∪ {0}, we define

Gn,c(f, x) = [n+ (m+ 1)c]
∞∑
k=1

pn+mc,k(x, c)

·
∫ ∞
0

pn+(m+2)c,k−1(t, c)f(t)dt+ pn+mc,0(x, c)f(0), (4)

where pn,k(x, c) is defined by (1). The above-mentioned two cases lead to the familiar
Phillips operators and the genuine Baskakov-Durrmeyer type operators for c = 0 and
c ∈ N, respectively. In the special case when m = 0 and c = 1, the operators Gn,c(f, x)
were considered by Finta [4]. who also estimated some converse results. Moreover, for
c = −1, the operators Gn,c(f, x) take the following form:

Gn,−1(f, x) = (n−m− 1)
n−m−1∑
k=1

pn−m,k(x,−1)

·
∫ 1

0
pn−m−2,k−1(t,−1)f(t)dt

+ pn−m,0(x,−1)f(0) + pn−m,n−m(x,−1)f(1), (5)

The moments of the operators Gn,c(f, x) of order r (r ∈ N) are given, in terms of the
Gauss hypergeometric function 2F1, as follows:



REFERENCES 578

Gn,c(er, x) =



r! · xΓ
(n
c
− r +m+ 1

)
cr−1 Γ

(n
c

+m
)

· 2F1

(n
c

+m+ 1, 1− r; 2;−cx
)

(c ∈ N ∪ {−1})

r! · (nx)

nr
1F1(1− r; 2;−nx) (c = 0).

(6)

Finally, by applying this last result (6), it can easily be verified that the operators
Gn,c(f, x) preserve not only the constant functions, but also linear functions.

3. Concluding Remarks and Observations

The present investigation was motivated essentially by the fact that the widely-studied
Srivastava-Gupta operator Vn,c(f, x) preserves only the constant functions, but not linear
functions. Here, in this paper, we have successfully provided a general sequence Gn,c(f, x)
of positive linear operators which preserve not only the constant functions, but also linear
functions. We have also considered several recent developments on the subject of positive
linear operators.

References

[1] T. Acar, L. N. Mishra and V. N. Mishra, Simultaneous approximation for generalized
Srivastava-Gupta operators, J. Funct. Spaces 2015 (2015), Article ID 936308, 1–11.

[2] Ç. Atakuta and İ. Büyükyazici, Rate of convergence for modified SrivastavaGupta
type operators, Turkish J. Math. Comput. Sci. 7 (2017), 10–15.

[3] N. Deo, Faster rate of convergence on Srivastava-Gupta operators, Appl. Math. Com-
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[4] Z. Finta, On converse approximation theorems, J. Math. Anal. Appl. 312 (2005),
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[5] V. Gupta and G. Tachev, Approximation with Positive Linear Operators and Linear
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[6] N. Ispir and İ. Yüksel, On the Bézier variant of Srivastava-Gupta operators, Appl.
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[7] A. Kumar, Approximation by Stancu type generalized Srivastava-Gupta operators
based on certain parameter, Khayaam J. Math. 3 (2017), 147–159.



REFERENCES 579

[8] P. Maheshwari, On modified Srivastava-Gupta operators, Filomat 29 (2015), 1173–
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[9] H. M. Srivastava and V. Gupta, A certain family of summation-integral type opera-
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[10] H. M. Srivastava and V. Gupta, Rate of convergence for the Bézier variant of the
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[11] H. M. Srivastava and X.-M. Zeng, Approximation by means of the Szász-Bézier inte-
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[12] D. K. Verma and P. N. Agrawal, Convergence in simultaneous approximation for
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[13] R. Yadav, Approximation by modified SrivastavaGupta operators, Appl. Math. Com-
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