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<h1>A Voronovskaja-type formula for the \(q\)-Meyer-König and Zeller operators</h1>
<p class="authors">
<span class="author">Tiberiu Trif\(^{\ast }\)</span>
</p>
<p class="date">May 17, 2011.</p>
</div>
<p>\(^{\ast }\)Babeş-Bolyai University, Faculty of Mathematics and Computer Science, Kogălniceanu no. 1, 400084 Cluj-Napoca, Romania, e-mail: <span class="ttfamily">ttrif@math.ubbcluj.ro</span> </p>

<div class="abstract"><p> A Voronovskaja-type formula for the \(q\)-Meyer-König and Zeller operators is presented. </p>
<p><b class="bf">MSC.</b> 41A36, 41A25. </p>
<p><b class="bf">Keywords.</b> Meyer-König and Zeller operators, rate of convergence, \(q\)-calculus. </p>
</div>
<h1 id="a0000000002">1 Introduction and notation</h1>
<p>Starting from the identity </p>
<div class="displaymath" id="a0000000003">
  \begin{equation*}  (1-x)^{n+1}\sum _{k=0}^\infty \tbinom {n+k}{k}x^k=1 \quad \mbox{for all } x\in [0,1), \end{equation*}
</div>
<p> W. Meyer-König and K. Zeller <span class="cite">
	[
	<a href="#MEYER-KOENIG" >12</a>
	]
</span> defined a sequence of linear positive operators associating with each continuous real-valued function defined on \([0,1]\) a so-called “Bernstein power series”. In the slight modification by E. W. Cheney and A. Sharma <span class="cite">
	[
	<a href="#CHENEY-SHARMA" >4</a>
	]
</span>, the Meyer-König and Zeller operators are defined for every \(n\in \mathbb {N}\) (the set of all positive integers) and every \(f\in C[0,1]\) by </p>
<div class="displaymath" id="a0000000004">
  \begin{eqnarray*}  M_nf(x) & :=&  \sum _{k=0}^\infty f\left(\tfrac {k}{n+k}\right) m_{n,k}(x) \quad \mbox{if } x\in [0,1), \\ M_nf(1) & :=&  f(1), \end{eqnarray*}
</div>
<p> where </p>
<div class="displaymath" id="a0000000005">
  \begin{equation*}  m_{n,k}(x):=\tbinom {n+k}{k}x^k(1-x)^{n+1}. \end{equation*}
</div>
<p>Let \(q{\gt}0\). For every \(n\in \{ 0,1,2,\ldots \} \) the \(q\)-<em>integer</em>  \([n]_q\) is defined by </p>
<div class="displaymath" id="a0000000006">
  \begin{equation*}  [0]_q := 0 \quad \mbox{and}\quad [n]_q := 1+q+\cdots +q^{n-1} \quad \mbox{if } n\geq 1. \end{equation*}
</div>
<p> The \(q\)-<em>factorial</em>  \([n]_q!\) is defined by </p>
<div class="displaymath" id="a0000000007">
  \begin{equation*}  [0]_q! := 1 \quad \mbox{and}\quad [n]_q! := [1]_q[2]_q\cdots [n]_q \quad \mbox{if } n\geq 1. \end{equation*}
</div>
<p> For all nonnegative integers \(n\) and \(k\) with \(n\geq k\), the <em>Gaussian binomial coefficient</em>  (or \(q\)-<em>binomial coefficient</em> ) \(\genfrac {[}{]}{0pt}{1}{n}{k}_q\) is defined by </p>
<div class="displaymath" id="a0000000008">
  \begin{equation*}  \genfrac {[}{]}{0pt}{1}{n}{k}_q:=\tfrac {[n]_q!}{[k]_q![n-k]_q!}\, . \end{equation*}
</div>
<p> Clearly, when \(q=1\) we have </p>
<div class="displaymath" id="a0000000009">
  \begin{equation*}  [n]_1=n, \quad [n]_1!=n! \quad \mbox{and}\quad \genfrac {[}{]}{0pt}{1}{n}{k}_1=\tbinom {n}{k}. \end{equation*}
</div>
<p>Throughout the rest of the paper \(q\) denotes a positive real number such that \(0{\lt}q{\lt}1\). In order to simplify the notation, whenever it is not necessary to mention explicitly \(q\), we write \([n]\), \([n]!\) and \(\genfrac {[}{]}{0pt}{1}{n}{k}\) instead of \([n]_q\), \([n]_q!\) and \(\genfrac {[}{]}{0pt}{1}{n}{k}_q\), respectively. Likewise, in order to emphasize the analogy with the classical \(M_n\)-operators, we make use (for any positive integer \(n\)) of the following notation: </p>
<div class="displaymath" id="a0000000010">
  \begin{equation*}  (a+b)_q^n:=(a+b)(a+qb)\cdots (a+q^{n-1}b). \end{equation*}
</div>
<p> We set also \((a+b)_q^0:=1\). </p>
<p>For every \(n\in \mathbb {N}\) one has (see, for instance, G. E. Andrews, R. Askey, R. Roy <span class="cite">
	[
	<a href="#ANDREWS-ASKEY-ROY" >2</a>
	, 
	Corollary 10.2.2
	]
</span>) </p>
<div class="equation" id="basic-q-identity">
<p>
  <div class="equation_content">
    \begin{equation} \label{basic-q-identity} (1-x)_q^{n+1}\sum _{k=0}^\infty \genfrac {[}{]}{0pt}{1}{n+k}{k}x^k=1 \quad \mbox{for all } x\in [0,1). \end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p> Starting from this identity, L. Lupaş <span class="cite">
	[
	<a href="#LUPAS" >9</a>
	]
</span> introduced a \(q\)-generalization of the \(M_n\)-operators. The \(q\)-Meyer-König and Zeller operators are defined for every \(n\in \mathbb {N}\) and every \(f\in C[0,1]\) by </p>
<div class="displaymath" id="a0000000011">
  \begin{eqnarray*}  M_{n,q}f(x) & :=&  \sum _{k=0}^\infty f\left(\tfrac {[k]}{[n+k]}\right) m_{{n},{k},q}(x) \quad \mbox{if } x\in [0,1), \\ M_{n,q}f(1) & :=&  f(1), \end{eqnarray*}
</div>
<p> where </p>
<div class="displaymath" id="a0000000012">
  \begin{equation*}  m_{{n},{k},q}(x):=\genfrac {[}{]}{0pt}{1}{n+k}{k}x^k(1-x)_q^{n+1}. \end{equation*}
</div>
<p> The \(M_{n,q}\)-operators and other similar ones have been intensively investigated in the last decade by many authors (see, for instance, <span class="cite">
	[
	<a href="#DOGRU-DUMAN" >5</a>
	]
</span>, <span class="cite">
	[
	<a href="#DOGRU-GUPTA" >6</a>
	]
</span>, <span class="cite">
	[
	<a href="#DOGRU-ORKCU" >7</a>
	]
</span>, <span class="cite">
	[
	<a href="#GOVIL-GUPTA" >8</a>
	]
</span>, <span class="cite">
	[
	<a href="#MAHMUDOV" >10</a>
	]
</span>, <span class="cite">
	[
	<a href="#OSTROVSKA1" >13</a>
	]
</span>, <span class="cite">
	[
	<a href="#OSTROVSKA2" >14</a>
	]
</span>, <span class="cite">
	[
	<a href="#OZARSLAN-DUMAN" >15</a>
	]
</span>, <span class="cite">
	[
	<a href="#SHARMA" >16</a>
	]
</span>, <span class="cite">
	[
	<a href="#TRIF" >19</a>
	]
</span>, <span class="cite">
	[
	<a href="#WANG1" >20</a>
	]
</span>, <span class="cite">
	[
	<a href="#WANG2" >21</a>
	]
</span>). We merely mention here that in <span class="cite">
	[
	<a href="#TRIF" >19</a>
	, 
	Lemma 2.1
	]
</span> it has been proved that for all \(n\in \mathbb {N}\), \(n\geq 3\) and all \(x\in [0,1]\) one has </p>
<div class="displaymath" id="a0000000013">
  \begin{equation*}  M_{n,q}e_2(x)=x^2+\tfrac {x(1-x)(1-q^nx)}{[n-1]}-R_{n,q}(x), \end{equation*}
</div>
<p> where </p>
<div class="displaymath" id="a0000000014">
  \begin{equation*}  0\leq R_{n,q}(x)\leq \tfrac {q^{n-1}(1+q)}{[n-1][n-2]}\,  x(1-x)(1-qx)(1-q^nx). \end{equation*}
</div>
<p> Here \(e_k\) \((k=0,1,2,\ldots )\) denotes, as usual, the monomial \(e_k(t):=t^k\). </p>
<h1 id="a0000000015">2 The second moment for the \(q\)-Meyer-König and Zeller operators</h1>
<p>J. A. H. Alkemade <span class="cite">
	[
	<a href="#ALKEMADE" >1</a>
	]
</span> was the first who derived an explicit expression for \(M_n e_2\) in terms of a hypergeometric series. More precisely, he proved that </p>
<div class="equation" id="Eq-Alkemade">
<p>
  <div class="equation_content">
    \begin{equation} \label{Eq-Alkemade} M_n e_2(x)=x^2+\tfrac {x(1-x)^2}{n+1}\, _2F_1(1,2;n+2;x) \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p> for all \(n\in \mathbb {N}\) and all \(x\in [0,1)\). Moreover, <a href="#Eq-Alkemade" class="eqref">2</a> holds also for \(x=1\) if \(n\geq 2\). In <a href="#Eq-Alkemade" class="eqref">2</a> the notation \(_2F_1(a,b;c;x)\) is used for the sum of the hypergeometric series </p>
<div class="equation" id="hypergeometric">
<p>
  <div class="equation_content">
    \begin{equation} \label{hypergeometric} \sum _{k=0}^\infty \tfrac {(a)_k(b)_k}{(c)_k k!}\, x^k, \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p> where </p>
<div class="displaymath" id="a0000000016">
  \begin{equation*}  (y)_0 := 1 \quad \mbox{and}\quad (y)_k := y(y+1)\cdots (y+k-1) \quad \mbox{if } k\geq 1. \end{equation*}
</div>
<p> The series <a href="#hypergeometric" class="eqref">3</a> converges for \(|x|{\lt}1\) and if \(c-a-b{\gt}0\) also for \(x=1\). </p>
<p>The \(q\)-analogue of the hypergeometric series <a href="#hypergeometric" class="eqref">3</a> is the basic \(q\)-hypergeometric series </p>
<div class="equation" id="q-hypergeometric">
<p>
  <div class="equation_content">
    \begin{equation} \label{q-hypergeometric} \sum _{k=0}^\infty \tfrac {(\alpha ;q)_k(\beta ;q)_k} {(\gamma ;q)_k(q;q)_k}\, x^k, \end{equation}
  </div>
  <span class="equation_label">4</span>
</p>
</div>
<p> where </p>
<div class="displaymath" id="a0000000017">
  \begin{equation*}  (y;q)_k:=(1-y)(1-qy)\cdots (1-q^{k-1}y)=(1-y)_q^k. \end{equation*}
</div>
<p> The series <a href="#q-hypergeometric" class="eqref">4</a> converges for \(|x|{\lt}1\). Its sum is usually denoted by </p>
<div class="displaymath" id="a0000000018">
  \[  _2\phi _1\left(\begin{array}{c} {\alpha },\, {\beta } \\ {\gamma } \end{array};\, q,\, {x}\right).  \]
</div>
<p> Set \([a]:=[a]_q:=\tfrac {1-q^a}{1-q}\) for every real number \(a\). Set also </p>
<div class="displaymath" id="a0000000019">
  \begin{equation*}  [a]_0 := [a]_{0,q}:= 1 \quad \mbox{and}\quad [a]_k := [a]_{k,q}:= [a][a+1]\cdots [a+k-1] \quad \mbox{if } k\geq 1, \end{equation*}
</div>
<p> and note that </p>
<div class="displaymath" id="a0000000020">
  \begin{equation*}  (q^a;q)_k=(1-q)^k[a]_k \end{equation*}
</div>
<p> for every real number \(a\) and every nonnegative integer \(k\). Therefore one has </p>
<div class="displaymath" id="a0000000021">
  \begin{equation*}  _2\phi _1\left(\begin{array}{c} {q^a},\, {q^b} \\ {q^c} \end{array};\, q,\, {x}\right)=\sum _{k=0}^\infty \tfrac {[a]_k[b]_k} {[c]_k[k]!}\, x^k \quad \mbox{for all } x\in (-1,1). \end{equation*}
</div>
<p>H. Wang <span class="cite">
	[
	<a href="#WANG2" >21</a>
	, 
	Theorem 1
	]
</span> proved that for every \(n\in \mathbb {N}\) and all \(x\in [0,1]\) one has </p>
<div class="equation" id="second-moment-Mnq-Wang">
<p>
  <div class="equation_content">
    \begin{equation} \label{second-moment-Mnq-Wang} M_{n,q}e_2(x)=x^2+\tfrac {x(1-x)}{[n+1]}\left(1- \tfrac {q^{n+2}[n]x}{[n+2]}\, _2\phi _1\left(\begin{array}{c} {q},\, {q^2} \\ {q^{n+3}} \end{array};\, q,\, {q^{n+1}x}\right) \right). \end{equation}
  </div>
  <span class="equation_label">5</span>
</p>
</div>
<p> But a simple computation shows that </p>
<div class="equation" id="Wang-Trif">
<p>
  <div class="equation_content">
    \begin{equation} \label{Wang-Trif} 1-\tfrac {q^{n+2}[n]x}{[n+2]}\, _2\phi _1\left(\begin{array}{c} {q},\, {q^2} \\ {q^{n+3}} \end{array};\, q,\, {q^{n+1}x}\right) = (1-q^nx)_2\phi _1\left(\begin{array}{c} {q},\, {q^2} \\ {q^{n+2}} \end{array};\, q,\, {q^n x}\right). \end{equation}
  </div>
  <span class="equation_label">6</span>
</p>
</div>
<p> By <a href="#second-moment-Mnq-Wang" class="eqref">5</a> and <a href="#Wang-Trif" class="eqref">6</a> we get </p>
<div class="equation" id="second-moment-Mnq">
<p>
  <div class="equation_content">
    \begin{equation} \label{second-moment-Mnq} M_{n,q}e_2(x)=x^2+\tfrac {x(1-x)(1-q^nx)}{[n+1]}\,  _2\phi _1\left(\begin{array}{c} {q},\, {q^2} \\ {q^{n+2}} \end{array};\, q,\, {q^n x}\right), \end{equation}
  </div>
  <span class="equation_label">7</span>
</p>
</div>
<p> a formula which is closer to <a href="#Eq-Alkemade" class="eqref">2</a> than <a href="#second-moment-Mnq-Wang" class="eqref">5</a>. </p>
<p>By using <a href="#second-moment-Mnq" class="eqref">7</a>, one can easily derive estimates for the second moment <br />\(M_{n,q}e_2(x)-x^2\). Set </p>
<div class="displaymath" id="a0000000022">
  \begin{equation*}  \Phi _m(x):=\sum _{k=m}^\infty \tfrac {[2]_k}{[n+2]_k}\, x^k \end{equation*}
</div>
<p> for every \(m\in \mathbb {N}\) and all \(x\in [0,1)\). </p>
<p><div class="theorem_thmwrapper " id="Theorem-Phi-m">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">1</span>
  </div>
  <div class="theorem_thmcontent">
  <p> For all \(n,m\in \mathbb {N}\) with \(n\geq 2\) and all \(x\in [0,q^{n-1}]\) one has </p>
<div class="equation" id="estimate-Phi-m">
<p>
  <div class="equation_content">
    \begin{equation} \label{estimate-Phi-m} \Phi _m(x)\leq \tfrac {[m+1]!}{[n-1][n+2]_{m-1}}\, x^m \end{equation}
  </div>
  <span class="equation_label">8</span>
</p>
</div>
<p> and </p>
<div class="displaymath" id="estimates-phi-21">
  \begin{eqnarray} \label{estimates-phi-21} & &  \sum _{k=0}^m \tfrac {[2]_k}{[n+2]_k}\, x^k \leq \, _2\phi _1\left(\begin{array}{c} {q},\, {q^2} \\ {q^{n+2}} \end{array};\, q,\, {x}\right) \leq \\ & &  \qquad \leq \sum _{k=0}^{m-1} \tfrac {[2]_k}{[n+2]_k}\, x^k +\tfrac {[m+1]!}{[n-1][n+2]_{m-1}}\, x^m. \nonumber \end{eqnarray}
</div>

  </div>
</div> </p>
<p><em>Proof</em> . We have </p>
<div class="displaymath" id="a0000000023">
  \begin{eqnarray*}  \Phi _m(x) & =&  \tfrac {[m+1]!}{[n+2]_m}\, x^m \sum _{k=m}^\infty \tfrac {[m+2]_{k-m}}{[n+m+2]_{k-m}}\, x^{k-m} \\ & =&  \tfrac {[m+1]!}{[n+2]_m}\, x^m \sum _{k=0}^\infty \tfrac {[m+2]_k}{[n+m+2]_k}\, x^k \\ & =&  \tfrac {[m+1]!}{[n+2]_m}\, x^m\, _2\phi _1\left(\begin{array}{c} {q},\, {q^{m+2}} \\ {q^{n+m+2}} \end{array};\, q,\, {x}\right) \\ & \leq &  \tfrac {[m+1]!}{[n+2]_m}\, x^m\,  _2\phi _1\left(\begin{array}{c} {q},\, {q^{m+2}} \\ {q^{n+m+2}} \end{array};\, q,\, {q^{n-1}}\right). \end{eqnarray*}
</div>
<p> But, for \(|\gamma /\alpha \beta |{\lt}1\) it holds that (see <span class="cite">
	[
	<a href="#ANDREWS-ASKEY-ROY" >2</a>
	, 
	Corollary 10.9.2
	]
</span>) </p>
<div class="displaymath" id="a0000000024">
  \begin{equation*}  _2\phi _1\left(\begin{array}{c} {\alpha },\, {\beta } \\ {\gamma } \end{array};\, q,\, {\gamma /\alpha \beta }\right)= \tfrac {(\gamma /\alpha ;q)_\infty (\gamma /\beta ;q)_\infty } {(\gamma ;q)_\infty (\gamma /\alpha \beta ;q)_\infty }\, , \end{equation*}
</div>
<p> where </p>
<div class="displaymath" id="a0000000025">
  \begin{equation*}  (y;q)_\infty :=\prod _{k=0}^\infty (1-q^ky). \end{equation*}
</div>
<p> Taking this into account, we have </p>
<div class="displaymath" id="a0000000026">
  \begin{equation*}  _2\phi _1\left(\begin{array}{c} {q},\, {q^{m+2}} \\ {q^{n+m+2}} \end{array};\, q,\, {q^{n-1}}\right)= \tfrac {(q^{n+m+1};q)_\infty (q^n;q)_\infty } {(q^{n+m+2};q)_\infty (q^{n-1};q)_\infty }= \tfrac {[n+m+1]}{[n-1]}\, . \end{equation*}
</div>
<p> Consequently, </p>
<div class="displaymath" id="a0000000027">
  \begin{equation*}  \Phi _m(x)\leq \tfrac {[m+1]!}{[n+2]_m}\cdot \tfrac {[n+m+1]}{[n-1]}\, x^m= \tfrac {[m+1]!}{[n-1][n+2]_{m-1}}\, x^m. \end{equation*}
</div>
<p> The left inequality in <a href="#estimates-phi-21" class="eqref">9</a> is obvious, while the right one follows immediately by <a href="#estimate-Phi-m" class="eqref">8</a>. \(\Box \) </p>
<p>By means of Theorem <a href="#Theorem-Phi-m">1</a> we deduce the following estimates of the second moment for the \(M_{n,q}\)-operators. These estimates are quite similar to those obtained by M. Becker and R. J. Nessel <span class="cite">
	[
	<a href="#BECKER-NESSEL" >3</a>
	]
</span> for the classical \(M_n\)-operators. </p>
<p><div class="corollary_thmwrapper " id="Corollary-estimates">
  <div class="corollary_thmheading">
    <span class="corollary_thmcaption">
    Corollary
    </span>
    <span class="corollary_thmlabel">2</span>
  </div>
  <div class="corollary_thmcontent">
  <p> For every \(n\in \mathbb {N}\), \(n\geq 2\) and all \(x\in [0,1]\) one has </p>
<div class="displaymath" id="a0000000028">
  \begin{align*} & \tfrac {x(1-x)(1-q^nx)}{[n+1]} \left( 1+\tfrac {(1+q)q^nx} {[n+2]}\right)\leq M_{n,q}e_2(x)-x^2 \leq \\ & \leq \tfrac {x(1-x)(1-q^nx)}{[n+1]} \left( 1+ \tfrac {(1+q)q^nx}{[n-1]}\right). \end{align*}
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000029">
  <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 Theorem <a href="#Theorem-Phi-m">1</a> with \(m=1\) it follows that </p>
<div class="displaymath" id="a0000000030">
  \begin{equation*}  1+\tfrac {1+q}{[n+2]}\, q^nx \leq \, _2\phi _1\left(\begin{array}{c} {q},\, {q^2} \\ {q^{n+2}} \end{array};\, q,\, {q^nx}\right) \leq 1+\tfrac {1+q}{[n-1]}\, q^nx. \end{equation*}
</div>
<p> This inequality and <a href="#second-moment-Mnq" class="eqref">7</a> yield the conclusion. </p>
<h1 id="a0000000031">3 A Voronovskaja-type formula for the \(q\)-Meyer-König and Zeller operators</h1>
<p>The goal of this section is to establish a Voronovskaja-type formula for the \(M_{n,q}\)-operators. Such a formula is lacking in the literature. In order to derive it, we need the following auxiliary results whose proofs are postponed to the end of the section. </p>
<p><div class="lemma_thmwrapper " id="Lemma-Mnq-e3">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">3</span>
  </div>
  <div class="lemma_thmcontent">
  <p> For every \(n\in \mathbb {N}\), \(n\geq 3\) and all \(x\in [0,1]\) one has </p>
<div class="equation" id="lemma6-eq1">
<p>
  <div class="equation_content">
    \begin{equation} \label{lemma6-eq1} M_{n,q}e_3(x)=xM_{n,q}e_2(x)+\tfrac {2qx^2(1-x)(1-q^nx)}{[n-1]} +R_{n,q}(x), \end{equation}
  </div>
  <span class="equation_label">10</span>
</p>
</div>
<p> where </p>
<div class="equation" id="lemma6-eq2">
<p>
  <div class="equation_content">
    \begin{equation} \label{lemma6-eq2} |R_{n,q}(x)|\leq \tfrac {9}{[n-1][n+2]}\, . \end{equation}
  </div>
  <span class="equation_label">11</span>
</p>
</div>

  </div>
</div> </p>
<p><div class="lemma_thmwrapper " id="Lemma-Mnq-e4">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">4</span>
  </div>
  <div class="lemma_thmcontent">
  <p> For every \(n\in \mathbb {N}\), \(n\geq 3\) and all \(x\in [0,1]\) one has </p>
<div class="displaymath" id="a0000000032">
  \begin{equation*}  M_{n,q}e_4(x)=xM_{n,q}e_3(x)+\tfrac {3q^2x^3(1-x)(1-q^nx)}{[n-1]} +\widetilde{R}_{n,q}(x), \end{equation*}
</div>
<p> where </p>
<div class="displaymath" id="a0000000033">
  \begin{equation*}  |\widetilde{R}_{n,q}(x)|\leq \tfrac {C}{[n-1][n+3]}\, , \end{equation*}
</div>
<p> \(C\) being an absolute constant (i.e., not depending on \(n\), \(q\) or \(x\)). </p>

  </div>
</div> </p>
<p>We notice that, for a fixed \(q\in (0,1)\), the sequence \((M_{n,q}f)_{n\geq 1}\) does not converge to \(f\) for every \(f\in C[0,1]\). For instance, by Corollary <a href="#Corollary-estimates">2</a> it follows that </p>
<div class="displaymath" id="a0000000034">
  \begin{equation*}  M_{n,q}e_2(x) \to x^2+(1-q)x(1-x) \quad \mbox{as } n\to \infty . \end{equation*}
</div>
<p> In order to obtain a convergent sequence of \(q\)-Meyer-König and Zeller operators we must replace \(q\) by a sequence \((q_n)_{n\geq 1}\) of numbers in \((0,1)\). If \((q_n)_{n\geq 1}\) satisfies </p>
<div class="equation" id="condition1-qn">
<p>
  <div class="equation_content">
    \begin{equation} \label{condition1-qn} q_n \to 1 \quad \mbox{and}\quad [n]_{q_n}=1+q_n+\cdots +q_n^{n-1} \to \infty \quad \mbox{as } n\to \infty , \end{equation}
  </div>
  <span class="equation_label">12</span>
</p>
</div>
<p> then the sequence \((M_{n,q_n}f)_{n\geq 1}\) converges uniformly to \(f\) on \([0,1]\) for all \(f\in C[0,1]\) (see <span class="cite">
	[
	<a href="#LUPAS" >9</a>
	, 
	Theorem 2
	]
</span> or <span class="cite">
	[
	<a href="#TRIF" >19</a>
	, 
	Theorem 2.2
	]
</span>). </p>
<p>In order to obtain a Voronovskaja-type formula for the \(M_{n,q_n}\)-operators, \((q_n)_{n\geq 1}\) must satisfy an additional condition, namely </p>
<div class="equation" id="condition2-qn">
<p>
  <div class="equation_content">
    \begin{equation} \label{condition2-qn} \mbox{there exists } \lim _{n\to \infty }q_n^n=:\alpha \in [0,1]. \end{equation}
  </div>
  <span class="equation_label">13</span>
</p>
</div>
<p> It is not difficult to construct a sequence \((q_n)_{n\geq 1}\), satisfying both <a href="#condition1-qn" class="eqref">12</a> and <a href="#condition2-qn" class="eqref">13</a>. Indeed, it suffices to take \(q_n\) such that </p>
<div class="displaymath" id="a0000000035">
  \begin{equation*}  1-\tfrac {1}{n}\leq q_n \leq 1-\tfrac {1}{n-1} \quad \mbox{for all } n\geq 3. \end{equation*}
</div>
<p> Then clearly \(q_n\to 1\) and \(q_n^n\to e^{-1}\) as \(n\to \infty \). Moreover, since </p>
<div class="displaymath" id="a0000000036">
  \begin{equation*}  1-\tfrac {r}{n}\leq q_n^r \quad \mbox{for } 1\leq r\leq n-1, \end{equation*}
</div>
<p> we have </p>
<div class="displaymath" id="a0000000037">
  \begin{equation*}  [n]_{q_n}\geq n-\tfrac {n(n-1)}{2n}=\tfrac {n+1}{2} \quad \mbox{for all } n\in \mathbb {N}. \end{equation*}
</div>
<p>We notice also that, if \((q_n)_{n\geq 1}\) is a sequence in \((0,1)\) satisfying <a href="#condition2-qn" class="eqref">13</a>, then </p>
<div class="equation" id="condition3-qn">
<p>
  <div class="equation_content">
    \begin{equation} \label{condition3-qn} \lim _{n\to \infty }\tfrac {[n]_{q_n}}{[n-1]_{q_n}}= \lim _{n\to \infty }\tfrac {[n]_{q_n}}{[n+1]_{q_n}}=1. \end{equation}
  </div>
  <span class="equation_label">14</span>
</p>
</div>
<p> Indeed, if \(\alpha \in [0,1)\), then </p>
<div class="displaymath" id="a0000000038">
  \begin{equation*}  \tfrac {[n]_{q_n}}{[n-1]_{q_n}}=\tfrac {1-q_n^n}{1-q_n^{n-1}} \to 1 \quad \mbox{as } n\to \infty \end{equation*}
</div>
<p> and, analogously, \([n]_{q_n}/[n+1]_{q_n}\to 1\) as \(n\to \infty \). On the other hand, if \(\alpha =1\), then it is easily seen that </p>
<div class="displaymath" id="a0000000039">
  \begin{equation*}  \lim _{n\to \infty }\tfrac {[n]_{q_n}}{n}= \lim _{n\to \infty }\tfrac {[n-1]_{q_n}}{n}= \lim _{n\to \infty }\tfrac {[n+1]_{q_n}}{n}=1, \end{equation*}
</div>
<p> whence <a href="#condition3-qn" class="eqref">14</a> holds also in this case. </p>
<p><div class="theorem_thmwrapper " id="a0000000040">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">5</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \((q_n)_{n\geq 1}\) be a sequence in \((0,1)\) satisfying <a href="#condition1-qn" class="eqref">12</a> and <a href="#condition2-qn" class="eqref">13</a>. Then for every \(x\in [0,1]\) and every function \(f\in C[0,1]\) which is twice differentiable at \(x\) one has </p>
<div class="displaymath" id="a0000000041">
  \begin{equation*}  \lim _{n\to \infty }[n]_{q_n}\Big(M_{n,q_n}f(x)-f(x)\Big)= \tfrac {x(1-x)(1-\alpha x)}{2}\, f”(x). \end{equation*}
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000042">
  <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 all \(n\in \mathbb {N}\) we have </p>
<div class="displaymath" id="a0000000043">
  \begin{equation*}  M_{n,q_n}e_k(x)=e_k(x), \quad k=0,1. \end{equation*}
</div>
<p> By Corollary <a href="#Corollary-estimates">2</a>, <a href="#condition1-qn" class="eqref">12</a>, <a href="#condition2-qn" class="eqref">13</a> and <a href="#condition3-qn" class="eqref">14</a> it follows that </p>
<div class="displaymath" id="a0000000044">
  \begin{equation*}  \lim _{n\to \infty }[n]_{q_n}\Big(M_{n,q_n}e_2(x)-x^2\Big)= x(1-x)(1-\alpha x). \end{equation*}
</div>
<p> Further, let \(\psi _x\) be the function defined by \(\psi _x(t):=t-x\). We have </p>
<div class="displaymath" id="a0000000045">
  \begin{equation*}  \Big(M_{n,q_n}\psi _x^4\Big)(x)=M_{n,q_n}e_4(x)-4xM_{n,q_n}e_3(x)+6x^2 M_{n,q_n}e_2(x)-3x^4. \end{equation*}
</div>
<p> By Lemma <a href="#Lemma-Mnq-e3">3</a>, Lemma <a href="#Lemma-Mnq-e4">4</a>, <a href="#condition1-qn" class="eqref">12</a>, <a href="#condition2-qn" class="eqref">13</a> and <a href="#condition3-qn" class="eqref">14</a> it follows that </p>
<div class="displaymath" id="a0000000046">
  \begin{equation*}  \lim _{n\to \infty }[n]_{q_n}\Big(M_{n,q_n}\psi _x^4\Big)(x)=0. \end{equation*}
</div>
<p> Now the conclusion of the theorem is an immediate consequence of a standard result (for instance <span class="cite">
	[
	<a href="#MAMEDOV" >11</a>
	, 
	Theorem 3
	]
</span> or <span class="cite">
	[
	<a href="#SIKKEMA1" >17</a>
	, 
	Theorem 1
	]
</span>). </p>
<p><em>Proof of Lemma <a href="#Lemma-Mnq-e3">3</a></em> . Clearly, the assertion holds for \(x=1\). For \(x\in [0,1)\) we have </p>
<div class="displaymath" id="a0000000047">
  \begin{eqnarray*}  M_{n,q}e_3(x) & =&  (1-x)_q^{n+1}\sum _{k=1}^\infty \genfrac {[}{]}{0pt}{1}{n+k}{k} x^k \tfrac {[k]^3}{[n+k]^3} \\ & =&  x(1-x)_q^{n+1}\sum _{k=0}^\infty \genfrac {[}{]}{0pt}{1}{n+k}{k} x^k \tfrac {[k+1]^2}{[n+k+1]^2}\, . \end{eqnarray*}
</div>
<p> Following P. C. Sikkema <span class="cite">
	[
	<a href="#SIKKEMA2" >18</a>
	]
</span> in his proof of Theorem 3, we notice that </p>
<div class="displaymath" id="a0000000048">
  \begin{equation*}  \tfrac {[k+1]^2}{[n+k+1]^2}=\tfrac {[k]^2}{[n+k]^2}+ \tfrac {2q^k[n][k]}{[n+k]^2[n+k+1]}+\tfrac {q^{2k}[n]^2} {[n+k]^2[n+k+1]^2}\, . \end{equation*}
</div>
<p> Taking this into account, we deduce that </p>
<div class="equation" id="lemma6-eq3">
<p>
  <div class="equation_content">
    \begin{equation} \label{lemma6-eq3} M_{n,q}e_3(x)=xM_{n,q}e_2(x)+S_{n,q}(x)+T_{n,q}(x), \end{equation}
  </div>
  <span class="equation_label">15</span>
</p>
</div>
<p> where </p>
<div class="displaymath" id="a0000000049">
  \begin{eqnarray*}  S_{n,q}(x) & :=&  2[n]x(1-x)_q^{n+1} \sum _{k=1}^\infty \genfrac {[}{]}{0pt}{1}{n+k}{k} (q x)^k \tfrac {[k]}{[n+k]^2[n+k+1]}\, , \\ T_{n,q}(x) & :=&  x(1-x)_q^{n+1} \sum _{k=0}^\infty \genfrac {[}{]}{0pt}{1}{n+k}{k} (q^2 x)^k \tfrac {[n]^2}{[n+k]^2[n+k+1]^2}\, . \end{eqnarray*}
</div>
<p> Since </p>
<div class="displaymath" id="a0000000050">
  \begin{equation*}  S_{n,q}(x)=2[n]qx^2(1-x)_q^{n+1}\sum _{k=0}^\infty \genfrac {[}{]}{0pt}{1}{n+k}{k} (q x)^k \tfrac {1}{[n+k+1][n+k+2]} \end{equation*}
</div>
<p> and </p>
<div class="displaymath" id="a0000000051">
  \begin{equation*}  \tfrac {1}{[n+k+1]}=\tfrac {1}{[n+k]}-\tfrac {q^{n+k}} {[n+k][n+k+1]}\, , \end{equation*}
</div>
<p> it follows that </p>
<div class="equation" id="lemma6-eq4">
<p>
  <div class="equation_content">
    \begin{equation} \label{lemma6-eq4} S_{n,q}(x)=U_{n,q}(x)-V_{n,q}(x), \end{equation}
  </div>
  <span class="equation_label">16</span>
</p>
</div>
<p> where </p>
<div class="displaymath" id="a0000000052">
  \begin{eqnarray*}  U_{n,q}(x) & :=&  2qx^2(1-x)_q^{n+1} \sum _{k=0}^\infty \genfrac {[}{]}{0pt}{1}{n+k}{k} \tfrac {(q x)^k\, [n]}{[n+k][n+k+2]}\, , \\ V_{n,q}(x) & :=&  2q^{n+1}x^2(1-x)_q^{n+1} \sum _{k=0}^\infty \genfrac {[}{]}{0pt}{1}{n+k}{k} \tfrac {(q^2 x)^k\, [n]}{[n+k][n+k+1][n+k+2]}\, . \end{eqnarray*}
</div>
<p> Since </p>
<div class="displaymath" id="a0000000053">
  \begin{equation*}  U_{n,q}(x)=2qx^2(1-x)_q^{n+1} \sum _{k=0}^\infty \genfrac {[}{]}{0pt}{1}{n-1+k}{k} (q x)^k \tfrac {1}{[n+k+2]} \end{equation*}
</div>
<p> and </p>
<div class="displaymath" id="a0000000054">
  \begin{equation*}  \tfrac {1}{[n+k+2]}=\tfrac {1}{[n-1+k]}-\tfrac {[3]q^{n-1+k}} {[n-1+k][n+k+2]}\, , \end{equation*}
</div>
<p> we deduce that </p>
<div class="displaymath" id="lemma6-eq5">
  \begin{eqnarray} \label{lemma6-eq5} U_{n,q}(x) & =&  \tfrac {2qx^2(1-x)(1-q^nx)(1-q x)_q^{n-1}}{[n-1]} \sum _{k=0}^\infty \genfrac {[}{]}{0pt}{1}{n-2+k}{k} (q x)^k \\ & &  \qquad -W_{n,q}(x) \nonumber \\ & =& \tfrac {2qx^2(1-x)(1-q^nx)}{[n-1]}-W_{n,q}(x), \nonumber \end{eqnarray}
</div>
<p> where </p>
<div class="displaymath" id="a0000000055">
  \begin{equation*}  W_{n,q}(x):=2[3]q^{n}x^2(1-x)_q^{n+1}\sum _{k=0}^\infty \genfrac {[}{]}{0pt}{1}{n-1+k}{k} \tfrac {(q^2 x)^k}{[n-1+k][n+k+2]}\, . \end{equation*}
</div>
<p> By <a href="#lemma6-eq3" class="eqref">15</a>, <a href="#lemma6-eq4" class="eqref">16</a> and <a href="#lemma6-eq5" class="eqref">17</a> we get </p>
<div class="displaymath" id="lemma6-eq6">
  \begin{eqnarray} \label{lemma6-eq6} M_{n,q}e_3(x) & =&  xM_{n,q}e_2(x)+ \tfrac {2qx^2(1-x)(1-q^nx)}{[n-1]} \\ & &  \qquad -W_{n,q}(x)-V_{n,q}(x)+T_{n,q}(x).\nonumber \end{eqnarray}
</div>
<p> Now we have </p>
<div class="displaymath" id="lemma6-eq7">
  \begin{eqnarray} \label{lemma6-eq7} 0 \leq W_{n,q}(x) & \leq &  \tfrac {6}{[n-1][n+2]}\, (1-x)_q^{n+1} \sum _{k=0}^\infty \genfrac {[}{]}{0pt}{1}{n-1+k}{k} (q^2 x)^k \\ & \leq &  \tfrac {6}{[n-1][n+2]}\, (1-x)_q^n \sum _{k=0}^\infty \genfrac {[}{]}{0pt}{1}{n-1+k}{k} x^k \nonumber \\ & =&  \tfrac {6}{[n-1][n+2]}\, , \nonumber \end{eqnarray}
</div>
<div class="displaymath" id="lemma6-eq8">
  \begin{eqnarray} \label{lemma6-eq8} 0 \leq V_{n,q}(x) & \leq &  \tfrac {2}{[n+1][n+2]}\, (1-x)_q^{n+1} \sum _{k=0}^\infty \genfrac {[}{]}{0pt}{1}{n-1+k}{k} (q^2 x)^k \\ & \leq &  \tfrac {2}{[n+1][n+2]}\, (1-x)_q^n \sum _{k=0}^\infty \genfrac {[}{]}{0pt}{1}{n-1+k}{k} x^k \nonumber \\ & =&  \tfrac {2}{[n+1][n+2]}\, , \nonumber \end{eqnarray}
</div>
<p> and </p>
<div class="displaymath" id="lemma6-eq9">
  \begin{eqnarray} \label{lemma6-eq9} 0 \leq T_{n,q}(x) & \leq &  \tfrac {1}{[n+1]^2}\, (1-x)_q^{n+1} \sum _{k=0}^\infty \genfrac {[}{]}{0pt}{1}{n+k}{k} (q^2 x)^k \\ & \leq &  \tfrac {1}{[n+1]^2}\, (1-x)_q^{n+1} \sum _{k=0}^\infty \genfrac {[}{]}{0pt}{1}{n+k}{k} x^k \nonumber \\ & =&  \tfrac {1}{[n+1]^2}\, . \nonumber \end{eqnarray}
</div>
<p> By <a href="#lemma6-eq6" class="eqref">18</a>, <a href="#lemma6-eq7" class="eqref">19</a>, <a href="#lemma6-eq8" class="eqref">20</a> and <a href="#lemma6-eq9" class="eqref">21</a> we conclude that <a href="#lemma6-eq1" class="eqref">10</a> and <a href="#lemma6-eq2" class="eqref">11</a> hold. \(\Box \) </p>

<p><em>Proof of Lemma <a href="#Lemma-Mnq-e4">4</a></em> . Since this proof is similar to that of Lemma <a href="#Lemma-Mnq-e3">3</a> and follows the same lines, we omit it. \(\Box \) </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
  <dt><a name="ALKEMADE">1</a></dt>
  <dd><p><i class="sc">Alkemade, J. A. H.</i>, <i class="it">The second moment for the Meyer-König and Zeller operators</i>, J. Approx. Theory, <b class="bf">40</b>, pp.&#160;261–273, 1984. </p>
</dd>
  <dt><a name="ANDREWS-ASKEY-ROY">2</a></dt>
  <dd><p><i class="sc">Andrews, G. E., Askey R.</i> and <i class="sc">Roy, R.</i>, <i class="it">Special Functions</i>, Cambridge University Press, Cambridge, 1999. </p>
</dd>
  <dt><a name="BECKER-NESSEL">3</a></dt>
  <dd><p><i class="sc">Becker, M.</i>and <i class="sc">Nessel, R. J.</i>, <i class="it">A global approximation theorem for Meyer-König and Zeller operators</i>, Math. Z, <b class="bf">160</b>, pp.&#160;195–206, 1978. </p>
</dd>
  <dt><a name="CHENEY-SHARMA">4</a></dt>
  <dd><p><i class="sc">Cheney, E. W.</i> and <i class="sc">Sharma, A.</i> <i class="it">Bernstein power series</i>, Canad. J. Math., <b class="bf">16</b>, pp.&#160;241–253, 1964. </p>
</dd>
  <dt><a name="DOGRU-DUMAN">5</a></dt>
  <dd><p><i class="sc">Doğru, O.</i> and <i class="sc">Duman, O.</i>, <i class="it">Statistical approximation of Meyer-König and Zeller operators based on \(q\)-integers</i>, Publ. Math. Debrecen, <b class="bf">68</b>, pp.&#160;199–214, 2006. </p>
</dd>
  <dt><a name="DOGRU-GUPTA">6</a></dt>
  <dd><p><i class="sc">Doğru, O.</i> and <i class="sc">Gupta, V.</i> <i class="it">Korovkin-type approximation properties of bivariate \(q\)-Meyer-König and Zeller operators</i>, Calcolo, <b class="bf">43</b>, pp.&#160;51–63, 2006. </p>
</dd>
  <dt><a name="DOGRU-ORKCU">7</a></dt>
  <dd><p><i class="sc">Doğru, O.</i> and <i class="sc">Örkcu, M.</i>, <i class="it">King type modification of Meyer-König and Zeller operators based on the \(q\)-integers</i>, Math. Comput. Modelling, <b class="bf">50</b>, pp.&#160;1245–1251, 2009. </p>
</dd>
  <dt><a name="GOVIL-GUPTA">8</a></dt>
  <dd><p><i class="sc">Govil, N. K.</i> and <i class="sc">Gupta, V.</i>, <i class="it">Convergence of \(q\)-Meyer-König-Zeller-Durrmeyer operators</i>, Adv. Stud. Contemp. Math. (Kyungshang), <b class="bf">19</b>, pp.&#160;97–108, 2009. </p>
</dd>
  <dt><a name="LUPAS">9</a></dt>
  <dd><p><i class="sc">Lupaş, L.</i>, <i class="it">A \(q\)-analogue of the Meyer-König and Zeller operator</i>, An. Univ. Oradea Fasc. Mat., <b class="bf">2</b>, pp.&#160;62–66, 1992. </p>
</dd>
  <dt><a name="MAHMUDOV">10</a></dt>
  <dd><p><i class="sc">Mahmudov, N. I.</i>, <i class="it">Korovkin-type theorems and applications</i>, Cent. Eur. J. Math., <b class="bf">7</b>, pp.&#160;348–356, 2009. </p>
</dd>
  <dt><a name="MAMEDOV">11</a></dt>
  <dd><p><i class="sc">Mamedov, R. G.</i>, <i class="it">Asymptotic approximation of differentiable functions by linear positive operators</i>, Dokl. Akad. Nauk SSSR, <b class="bf">128</b>, pp.&#160;471–474, 1959. </p>
</dd>
  <dt><a name="MEYER-KOENIG">12</a></dt>
  <dd><p><i class="sc">Meyer-König, W.</i> and <i class="sc">Zeller, K.</i>, <i class="it">Bernsteinsche Potenzreihen</i>, Studia Math., <b class="bf">19</b>, pp.&#160;89–94, 1960. </p>
</dd>
  <dt><a name="OSTROVSKA1">13</a></dt>
  <dd><p><i class="sc">Ostrovska, S.</i>, <i class="it">On the improvement of analytic properties under the limit \(q\)-Bernstein operator</i>, J. Approx. Theory, <b class="bf">138</b>, pp.&#160;37–53, 2006. </p>
</dd>
  <dt><a name="OSTROVSKA2">14</a></dt>
  <dd><p><i class="sc">Ostrovska, S.</i>, <i class="it">The unicity theorems for the limit \(q\)-Bernstein operator</i>, Applicable Anal., <b class="bf">68</b>, pp.&#160;161–167, 2009. </p>
</dd>
  <dt><a name="OZARSLAN-DUMAN">15</a></dt>
  <dd><p><i class="sc">Özarslan, M. A.</i> and <i class="sc">Duman, O.</i>, <i class="it">Approximation theorems by Meyer-König and Zeller type operators</i>, Chaos, Solitons and Fractals, <b class="bf">41</b>, pp.&#160;451–456, 2009. </p>
</dd>
  <dt><a name="SHARMA">16</a></dt>
  <dd><p><i class="sc">Sharma, H.</i>, <i class="it">Properties of \(q\)-Meyer-König-Zeller Durrmeyer operators</i>, JIPAM. J. Inequal. Pure Appl. Math., <b class="bf">10</b>, no. 4, Article 105, 10 pp. (electronic), 2009. </p>
</dd>
  <dt><a name="SIKKEMA1">17</a></dt>
  <dd><p><i class="sc">Sikkema, P. C.</i>, <i class="it">On some linear positive operators</i>, Indag. Math, <b class="bf">32</b>, pp.&#160;327–337, 1970. </p>
</dd>
  <dt><a name="SIKKEMA2">18</a></dt>
  <dd><p><i class="sc">Sikkema, P. C.</i>, <i class="it">On the asymptotic approximation with operators of Meyer-König and Zeller</i>, Indag. Math., <b class="bf">32</b>, pp.&#160;428–440, 1970. </p>
</dd>
  <dt><a name="TRIF">19</a></dt>
  <dd><p><a href ="http://ictp.acad.ro/jnaat/journal/article/view/2000-vol29-no2-art13"> <i class="sc">Trif, T.</i>, <i class="it">Meyer-König and Zeller operators based on the \(q\)-integers</i>, Rev. Anal. Numér. Théor. Approx., <b class="bf">29</b>, pp.&#160;221–229, 2000. <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="WANG1">20</a></dt>
  <dd><p><i class="sc">Wang, H.</i>, <i class="it">Korovkin-type theorem and application</i>, J. Approx. Theory, <b class="bf">132</b>, pp.&#160;258–264, 2005. </p>
</dd>
  <dt><a name="WANG2">21</a></dt>
  <dd><p><i class="sc">Wang, H.</i>, <i class="it">Properties of convergence for the \(q\)-Meyer-König and Zeller operators</i>, J. Math. Anal. Appl., <b class="bf">335</b>, pp.&#160;1360–1373, 2007. </p>
</dd>
</dl>


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