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on Voronovskaja-type asymptotic formula for Bézier–Bernstein operators with shape parameter

The statement

Fix a real number $\alpha$ with $\alpha > 0$ and $\alpha \neq 1$ . For each $n \in \mathbb{N}$ , the Bernstein basis polynomial of degree $n$ and index $j$ is

p_{n,j}(x) = \binom{n}{j} x^j (1-x)^{n-j}.

For $0 \leq k \leq n$ , define the cumulative sum

J_{n,k}(x) = \sum_{j=k}^{n} p_{n,j}(x),

and set $J_{n,n+1}(x) = 0$ . The Bézier-type Bernstein operator $B_{n,\alpha}$ acts on a function $f: [0,1] \to \mathbb{R}$ by

(B_{n,\alpha} f)(x) = \sum_{k=0}^{n} f\!\left(\frac{k}{n}\right) \Bigl( J_{n,k}(x)^{\alpha} - J_{n,k+1}(x)^{\alpha} \Bigr).

Determine whether there exists a function $L: (\mathbb{R} \to \mathbb{R}) \times [0,1] \to \mathbb{R}$ and an integer $m_0 \in \mathbb{N}$ such that for every integer $m \geq m_0$ , every function $f: \mathbb{R} \to \mathbb{R}$ that is $C^m$ on $[0,1]$ , and every $x \in [0,1]$ , the following asymptotic formula holds:

\lim_{n \to \infty} \sqrt{n}\, \bigl( (B_{n,\alpha} f)(x) - f(x) \bigr) = L(f, x).

If such a limit formula exists, find an explicit expression for $L(f,x)$ in terms of $f$ , $x$ , and $\alpha$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.