From (
4), it is easy to prove
\(\sum_{k=0}^{n}\tilde {b}_{n,k}(\lambda;x)=1\), then we can obtain (
5). Next,
$$\begin{aligned}& B_{n,\lambda}(t;x) \\& \quad = \sum_{k=0}^{n}\frac{k}{n} \tilde{b}_{n,k}(\lambda;x) \\& \quad = \sum_{k=0}^{n-1}\frac{k}{n} \biggl[b_{n,k}(x)+\lambda \biggl(\frac {n-2k+1}{n^{2}-1}b_{n+1,k}(x)- \frac{n-2k-1}{n^{2}-1}b_{n+1,k+1}(x) \biggr) \biggr] \\& \qquad {} +b_{n,n}(x)-\frac{\lambda}{n+1}b_{n+1,n}(x) \\& \quad = \sum_{k=0}^{n}\frac{k}{n}b_{n,k}(x)+ \lambda \Biggl(\sum_{k=0}^{n} \frac {k}{n}\frac{n-2k+1}{n^{2}-1}b_{n+1,k}(x)-\sum _{k=1}^{n-1}\frac{k}{n}\frac {n-2k-1}{n^{2}-1}b_{n+1,k+1}(x) \Biggr), \end{aligned}$$
as is well known, the Bernstein operators (
1) preserve linear functions, that is to say,
\(B_{n}(at+b;x)=ax+b\). We denote the latter two parts in the bracket of the last formula by
\(\triangle_{1}(n;x)\) and
\(\triangle_{2}(n;x)\), then we have
$$ B_{n,\lambda}(t;x) = x+\lambda \bigl(\triangle_{1}(n;x)+\triangle _{2}(n;x) \bigr). $$
(10)
Now, we will compute
\(\triangle_{1}(n;x)\) and
\(\triangle_{2}(n;x)\),
$$\begin{aligned} \triangle_{1}(n;x) =&\sum_{k=0}^{n} \frac{k}{n}\frac {n-2k+1}{n^{2}-1}b_{n+1,k}(x) \\ =&\frac{1}{n-1}\sum_{k=0}^{n} \frac{k}{n}b_{n+1,k}(x)-\frac {2}{n^{2}-1}\sum _{k=0}^{n}\frac{k^{2}}{n}b_{n+1,k}(x) \\ =&\frac{(n+1)x}{n(n-1)}\sum_{k=0}^{n-1}b_{n,k}(x)- \frac{2x^{2}}{n-1}\sum_{k=0}^{n-2}b_{n-1,k}(x)- \frac{2x}{n(n-1)}\sum_{k=0}^{n-1}b_{n,k}(x) \\ =&\frac{(n+1)x}{n(n-1)} \bigl(1-x^{n} \bigr)-\frac{2x^{2}}{n-1} \bigl(1-x^{n-1} \bigr)-\frac{2x}{n(n-1)} \bigl(1-x^{n} \bigr) \\ =&\frac{x}{n}-\frac{2x^{2}}{n-1}+\frac{x^{n+1}}{n}+ \frac {2x^{n+1}}{n(n-1)}, \end{aligned}$$
(11)
and
$$\begin{aligned} \triangle_{2}(n;x) =&-\sum_{k=1}^{n-1} \frac{k}{n}\frac {n-2k-1}{n^{2}-1}b_{n+1,k+1}(x) \\ =&-\frac{x}{n}\sum_{k=1}^{n-1}b_{n,k}(x)+ \frac{1}{n(n+1)}\sum_{k=1}^{n-1}b_{n+1,k+1}(x)+ \frac{2x^{2}}{n-1}\sum_{k=0}^{n-2}b_{n-1,k}(x) \\ &{}-\frac{2x}{n(n-1)}\sum_{k=1}^{n-1}b_{n,k}(x)+ \frac{2}{n(n^{2}-1)}\sum_{k=1}^{n-1}b_{n+1,k+1}(x) \\ =&-\frac{x [1-(1-x)^{n}-x^{n} ]}{n}+\frac{ [1-(1-x)^{n+1}-(n+1)x(1-x)^{n}-x^{n+1} ]}{n(n+1)} \\ &{}+\frac{2x^{2} (1-x^{n-1} )}{n-1}-\frac{2x [1-(1-x)^{n}-x^{n} ]}{n(n-1)} \\ &{}+\frac{2 [1-(1-x)^{n+1}-(n+1)x(1-x)^{n}-x^{n+1} ]}{n(n^{2}-1)} \\ =&\frac{2x^{2}-x-x^{n+1}}{n-1}+\frac{1-(1-x)^{n+1}-x}{n(n-1)}. \end{aligned}$$
(12)
Combining (
10), (
11) and (
12), we have
$$ B_{n,\lambda}(t;x)=x+\frac{1-2x+x^{n+1}-(1-x)^{n+1}}{n(n-1)}\lambda. $$
Hence, (
6) is proved. Finally, by (
4), we have
$$ \begin{aligned} &B_{n,\lambda}\bigl(t^{2};x\bigr) \\ &\quad = \sum_{k=0}^{n}\frac{k^{2}}{n^{2}} \tilde{b}_{n,k}(\lambda;x) \\ &\quad = \sum_{k=0}^{n-1}\frac{k^{2}}{n^{2}} \biggl[b_{n,k}(x)+\lambda \biggl(\frac {n-2k+1}{n^{2}-1}b_{n+1,k}(x)- \frac{n-2k-1}{n^{2}-1}b_{n+1,k+1}(x) \biggr) \biggr] \\ &\qquad {} +b_{n,n}(x)-\frac{\lambda}{n+1}b_{n+1,n}(x) \\ &\quad = \sum_{k=0}^{n}\frac{k^{2}}{n^{2}}b_{n,k}(x)+ \lambda \Biggl(\sum_{k=0}^{n} \frac{k^{2}}{n^{2}}\frac{n-2k+1}{n^{2}-1}b_{n+1,k}(x)-\sum _{k=1}^{n-1}\frac{k^{2}}{n^{2}}\frac{n-2k-1}{n^{2}-1}b_{n+1,k+1}(x) \Biggr), \end{aligned} $$
since
\(B_{n}(t^{2};x)=\sum_{k=0}^{n}\frac{k^{2}}{n^{2}}b_{n,k}(x)=x^{2}+\frac {x(1-x)}{n}\), and we denote the latter two parts in the bracket of last formula by
\(\triangle_{3}(n;x)\) and
\(\triangle_{4}(n;x)\), then we have
$$ B_{n,\lambda}\bigl(t^{2};x\bigr)=x^{2}+\frac{x(1-x)}{n}+ \lambda \bigl(\triangle _{3}(n;x)+\triangle_{4}(n;x) \bigr). $$
(13)
On the one hand,
$$\begin{aligned} \triangle_{3}(n;x) =&\sum_{k=0}^{n} \frac{k^{2}}{n^{2}}\frac{n-2k+1}{n^{2}-1}b_{n+1,k}(x) \\ =&\frac{1}{n-1}\sum_{k=0}^{n} \frac{k^{2}}{n^{2}}b_{n+1,k}(x)-\frac {2}{n^{2}-1}\sum _{k=0}^{n}\frac{k^{3}}{n^{2}}b_{n+1,k}(x) \\ =&\frac{(n+1)x^{2}}{n(n-1)}\sum_{k=0}^{n-2}b_{n-1,k}(x)+ \frac {(n+1)x}{n^{2}(n-1)}\sum_{k=0}^{n-1}b_{n,k}(x)- \frac{2x^{3}}{n}\sum_{k=0}^{n-3}b_{n-2,k}(x) \\ &{}-\frac{6x^{2}}{n(n-1)}\sum_{k=0}^{n-2}b_{n-1,k}(x)- \frac {2x}{n^{2}(n-1)}\sum_{k=0}^{n-1}b_{n,k}(x) \\ =&\frac{(n+1)x^{2} (1-x^{n-1} )}{n(n-1)}+\frac{(n+1)x (1-x^{n} )}{n^{2}(n-1)}-\frac{2x^{3} (1-x^{n-2} )}{n} \\ &{}-\frac{6x^{2} (1-x^{n-1} )}{n(n-1)}-\frac{2x (1-x^{n} )}{n^{2}(n-1)} \\ =&\frac{2x^{n+1}-2x^{3}}{n}+\frac{x^{2}-x^{n+1}}{n-1}+\frac {x-5x^{2}+4x^{n+1}}{n(n-1)}+ \frac{x^{n+1}-x}{n^{2}(n-1)}. \end{aligned}$$
(14)
On the other hand,
$$\begin{aligned} \triangle_{4}(n;x) =&-\sum_{k=1}^{n-1} \frac{k^{2}}{n^{2}}\frac {n-2k-1}{n^{2}-1}b_{n+1,k+1}(x) \\ =&-\frac{1}{n+1}\sum_{k=1}^{n-1} \frac{k^{2}}{n^{2}}b_{n+1,k+1}(x)+\frac {2}{n^{2}-1}\sum _{k=1}^{n-1}\frac{k^{3}}{n^{2}}b_{n+1,k+1}(x) \\ =&-\frac{x^{2}}{n}\sum_{k=0}^{n-2}b_{n-1,k}(x)+ \frac{x}{n^{2}}\sum_{k=1}^{n-1}b_{n,k}(x)- \frac{1}{n^{2}(n+1)}\sum_{k=1}^{n-1}b_{n+1,k+1}(x) \\ &{}+\frac{2x^{3}}{n}\sum_{k=0}^{n-3}b_{n-2,k}(x)+ \frac{2x}{n^{2}(n-1)}\sum_{k=1}^{n-1}b_{n,k}(x)- \frac{2}{n^{2} (n^{2}-1 )}\sum_{k=1}^{n-1}b_{n+1,k+1}(x) \\ =&-\frac{x^{2} (1-x^{n-1} )}{n}+\frac{x [1-(1-x)^{n}-x^{n} ]}{n^{2}} \\ &{}-\frac{1-(1-x)^{n+1}-(n+1)x(1-x)^{n}-x^{n+1}}{n^{2}(n+1)} \\ &{}+\frac{2x^{3} (1-x^{n-2} )}{n}+\frac{2x [1-(1-x)^{n}-x^{n} ]}{n^{2}(n-1)} \\ &{}-\frac{2 [1-(1-x)^{n+1}-(n+1)x(1-x)^{n}-x^{n+1} ]}{n^{2} (n^{2}-1 )} \\ =&\frac{(2x-1)x^{2}}{n}+\frac{x}{n(n-1)}+\frac {x-1+(1-x)^{n+1}}{n^{2}(n-1)}- \frac{x^{n+1}}{n-1}. \end{aligned}$$
(15)
Combining (
13), (
14) and (
15), we obtain
$$ B_{n,\lambda}\bigl(t^{2};x\bigr) = x^{2}+ \frac{x(1-x)}{n}+\lambda \biggl[\frac {2x-4x^{2}+2x^{n+1}}{n(n-1)}+\frac{x^{n+1}+(1-x)^{n+1}-1}{n^{2}(n-1)} \biggr], $$
therefore, we get (
7). Thus, Lemma
2.1 is proved.