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<h1>A Summation-Integral type modification of Szász-Mirakjan-Stancu operators</h1>
<p class="authors">
<span class="author">Vishnu Narayan Mishra\(^\ast \), Rajiv B. Gandhi\(^{\S }\) Ram N. Mohapatra\(^\bullet \)</span>
</p>
<p class="date">January 12, 2016.</p>
</div>
<p>\(^\ast , ^{\S }\)Applied Mathematics &amp; Humanities Department, Sardar Vallabhbhai National Institute of Technology, Ichchhanath Mahadev, Dumas Road, Surat - 395 007, (Gujarat), India </p>
<p>\(^\ast \)L. 1627 Awadh Puri Colony Beniganj, Phase-III, Opposite - Industrial Training Institute (ITI), Ayodhya Main Road, Faizabad, Uttar Pradesh 224 001, India, e-mail: <span class="tt">vishnunarayanmishra@gmail.com</span>, <span class="tt">vishnu_narayanmishra@yahoo.co.in</span> </p>
<p>\(^\S \)Department of Mathematics, BVM Engineering College, Vallabh Vidyanagar - 388 120, (Gujarat), India, e-mail: <span class="tt">rajiv55in@yahoo.com</span>. \(^\bullet \)Department of Mathematics, University of Central Florida, Orlando, FL. 32816, USA, e-mail: <span class="tt">Ram.mohapatra@ucf.edu</span> </p>

<div class="abstract"><p> In this paper, we introduce a summation-integral type modification of Szasz-Mirakjan-Stancu operators. Calculation of moments, density theorem, a direct result and a Voronovskaja-type result are obtained for the operators. </p>
<p><b class="bf">MSC.</b> 41A10, 41A25, 41A36, 40A30 </p>
<p><b class="bf">Keywords.</b> Szász-Mirakjan operators, the Korovkin-type approximation result, K-functional, modulus of smoothness, Voronovskaja-type result. </p>
</div>
<h1 id="a0000000002">1 Introduction</h1>
<p>In 1941, G.M. Mirakjan <span class="cite">
	[
	<a href="#Mir" >17</a>
	]
</span> defined the operators \(SM_n: C_2[0,\infty ) \rightarrow C[0,\infty )\) for any \(x \in [0, \infty )\) and for any \(n \in \mathbb {N}\) given by, </p>
<div class="equation" id="E1">
<p>
  <div class="equation_content">
    \begin{equation} \label{E1} SM_n(f;x)=\sum _{k=0}^{\infty }s_{n,k}(x) f\left(\tfrac {k}{n}\right), \end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p> where </p>
<div class="equation" id="E2">
<p>
  <div class="equation_content">
    \begin{equation} \label{E2} s_{n,k}(x)=e^{-nx}\tfrac {(nx)^k}{k!}, \qquad 0 \leq x < \infty . \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p> and </p>
<div class="displaymath" id="a0000000003">
  \begin{equation*}  C_2[0,\infty ) = \left\lbrace f \in C[0,\infty ) : \lim _{x \rightarrow \infty } \tfrac {f(x)}{1+x^2} \,  \text{exists and is finite} \right\rbrace . \end{equation*}
</div>
<p>The operators \(\left(SM_n \right)_{n \in \mathbb {N}}\) are named Szász-Mirakjan operators, where \(s_{n,k}\)’s are Szász basis functions. They were extensively studied in 1950 by O. Szász <span class="cite">
	[
	<a href="#Sza" >19</a>
	]
</span>. A modification of operators (<a href="#E1">1</a>) was introduced by P.L. Butzer <span class="cite">
	[
	<a href="#But" >7</a>
	]
</span>, in order to obtained an approximation process for spaces of integrable functions, on unbounded intervals, which are now known as Szász-Mirakjan-Kantorovich operators. </p>
<p>Durrmeyer <span class="cite">
	[
	<a href="#Dur" >9</a>
	]
</span> defined the summation-integral type approximation process, using the Bernstein polynomials, as </p>
<div class="equation" id="Dur">
<p>
  <div class="equation_content">
    \begin{equation} \label{Dur} D_n(f;x)=(n+1)\sum _{k=0}^n b_{n,k}(x)\left(\int _{0}^1 b_{n,k}(t)f(t)\, dt \right), \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p> where Bernstein polynomial are given by </p>
<div class="displaymath" id="a0000000004">
  \begin{equation*}  B_n(f;x)=\sum _{k=0}^n b_{n,k}(x)f\left(\tfrac {k}{n} \right), \end{equation*}
</div>
<p> and </p>
<div class="displaymath" id="a0000000005">
  \begin{equation*}  b_{n,k}(x) = \tbinom {n}{k}x^k(1-x)^{n-k}, \end{equation*}
</div>
<p> \(0 \leq x \leq 1\), \(k= 0,1, \ldots , n\) and \(n \in \mathbb {N}\). </p>
<p>Derriennic <span class="cite">
	[
	<a href="#Der" >8</a>
	]
</span> studied the operators given by (<a href="#Dur">3</a>) extensively. Motivated by Derriennic, Sahai and Prasad <span class="cite">
	[
	<a href="#Sah" >18</a>
	]
</span> studied many properties of the modified Lupaş operators of the type </p>
<div class="equation" id="Sah">
<p>
  <div class="equation_content">
    \begin{equation} \label{Sah} M_n(f;x)=(n-1)\sum _{k=0}^{\infty }p_{n,k}(x)\left(\int _0^{\infty } p_{n,k}(t)f(t)\,  dt \right), \end{equation}
  </div>
  <span class="equation_label">4</span>
</p>
</div>
<p> where </p>
<div class="displaymath" id="a0000000006">
  \begin{equation*}  p_{n,k}(x)=\tbinom {n+k-1}{k}\tfrac {x^k}{(1+x)^{n+k}}, \end{equation*}
</div>
<p> \(0 \leq x {\lt} \infty \), \(k= 0,1,2, \ldots \) and \(n \in \mathbb {N}\). </p>
<p>Mazhar and Totik <span class="cite">
	[
	<a href="#Maz" >16</a>
	]
</span> introduced two Durrmeyer type modifications of Szász-Mirakjan operators (<a href="#E1">1</a>) as </p>
<div class="displaymath" id="a0000000007">
  \begin{equation*}  \bar{S}_n(f;x)=f(0)s_{n,0}(x)+ n\sum _{k=1}^{\infty }s_{n,k}(x)\int _{0}^{\infty }s_{n,k-1}(t)f(t)\, dt \end{equation*}
</div>
<p> and </p>
<div class="equation" id="Maz1">
<p>
  <div class="equation_content">
    \begin{equation} \label{Maz1} S_n(f;x)= n\sum _{k=0}^{\infty }s_{n,k}(x)\int _{0}^{\infty }s_{n,k}(t)f(t)\, dt, \end{equation}
  </div>
  <span class="equation_label">5</span>
</p>
</div>
<p> where \(s_{n,k}\)’s are as given by (<a href="#E2">2</a>). </p>
<p>Various properties, like global approximation in weight spaces, uniform approximation, simultaneous approximation, weighted approximations, of these operators, their generalizations and modifications are studied over the years. We can mention some important studies of this type (see <span class="cite">
	[
	<a href="#Aca" >1</a>
	]
</span>–<span class="cite">
	[
	<a href="#Agr1" >3</a>
	]
</span>, <span class="cite">
	[
	<a href="#Bec" >5</a>
	]
</span>–<span class="cite">
	[
	<a href="#Ibr" >15</a>
	]
</span>, <span class="cite">
	[
	<a href="#Tot" >21</a>
	]
</span>–<span class="cite">
	[
	<a href="#Vec2" >23</a>
	]
</span>, <span class="cite">
	[
	<a href="#RBG1" >25</a>
	]
</span>–<span class="cite">
	[
	<a href="#Zho" >32</a>
	]
</span>). </p>
<p>In 2015, Mishra <i class="it">et al.</i> <span class="cite">
	[
	<a href="#RBG" >24</a>
	]
</span> introduced Szász-Mirakjan-Durrmeyer-type generalization of (<a href="#Maz1">5</a>) given by </p>
<div class="equation" id="RBG">
<p>
  <div class="equation_content">
    \begin{equation} \label{RBG} S_n^{*}(f;x) = b_n\sum _{k=0}^{\infty } s_{b_n,k}(x)\int _0^{\infty }s_{b_n,k}(t)f\left(t\right)\,  dt, \end{equation}
  </div>
  <span class="equation_label">6</span>
</p>
</div>
<p> where </p>
<div class="equation" id="Sza">
<p>
  <div class="equation_content">
    \begin{equation} \label{Sza} s_{b_n,k}(x)=e^{-b_nx}\tfrac {(b_nx)^k}{k!}, \quad k = 0,1,2, \ldots ; n \in \mathbb {N}, \end{equation}
  </div>
  <span class="equation_label">7</span>
</p>
</div>
<p> \((b_n)\) is an increasing sequence of positive real numbers, \( b_n \rightarrow \infty \) as \(n \rightarrow \infty \), \(b_1 \geq 1\) and studied the simultaneous approximation properties of the operators (<a href="#RBG">6</a>). </p>
<p>In this article, we introduce Stancu <span class="cite">
	[
	<a href="#Sta" >20</a>
	]
</span> type summation integral type of modification for \(0 \leq \alpha \leq \beta \), for any \(x \in [0, \infty )\) and for any \(n \in \mathbb {N}\) of (\(\ref{E1}\)) given by </p>
<div class="equation" id="E4">
<p>
  <div class="equation_content">
    \begin{equation} \label{E4} S_n^{*,\alpha , \beta }(f;x)= b_n \sum _{k=0}^{\infty }s_{b_n,k}(x)\int _{0}^{\infty }s_{b_n,k}(t) f\big(\tfrac {b_nt + \alpha }{b_n + \beta }\big)\,  dt, \end{equation}
  </div>
  <span class="equation_label">8</span>
</p>
</div>
<p> where \(s_{b_n,k}\)’s are as given in (\(\ref{Sza}\)), \((b_n)\) is an increasing sequence of positive real numbers with \(b_n \rightarrow \infty \) as \(n \rightarrow \infty \) and \(b_1 + \beta \geq 1\). The operators \(\big(S_n^{*,\alpha , \beta }\big)_{n \in \mathbb {N}}\), given by (<a href="#E4">8</a>), are positive linear operators. Here we studied the direct result related properties of the operators (<a href="#E4">8</a>). </p>
<h1 id="a0000000008">2 Estimation of moments</h1>
<p> Let us denote by \(\mu _{n,m}^{*,\alpha ,\beta }\), the \(m^{th}\) moments of the operators given in (<a href="#E4">8</a>), defined as </p>
<div class="equation" id="E5">
<p>
  <div class="equation_content">
    \begin{equation} \label{E5} \mu _{n,m}^{*,\alpha ,\beta }(x)= S_n^{*,\alpha , \beta }((t-x)^m;x),\qquad m = 0,1,2,\ldots . \end{equation}
  </div>
  <span class="equation_label">9</span>
</p>
</div>
<p><div class="lemma_thmwrapper " id="L1">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">1</span>
  </div>
  <div class="lemma_thmcontent">
  <p> For \(m=1,2, \ldots \), the following relation holds: </p>
<div class="displaymath" id="E11">
  \begin{eqnarray}  \label{E11} S_n^{*,\alpha , \beta }(t^m;x)=\sum _{j=0}^{m} \tbinom {m}{j}\tfrac {b_n^j \alpha ^{m-j}}{(b_n + \beta )^m}S_n^{*}(t^j;x), \end{eqnarray}
</div>
<p> where \(S_n^{*}(f;x)\) and \(S_n^{*,\alpha , \beta }(f;x)\) are as given by <span class="rm">(\(\ref{RBG}\))</span> and <span class="rm">(\(\ref{E4}\)),</span> respectively. </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000009">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Using <span class="rm">(\(\ref{RBG}\))</span> and <span class="rm">(<a href="#E4">8</a>)</span>, </p>
<div class="displaymath" id="a0000000010">
  \begin{eqnarray*}  S_n^{*,\alpha , \beta }(t^m;x)& =&  b_n \sum _{k=0}^{\infty }s_{b_n,k}(x)\int _{0}^{\infty }s_{b_n,k}(t) \left(\tfrac {b_nt + \alpha }{b_n + \beta }\right)^m\,  dt\\ & =&  b_n \sum _{k=0}^{\infty }s_{b_n,k}(x)\int _{0}^{\infty }s_{b_n,k}(t) \left(\sum _{j=0}^{m} \tbinom {m}{j} \tfrac {b_n^j \alpha ^{m-j}}{(b_n + \beta )^m} t^j \right)\,  dt \\ & =& \sum _{j=0}^{m} \tbinom {m}{j} \tfrac {b_n^j \alpha ^{m-j}}{(b_n + \beta )^m}\left( b_n \sum _{k=0}^{\infty }s_{b_n,k}(x)\int _{0}^{\infty }s_{b_n,k}(t) t^j \,  dt \right) \\ & =& \sum _{j=0}^{m} \tbinom {m}{j} \tfrac {b_n^j \alpha ^{m-j}}{(b_n + \beta )^m}S_n^{*}(t^j;x). \end{eqnarray*}
</div>
<p> <div class="lemma_thmwrapper " id="L2">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">2</span>
  </div>
  <div class="lemma_thmcontent">
  <p> For \(m=1,2, \ldots \), the following holds: </p>
<div class="displaymath" id="a0000000011">
  \begin{eqnarray*}  S_n^{*,\alpha , \beta }((t-x)^m;x)= \sum _{j=0}^{m} \tbinom {m}{j} (-x)^{m-j}S_n^{*,\alpha , \beta }(t^j;x), \end{eqnarray*}
</div>
<p> where \(S_n^{*,\alpha , \beta }(f;x)\) is as given in <span class="rm">(\(\ref{E4}\))</span>. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000012">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Using (\(\ref{E4}\)), </p>
<div class="displaymath" id="a0000000013">
  \begin{align*} &  S_n^{*,\alpha , \beta }((t-x)^m;x)=\\ & = b_n \sum _{k=0}^{\infty }s_{b_n,k}(x)\int _{0}^{\infty }s_{b_n,k}(t) \left(\tfrac {b_nt + \alpha }{b_n + \beta } - x\right)^m\,  dt \\ & = b_n \sum _{k=0}^{\infty }s_{b_n,k}(x)\int _{0}^{\infty }s_{b_n,k}(t) \left\lbrace \sum _{j=0}^{m} \tbinom {m}{j} (-x)^{m-j} \left(\tfrac {b_nt+\alpha }{b_n+\beta } \right)^j \right\rbrace \,  dt \\ & =\sum _{j=0}^{m} \tbinom {m}{j} (-x)^{m-j}\left\lbrace b_n \sum _{k=0}^{\infty }s_{b_n,k}(x)\int _{0}^{\infty }s_{b_n,k}(t) \left(\tfrac {b_nt+\alpha }{b_n+\beta } \right)^j \,  dt \right\rbrace \\ & =\sum _{j=0}^{m} \tbinom {m}{j} (-x)^{m-j}S_n^{*,\alpha , \beta }(t^j;x). \end{align*}
</div>

<p><div class="lemma_thmwrapper " id="L3">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">3</span>
  </div>
  <div class="lemma_thmcontent">
  <p> For \(e_i(t)=t^i, \quad i = 0,1,2,3,4\), the following holds: </p>
<ol class="enumerate">
  <li><p>\(S_n^{*}(e_0;x)=1,\) </p>
</li>
  <li><p>\(S_n^{*}(e_1;x)=\tfrac {1}{b_n}(b_nx+1),\) </p>
</li>
  <li><p>\(S_n^{*}(e_2;x)=\tfrac {1}{b_n^2}(b_n^2x^2+4b_nx+2),\) </p>
</li>
  <li><p>\(S_n^{*}(e_3;x)=\tfrac {1}{b_n^3}(b_n^3x^3+9b_n^2x^2+18b_nx+6),\) </p>
</li>
  <li><p>\(S_n^{*}(e_4;x)=\tfrac {1}{b_n^4}(b_n^4x^4 + 16b_n^3x^3+72b_n^2x^2+96b_nx+24).\) </p>
</li>
</ol>

  </div>
</div> <div class="proof_wrapper" id="a0000000014">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>By (\(\ref{RBG}\)), </p>
<div class="displaymath" id="a0000000015">
  \begin{eqnarray*}  S_n^{*}(e_2;x)& =&  \sum _{k=0}^{\infty }s_{b_n,k}(x) b_n \int _{0}^{\infty }s_{b_n,k}(t) t^2\,  dt \\ & =&  \sum _{k=0}^{\infty }s_{b_n,k}(x) b_n \int _{0}^{\infty }e^{-b_nt} \tfrac {(b_nt)^k}{k!} t^2\,  dt \\ & =&  \sum _{k=0}^{\infty }s_{b_n,k}(x) \tfrac {(k+1)(k+2)}{b_n^2}\\ & =&  \tfrac {1}{b_n^2}\sum _{k=0}^{\infty }e^{-b_nx} \tfrac {(b_nx)^k}{k!} [k(k-1)+4k+2] \\ & =& \tfrac {1}{b_n^2}(b_n^2x^2+4b_nx+2). \end{eqnarray*}
</div>
<p> This proves (c). </p>
<p>Other relations follow on the same line. <div class="lemma_thmwrapper " id="L4">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">4</span>
  </div>
  <div class="lemma_thmcontent">
  <p> For \(e_i(t)=t^i, \quad i = 0,1,2,3,4\), the following holds: </p>
<ol class="enumerate">
  <li><p>\(S_n^{*,\alpha , \beta }(e_0;x)=1,\) </p>
</li>
  <li><p>\(S_n^{*,\alpha , \beta }(e_1;x)=\tfrac {1}{b_n+\beta }(b_nx+\alpha +1),\) </p>
</li>
  <li><p>\(S_n^{*,\alpha , \beta }(e_2;x)=\tfrac {1}{(b_n+\beta )^2}[b_n^2x^2+2b_nx(\alpha +2)+(\alpha ^2 + 2\alpha + 2)],\) </p>
</li>
  <li><p>\(S_n^{*,\alpha , \beta }(e_3;x)=\tfrac {1}{(b_n+\beta )^3}\Big[b_n^3x^3+3b_n^2x^2(\alpha +3)+3b_nx(\alpha ^2+4\alpha +6)+(\alpha ^3+3\alpha ^2 + 6\alpha + 6)\Big],\) </p>
</li>
  <li><p>\(S_n^{*,\alpha , \beta }(e_4;x)=\tfrac {1}{(b_n+\beta )^4}\Big[b_n^4x^4+ 4b_n^3x^3(\alpha +4)+6b_n^2x^2(\alpha ^2+6\alpha +12)+4b_nx(\alpha ^3+6\alpha ^2+18\alpha +24)+(\alpha ^4+4\alpha ^3+12\alpha ^2 + 24\alpha + 24)\Big].\) </p>
</li>
</ol>

  </div>
</div> <div class="proof_wrapper" id="a0000000016">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Using (<a href="#E11">10</a>), </p>
<div class="displaymath" id="a0000000017">
  \begin{eqnarray*}  S_n^{*,\alpha , \beta }(e_2;x)& =& \sum _{j=0}^{2} \tbinom {2}{j} \tfrac {b_n^j \alpha ^{2-j}}{(b_n + \beta )^2}S_n^{*}(t^j;x)\\ & =&  \tfrac {1}{(b_n+\beta )^2}[\alpha ^2 S_n^{*}(1;x)+2b_n\alpha S_n^{*}(t;x)+b_n^2 S_n^{*}(t^2;x)]\\ & =&  \tfrac {1}{(b_n+\beta )^2}[b_n^2x^2+2b_nx(\alpha +2)+(\alpha ^2 + 2\alpha + 2)]. \end{eqnarray*}
</div>
<p> This proves (c). </p>
<p>Other relations follow on the same line. Consider the Banach lattice </p>
<div class="displaymath" id="a0000000018">
  \begin{equation*}  C_\gamma [0,\infty ) = \Big\{ f \in C[0,\infty ): \left|f(t) \right| \leq M (1+t)^{\gamma }\Big\}  \end{equation*}
</div>
<p> for some \(M{\gt}0, \gamma {\gt}0\). </p>
<p><div class="theorem_thmwrapper " id="T1">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">5</span>
  </div>
  <div class="theorem_thmcontent">
  <p> \(\lim _{n \rightarrow \infty } S_n^{*,\alpha , \beta }(f;x)= f(x)\) uniformly for \(x \in [0,a]\), provided \(f \in C_\gamma [0,\infty )\), \(\gamma \geq 2\) and \(a {\gt} 0\). </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000019">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>For fix \(a {\gt}0\), consider the lattice homomorphism \(T_a: C[0, \infty ) \rightarrow C[0,a]\) defined by \(T_a(f):= \left.f\right|_{[0,a]}\) for every \(f \in C[0, \infty )\), where \(\left.f\right|_{[0,a]}\) denotes the restriction of the domain of \(f\) to the interval \([0,a]\). In this case, we see that, for each \(i = 0,1,2\) and by (a)–(c) of Lemma <a href="#L4">4</a>, </p>
<div class="equation" id="E13">
<p>
  <div class="equation_content">
    \begin{equation} \label{E13} \lim _{n \rightarrow \infty } T_a \left(S_n^{\alpha , \beta }(e_i;x)\right)= T_a( e_i(x)), \;  \text{uniformly on} \; [0,a]. \end{equation}
  </div>
  <span class="equation_label">11</span>
</p>
</div>
<p>Thus, by using (<a href="#E13">11</a>) and with the universal Korovkin-type property with respect to positive linear operators (see Theorem 4.1.4 (vi) of <span class="cite">
	[
	<a href="#Alt" >4</a>
	]
</span>, p.199) we have the result. </p>
<p><div class="lemma_thmwrapper " id="L5">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">6</span>
  </div>
  <div class="lemma_thmcontent">
  <p> For the moments defined in (\(\ref{E5}\)), the following holds: </p>
<ol class="enumerate">
  <li><p>\(\mu _{n,1}^{*,\alpha ,\beta }(x)= S_n^{*,\alpha , \beta }((t-x);x)=\tfrac {1}{b_n+\beta }(\alpha +1-\beta x) ,\) </p>
</li>
  <li><p>\(\mu _{n,2}^{*,\alpha ,\beta }(x)= S_n^{*,\alpha , \beta }((t-x)^2;x)=\tfrac {1}{(b_n+\beta )^2}\Big(\beta ^2x^2+2x(b_n-\alpha \beta -\beta )+(\alpha ^2 + 2\alpha + 2)\Big),\) </p>
</li>
  <li><p>\(\mu _{n,3}^{*,\alpha ,\beta }(x)= S_n^{*,\alpha , \beta }((t-x)^3;x)=\tfrac {1}{(b_n+\beta )^3}\Big[-\beta ^3x^3+3\beta x^2 \{ (2b_n+\beta )(\alpha +1)-2(\alpha +2)\} +3x \{ 2b_n(\alpha +2)-\beta (\alpha ^2+2\alpha +2)\} +(\alpha ^3+3\alpha ^2 + 6\alpha + 6)\Big],\) </p>
</li>
  <li><p>\(\mu _{n,4}^{*,\alpha ,\beta }(x)= S_n^{*,\alpha , \beta }((t-x)^4;x)=\tfrac {1}{(b_n+\beta )^4}\Big[\beta ^4 x^4 + 4\beta ^2x^3 \{ 3b_n - \beta (\alpha +1)\}  +6x^2 \{ 2b_n^2 -4b_n \beta (\alpha +3) + \beta ^2 (\alpha ^2 + 2\alpha + 2)\}  + 4x \{  3b_n(\alpha ^2+4\alpha +6)-\beta (\alpha ^3 + 3 \alpha ^2 + 6 \alpha + 6)\} + (\alpha ^4 +4\alpha ^3 + 12 \alpha ^2 + 24 \alpha + 24)\Big].\) </p>
</li>
</ol>

  </div>
</div> <div class="proof_wrapper" id="a0000000020">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div></p>
<p>The results follow from linearity of the operators \(S_n^{*,\alpha , \beta }\) and lemma (\(\ref{L4}\)). </p>
<h1 id="a0000000021">3 Direct result</h1>
<p> Let us consider the space \(C_B[0,\infty )\) of all continuous and bounded functions on \([0, \infty )\) and for \(f \in C_B[0,\infty )\), consider the supremum norm \(\| f\| =\sup \{ |f(x)|: x \in [0,\infty )\}  \). Also, consider the \(K\)-functional </p>
<div class="equation" id="E7">
<p>
  <div class="equation_content">
    \begin{equation} \label{E7} K_2(f;\delta )=\inf _{g \in W^2}\left\lbrace \| f-g \| +\delta \| g'' \|  \right\rbrace , \end{equation}
  </div>
  <span class="equation_label">12</span>
</p>
</div>
<p> where \(\delta {\gt}0\) and \(W^2=\left\lbrace g \in C_B[0,\infty ): g', g'' \in C_B[0, \infty ) \right\rbrace \). For a constant \(C {\gt}0\), the following relationship exists: </p>
<div class="displaymath" id="E8">
  \begin{align} \label{E8} K_2(f;\delta )\leq C \omega _2(f, \sqrt{\delta }), \end{align}
</div>
<p> where </p>
<div class="displaymath" id="E9">
  \begin{align} \label{E9} \omega _2(f, \sqrt{\delta })=\sup _{0{\lt}h{\lt}\sqrt{\delta }} \;  \sup _{x \in [0, \infty )}|f(x+2h)-2f(x+h)+f(x)| \end{align}
</div>
<p> is the second order modulus of smoothness of \(f \in C_B[0,\infty )\); and for \(f \in C_B[0,\infty )\), let the modulus of continuity be given by </p>
<div class="displaymath" id="E10">
  \begin{align} \label{E10} \omega _1(f, \sqrt{\delta })=\sup _{0{\lt}h{\lt}\sqrt{\delta }} \;  \sup _{x \in [0, \infty )}|f(x+h)-f(x)|. \end{align}
</div>
<p><div class="theorem_thmwrapper " id="a0000000022">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">7</span>
  </div>
  <div class="theorem_thmcontent">
  <p>For \(f \in C_B[0, \infty )\), we have </p>
<div class="displaymath" id="a0000000023">
  \begin{equation*}  |S_n^{*,\alpha , \beta }(f;x) - f(x)| \leq \omega _1 \left(f, \tfrac {\left|\alpha +1-\beta x\right|}{b_n+\beta } \right)+ C\omega _2 \left(f,\sqrt{\mu _{n,2}^{*,\alpha ,\beta }(x) + \big(\tfrac {\alpha +1-\beta x}{b_n+\beta }\big)^2} \right), \end{equation*}
</div>
<p> where \(C\) is a positive constant. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000024">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Let the auxiliary operator denoted by \(\bar{S}_n^{*,\alpha , \beta }\) be defined as </p>
<div class="displaymath" id="a0000000025">
  \begin{equation*}  \bar{S}_n^{*,\alpha , \beta }(f;x) = S_n^{*,\alpha , \beta }(f;x)-f\left(\tfrac {b_nx+\alpha +1}{b_n+\beta } \right)+f(x) \end{equation*}
</div>
<p> for every \(x \in [0, \infty )\). It is a linear operator which preserves the linear functions as \(\bar{S}_n^{*,\alpha , \beta }(1;x)=1\) and \(\bar{S}_n^{*,\alpha , \beta }(t;x)=x\). This gives us \(\bar{S}_n^{*,\alpha , \beta }(t-x;x)=0\). </p>
<p>For \(g \in W^2\), \(x \in [0, \infty )\) and by Taylor’s expansion, we have </p>
<div class="displaymath" id="a0000000026">
  \begin{equation*}  g\left(\tfrac {b_nt+\alpha }{b_n+\beta }\right) = g(x) + \left(\tfrac {b_nt+\alpha }{b_n+\beta }-x\right)g’(x)+\int _x^{\frac{b_nt+\alpha }{b_n+\beta }}\left(\tfrac {b_nt+\alpha }{b_n+\beta }-u\right) g”(u)du. \end{equation*}
</div>
<p> Operating \(\bar{S}_n^{*,\alpha , \beta }\) on both the sides, </p>
<div class="displaymath" id="a0000000027">
  \begin{eqnarray*}  \left|\bar{S}_n^{*,\alpha , \beta }(g;x)-g(x)\right| & =& \left| \bar{S}_n^{*,\alpha , \beta }\left( \int _x^t(t-u)g”(u)du;x\right)\right| \\ & \leq &  \left| S_n^{*,\alpha , \beta }\left( \int _x^t(t-u)g”(u)du;x\right)\right| \\ & &  + \left|\int _x^{\tfrac {b_nx+\alpha +1}{b_n+\beta }} \left(\tfrac {b_nx+\alpha +1}{b_n+\beta }-u \right) g”(u)du\right| \\ & \leq &  \Vert g” \Vert S_n^{*,\alpha , \beta }((t-x)^2;x)+ \Vert g” \Vert \left(\tfrac {b_nx+\alpha +1}{b_n+\beta } -x \right)^2 \\ &  = &  \Vert g” \Vert \left[ \mu _{n,2}^{*,\alpha , \beta }(x)+ \left(\tfrac {- \beta x+\alpha +1}{b_n+\beta }\right)^2 \right]. \end{eqnarray*}
</div>
<p> Also, we have \(\left|S_n^{*,\alpha , \beta }(f;x)\right| \leq \Vert f \Vert \). Using these, we get </p>
<div class="displaymath" id="a0000000028">
  \begin{eqnarray*}  \left|S_n^{*,\alpha , \beta }(f;x)- f(x)\right|&  \leq &  \left|\bar{S}_n^{*,\alpha , \beta }(f-g;x)- (f-g)(x)\right| \\ & &  + \left|\bar{S}_n^{*,\alpha , \beta }(g;x)- g(x)\right|+ \left|f\left( \tfrac {b_nx+\alpha +1}{b_n+\beta }\right)- f(x)\right| \\ &  \leq &  2 \Vert f-g \Vert + \Vert g” \Vert \left[ \mu _{n,2}^{*,\alpha , \beta }(x)+ \left(\tfrac {- \beta x+\alpha +1}{b_n+\beta }\right)^2 \right] \\ & &  + \omega _1 \left(f, \tfrac {\left|-\beta x+\alpha +1\right|}{b_n+\beta } \right). \end{eqnarray*}
</div>
<p> Taking infimum on the right hand side for all \(g \in W^2\), we get </p>
<div class="displaymath" id="a0000000029">
  \begin{eqnarray*}  \left|S_n^{*,\alpha , \beta }(f;x)- f(x)\right| &  \leq &  2K_2\left(f,\tfrac {1}{2} \left[ \mu _{n,2}^{*,\alpha , \beta }(x)+ \left(\tfrac {- \beta x+\alpha +1}{b_n+\beta }\right)^2 \right]\right) \\ & &  + \omega _1 \left(f, \tfrac {\left|-\beta x+\alpha +1\right|}{b_n+\beta } \right). \end{eqnarray*}
</div>
<p> Using (<a href="#E8">13</a>) and \(\omega _2(f, \lambda \delta ) \leq (\lambda + 1)^2 \omega _2(f, \delta )\) for \(\lambda {\gt}0\), we get </p>
<div class="displaymath" id="a0000000030">
  \begin{eqnarray*}  \left|S_n^{*,\alpha , \beta }(f;x)- f(x)\right| &  \leq &  C \omega _2\left(f, \sqrt{\mu _{n,2}^{*,\alpha , \beta }(x)+ \left(\tfrac {- \beta x+\alpha +1}{b_n+\beta }\right)^2} \right) \\ & &  + \omega _1 \left(f, \tfrac {\left|-\beta x+\alpha +1\right|}{b_n+\beta } \right). \end{eqnarray*}
</div>

<h1 id="a0000000031">4 A Voronovskaja-type result</h1>
<p> In this section we prove a Voronovskaja-type theorem for the operators \(S_n^{*,\alpha , \beta }\) given in (<a href="#E4">8</a>). <div class="lemma_thmwrapper " id="L6">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">8</span>
  </div>
  <div class="lemma_thmcontent">
  <p> \(\lim _{n \rightarrow \infty } (b_n+\beta )^2 \mu _{n,4}^{*,\alpha , \beta }(x) = 12 x^2\) uniformly with respect to \(x \in [0, a],\,  a {\gt} 0\). </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000032">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div></p>
<p>By lemma (<a href="#L5">6</a>)(d), we may write that </p>
<div class="displaymath" id="a0000000033">
  \begin{eqnarray*}  (b_n+\beta )^2 \mu _{n,4}^{*,\alpha ,\beta }(x)& =& \tfrac {1}{(b_n+\beta )^2}\Big[\beta ^4 x^4 + 4\beta ^2x^3 \{ 3b_n - \beta (\alpha +1)\}  \\ & &  +6x^2 \{ 2b_n^2 -4b_n \beta (\alpha +3) + \beta ^2 (\alpha ^2 + 2\alpha + 2)\}  \\ & &  + 4x \{  3b_n(\alpha ^2+4\alpha +6)-\beta (\alpha ^3 + 3 \alpha ^2 + 6 \alpha + 6)\}  \\ & &  + (\alpha ^4 +4\alpha ^3 + 12 \alpha ^2 + 24 \alpha + 24)\Big] \end{eqnarray*}
</div>
<p> Taking limits on both sides, as \(n \rightarrow \infty \), the Lemma is proved. </p>
<p><div class="theorem_thmwrapper " id="T2">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">9</span>
  </div>
  <div class="theorem_thmcontent">
  <p> For every \(f \in C_\gamma [0,\infty )\) such that \(f', f'' \in C_\gamma [0,\infty ), \gamma \geq 4\), we have </p>
<div class="displaymath" id="a0000000034">
  \begin{equation*}  \lim _{n \rightarrow \infty } (b_n+\beta ) \left[S_n^{*,\alpha , \beta }(f;x)-f(x) - \tfrac {-\beta x + \alpha + 1}{b_n } f’(x) \right]= xf”(x) \end{equation*}
</div>
<p> uniformly with respect to \(x \in \left[\tfrac {\alpha }{b_n + \beta }, a\right] \,  \left(a {\gt} \tfrac {\alpha }{b_n + \beta }\right)\). </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000035">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Let \(f,f', f'' \in C_\gamma [0,\infty )\) and \(x \geq \tfrac {\alpha }{b_n + \beta }\). Define, for \(t \in [0, \infty )\) with \(\tfrac {b_nt+ \alpha }{b_n + \beta } \neq x\), </p>
<div class="displaymath" id="a0000000036">
  \begin{align*} & \Psi \left(t,x\right) = \\ & = \tfrac {1}{\left(\frac{b_nt+\alpha }{b_n+\beta }-x\right)^2} \left[f\left(\tfrac {b_nt+\alpha }{b_n+\beta }\right)\! -\! f(x)-\left(\tfrac {b_nt+\alpha }{b_n+\beta }\! -\! x\right) f’(x) \! -\! \tfrac {1}{2}\left(\tfrac {b_nt+\alpha }{b_n+\beta }\! -\! x\right)^2 f”(x)\right] , \end{align*}
</div>
<p> and \(\Psi \Big(\tfrac {\left(b_n + \beta \right)x - \alpha }{b_n},x\Big)=0\). The function \(\Psi (\cdot ,x) \in C_\gamma [0,\infty )\). Also, for \(n \rightarrow \infty \), \(\Psi \Big(\tfrac {\left(b_n + \beta \right)x - \alpha }{b_n},x\Big)=\Psi \left( x,x\right)\), so \(\Psi (x,x)=0\), as \(n \rightarrow \infty \). By Taylor’s theorem we get </p>
<div class="displaymath" id="a0000000037">
  \begin{align*} & f\left(\tfrac {b_nt+\alpha }{b_n+\beta }\right)= \\ & = f(x)+\left(\tfrac {b_nt+\alpha }{b_n+\beta }-x\right)f’(x)+\tfrac {1}{2}\left(\tfrac {b_nt+\alpha }{b_n+\beta }-x\right)^2 f”(x) +\left(\tfrac {b_nt+\alpha }{b_n+\beta }-x\right)^2\Psi (t,x). \end{align*}
</div>
<p>Now from Lemma <a href="#L5">6</a>(a)–(b) </p>
<div class="displaymath" id="E12">
  \begin{align} \label{E12} (b_n\! +\! \beta ) \left[S_n^{*,\alpha , \beta }(f;x)\! -\!  f(x)\right] = &  (b_n\! +\! \beta )f’(x)\mu _{n,1}^{*,\alpha , \beta }(x) \! +\!  \tfrac {1}{2} (b_n\! +\! \beta )f”(x)\mu _{n,2}^{*,\alpha , \beta }(x) \\ &  + (b_n+\beta ) S_{n}^{*,\alpha , \beta }((t-x)^2 \Psi (t,x)). \nonumber \end{align}
</div>
<p> If we apply the Cauchy-Schwarz inequality to the third term on the right hand side of (<a href="#E12">16</a>), then </p>
<div class="displaymath" id="a0000000038">
  \begin{equation*}  (b_n+\beta ) S_{n}^{*,\alpha , \beta }((t-x)^2 \Psi (t,x);x) \leq \left( (b_n+\beta )^2 \mu _{n,4}^{*,\alpha , \beta }(x)\right)^{\frac{1}{2}} ( S_{n}^{*,\alpha , \beta }(\Psi ^2(t,x);x))^{\frac{1}{2}} \end{equation*}
</div>
<p> Now \(\Psi ^2(\cdot ,x) \in C_{2\gamma }[0,\infty )\), using Theorem <a href="#T1">5</a>, we have \(S_{n}^{*,\alpha , \beta }(\Psi ^2(t,x);x) \rightarrow \Psi ^2(x,x)=0\), as \(n \rightarrow \infty \) and using Lemma <a href="#L6">8</a>, this third term on the right tends to zero for \(x \in \left[\tfrac {\alpha }{b_n + \beta }, a\right]\) and we get </p>
<div class="displaymath" id="a0000000039">
  \begin{equation*}  \lim _{n \rightarrow \infty } (b_n+\beta ) \left[S_n^{*,\alpha , \beta }(f;x)-f(x)\right]= (1+\alpha - \beta x)f’(x) + xf”(x). \end{equation*}
</div>
<p> for \(x \in \left[\tfrac {\alpha }{b_n + \beta }, a\right] \,  \left(a {\gt} \tfrac {\alpha }{b_n + \beta }\right)\). </p>
<p><div class="acknowledgement_thmwrapper " id="a0000000040">
  <div class="acknowledgement_thmheading">
    <span class="acknowledgement_thmcaption">
    Acknowledgement
    </span>
  </div>
  <div class="acknowledgement_thmcontent">
  <p>The authors are extremely grateful to the learned referee(s) and the editor for their careful reading of the manuscript, valuable suggestions and constructive comments, which has greatly contributed to improving the quality of this research article. </p>

  </div>
</div> </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
  <dt><a name="Aca">1</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1155/2015/936308"> <i class="sc">T. Acar</i>, <i class="sc">L.N. Mishra</i> and <i class="sc">V.N. Mishra</i>, <i class="it">Simultaneous approximation for generalized Srivastava-Gupta operator</i>, J. Function Spaces, Article ID 936308, 12 pages, 2015. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Agratini">2</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1016/j.amc.2014.03.052"> <i class="sc">O. Agratini</i>, <i class="it">On approximation process of integral type</i>, Appl. Math. Comput., <b class="bf">236</b> (2014) pp.&#160;195–201. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Agr1">3</a></dt>
  <dd><p><i class="sc">P.N. Agrawal</i> and <i class="sc">H.S. Kasana</i>, <i class="it">On simultaneous approximation by Szász-Mirakjan operators</i>. Bull. Inst. Math. Acad. Sinica <b class="bf">22</b> (1994), pp.&#160;181–188. </p>
</dd>
  <dt><a name="Alt">4</a></dt>
  <dd><p><i class="sc">P. Altomare</i> and <i class="sc">M. Campiti</i>, <i class="it">Korovkin-type Approximation Theory and its Applications</i>, Walter de Gruyter, Berlin - New York, 1994. </p>
</dd>
  <dt><a name="Bec">5</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1512/iumj.1978.27.27011"> <i class="sc">M. Becker</i>, <i class="it">Global approximation theorems for Szász-Mirakjan and Baskakov operators in polynomial weight spaces</i>, Indiana Univ. Math. J., <b class="bf">27</b> (1978), pp.&#160;127–142. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Bec1">6</a></dt>
  <dd><p><i class="sc">M. Becker</i>, <i class="sc">D. Kucharski</i> and <i class="sc">R.J. Nessel</i>, <i class="it">Global approximation theorems for Szász-Mirakjan and Baskakov operators in exponential weight spaces</i>, In: Linear Spaces and Approximation (Proc. Conf. Oberwolfach, 1977), Birkhãuser Verlag, Basel. </p>
</dd>
  <dt><a name="But">7</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.2307/2032032"> <i class="sc">P.L. Butzer</i>, <i class="it">On the extension of Bernstein polynomials to the infinite interval.</i> Proc. Amer. Math. Soc., <b class="bf">5</b> (1954), pp.&#160;547–553. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Der">8</a></dt>
  <dd><p><i class="sc">M.M. Derriennic</i>, <i class="it">Sur l’approximation des fonctions d’une ou plusieurs variables par des polynomes de Bernstein modifiés et application au probléme des moments</i>, Thése de 3e cycle, Université de Rennes, 1978. </p>
</dd>
  <dt><a name="Dur">9</a></dt>
  <dd><p><i class="sc">J.L. Durrmeyer</i>, <i class="it">Une formule d’inversion de la transformée de Laplace: Applications á la théorie des moments</i>, Thése de 3e cycle, Faculté des Sciences de l’Université de Paris, 1967. </p>
</dd>
  <dt><a name="Gai">10</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1007/s40010-016-0267-z"> <i class="sc">A.R. Gairola, Deepmala</i>, <i class="sc">L.N. Mishra</i>, <i class="it">Rate of approximation by finite iterates of q-Durrmeyer operators</i>, Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, <b class="bf">86</b> (2016) no. 2, pp.&#160;229–234. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Gur">11</a></dt>
  <dd><p><i class="sc">I. Gürhan</i> and <i class="sc">R.N. Mohapatra</i>, <i class="it">Approximation properties by \(q\)-Durrmeyer-Stancu operators</i>, Anal. Theory Appl. <b class="bf">29</b>(4) (2013), pp.&#160;373–383. </p>
</dd>
  <dt><a name="Her">12</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1007/BF01902211"> <i class="sc">T. Hermann</i>, <i class="it">On the Szász-Mirakjan operator</i>, Acta Math. Sci. Acad. Hungar <b class="bf">32</b>(1-2) (1978), pp.&#160;163–173. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="HMS1">13</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1016/j.amc.2007.01.039 "> <i class="sc">H.M. Srivastava</i>, <i class="sc">Z. Finta</i> and <i class="sc">V. Gupta</i>, <i class="it">Direct results for a certain family of summation-integral type operators</i>, Appl. Math. Comput. <b class="bf">190</b> (2007), pp.&#160;449–457. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="HMS2">14</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1016/S0895-7177(03)90042-2"> <i class="sc">H.M. Srivastava</i> and <i class="sc">V. Gupta</i>, <i class="it">A certain family of summation-integral type operators</i>, Math. Comput. Modelling <b class="bf">37</b>(2003), pp.&#160;1307–1315. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Ibr">15</a></dt>
  <dd><p><i class="sc">B. Ibrahim</i> and <i class="sc">E. Ibikli</i>, <i class="it">Inverse theorems for Bernstein-Chlodowsky type polynomials</i>, J. Math. Kyoto Univ., <b class="bf">46</b> (2006) 1, pp.&#160;21–29. </p>
</dd>
  <dt><a name="Maz">16</a></dt>
  <dd><p><i class="sc">S.M. Mazhar</i> and <i class="sc">V. Totik</i>, <i class="it">Approximation by modified Szász operators</i>, Acta Sci. Math., <b class="bf">49</b> (1985), pp.&#160;257–269. </p>
</dd>
  <dt><a name="Mir">17</a></dt>
  <dd><p><i class="sc">G.M. Mirakjan</i>, <i class="it">Approximation of continuous functions with the aid of polynomials \({\rm e}^{-nx}\sum _{k=0}^mC_{k,n}\chi ^k\)</i>, Dokl. Akad. Nauk SSSR, <b class="bf">31</b> (1941), pp.&#160;201–205. </p>
</dd>
  <dt><a name="Sah">18</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1016/0021-9045(85)90039-5"> <i class="sc">A. Sahai</i> and <i class="sc">G. Prasad</i>, <i class="it">On simultaneous approximation by modified Lupaş operators</i>, J. Approx. Theory, <b class="bf">45</b> (1985), 122–128. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Sza">19</a></dt>
  <dd><p><i class="sc">O. Szász</i>, <i class="it">Generalization of S. Bernstein’s polynomials to the infinite interval</i>, J. Res. Nat. Bur. Standards Sect. B. <b class="bf">45</b> (1950), pp.&#160;239–245. </p>
</dd>
  <dt><a name="Sta">20</a></dt>
  <dd><p><i class="sc">D.D. Stancu</i>, <i class="it">Asupra unei generalizări a polinoamelor lui Bernstein</i>, Studia Universitatis Babes-Bolyai, <b class="bf">14</b> (1969) 2, pp.&#160;31–45 (in Romanian). </p>
</dd>
  <dt><a name="Tot">21</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1007/BF01961317"> <i class="sc">V. Totik, V.</i>, <i class="it">Uniform approximation by Szász-Mirakjan type operators</i>, Acta Math. Hungar. <b class="bf">41</b> (1983), pp.&#160;291–307. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Vec1">22</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1007/s10474-006-0057-1"> <i class="sc">B. Della Vecchia</i>, <i class="sc">G. Mastroianni</i> and <i class="sc">J. Szabados</i>, <i class="it">Weighted approximation of functions by Szász-Mirakyan-type operators</i>, Acta Math. Hungar., <b class="bf">111</b> (2006), pp.&#160;325–345. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Vec2">23</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1007/s00009-014-0413-2"> <i class="sc">B. Della Vecchia</i>, <i class="sc">G. Mastroianni</i> and <i class="sc">J. Szabados</i>, <i class="it">A weighted generalization of Szász-Mirakyan and Butzer operators</i>, Mediterr. J. Math., <b class="bf">12</b> (2015) no. 2, <br />pp.&#160;433–454. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="RBG">24</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1007/s40574-015-0045-x"> <i class="sc">V.N. Mishra</i>, <i class="sc">R.B. Gandhi</i> and <i class="sc">F. Nasaireh</i>, <i class="it">Simultaneous <span class="mbox" style="width: ">approximation by Szász-</span> Mirakjan-Durrmeyer-type operators</i>, Boll. Unione Mat. Ital., <b class="bf">8</b> (2016) 4, pp 297–305. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="RBG1">25</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1007/s10998-016-0145-0"> <i class="sc">V.N. Mishra</i> and <i class="sc">R.B. Gandhi</i>, <i class="it">Simultaneous approximation by Szász-Mirakjan-Stancu-Durrmeyer type operators</i>, Periodica Mathematica Hungarica, 2016. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="VNM1">26</a></dt>
  <dd><p><i class="sc">V.N. Mishra</i>, <i class="sc">H.H. Khan</i>, <i class="sc">K. Khatri</i> and <i class="sc">L.N. Mishra</i>, <i class="it">Hypergeometric representation for Baskakov-Durrmeyer-Stancu type operators</i>, Bulletin of Mathematical Analysis and Applications, <b class="bf">5</b> (2013) 3, pp.&#160;18–26. </p>
</dd>
  <dt><a name="VNM2">27</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1186/1029-242X-2013-586"> <i class="sc">V.N. Mishra</i>, <i class="sc">K. Khatri</i>, <i class="sc">L.N. Mishra</i> and <i class="sc">Deepmala</i>, <i class="itshape">Inverse result in simultaneous approximation by Baskakov-Durrmeyer-Stancu operators</i>, Journal of Inequalities and Applications 2013, 2013:586. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="VNM3">28</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1155/2013/814824"> <i class="sc">V.N. Mishra</i>, <i class="sc">K. Khatri</i> and <i class="sc">L.N. Mishra</i>, <i class="itshape">Some approximation properties of q-Baskakov-Beta-Stancu type operators</i>, Journal of Calculus of Variations, Volume 2013, Article ID 814824, 8 pp. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="VNM4">29</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1186/10.1186/1687-1847-2013-345"> <i class="sc">V.N. Mishra</i>, <i class="sc">K. Khatri</i> and <i class="sc">L.N. Mishra</i>, <i class="itshape">Statistical approximation by Kantorovich type discrete \(q-\)beta operators</i>, Advances in Difference Equations 2013, 2013:345. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="VNM5">30</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1016/j.joems.2015.07.005"> <i class="sc">V.N. Mishra</i>, <i class="sc">P. Sharma</i> and <i class="sc">L.N. Mishra</i>, <i class="itshape">On statistical approximation properties of q-Baskakov-Szász-Stancu operators</i>, Journal of Egyptian Mathematical Society, <b class="bf">24</b> (2016) 3, pp.&#160;396–401. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Waf">31</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.18576/amisl/040303"> <i class="sc">A. Wafi</i>, <i class="sc">N. Rao</i> and <i class="sc">Deepmala Rai</i>, <i class="it">Approximation properties by generalized-Baskakov-Kantorovich-Stancu type operators</i>, Appl. Math. Inf. Sci. Lett., <b class="bf">4</b> (2016) 3, pp. 111–118. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="Zho">32</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1006/jath.1994.1025"> <i class="sc">D.X. Zhou</i>, <i class="it">Weighted approximation by Szász-Mirakjan operators</i>, J. Approx. Theory <b class="bf">76</b> (1994), pp.&#160;393–402. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
</dl>


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