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For given $q\in (0,1),$ coefficients $|c_k|\leq Cq^{k^2/3},$ and non-negative non-decreasing convergent sequences $\{a_k\}_{k=0}^\infty$ and $\{b_k\}_{k=0}^\infty$ satisfying $a_k\geqslant b_k,\;k=0,1,\dots $ consider two entire functions: \begin{equation*} f(z)=\sum_{k=0}^\infty c_k \prod_{j=0}^{k-1}\left(\frac{z+a_j}{1+a_j}\right)\quad \mathrm{and}\quad g(z)=\sum_{k=0}^\infty c_k \prod_{j=0}^{k-1}\left(\frac{z+b_j}{1+b_j}\right). \end{equation*}

Question 1. If $f(z)=g(z),\;z\in \mathbb{C},$ does it follow that either $f(z)$ is a polynomial or that all $a_k=b_k$?

Question 2. If the answer is "no", what can be stated about $a_k$ and $b_k$ under the condition $f(z)=g(z)$?

The problem arises in the investigation of the limit behavior of the $q$-Stancu polynomials as defined by G. Nowak in:

G. Nowak, Approximation properties for generalized $q$-Bernstein polynomials, J. Math. Anal. Appl., 350(1) (2009), 50--55.

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    $\begingroup$ Shouldn't it be each $c_k$ term to power $k$ or the like? $\endgroup$
    – Conrad
    Commented Oct 18 at 12:45
  • $\begingroup$ Please double-check check formulae. As written, your $f$ and $g$ are linear functions. My guess is that the expressions in parenthesis should be raised to power $k$. $\endgroup$
    – Oleg Eroshkin
    Commented Oct 18 at 13:09
  • $\begingroup$ @Oleg Eroshkin Thank you for your remark. Yes, there was a misprint (Product is missing). I apologize for the error. $\endgroup$
    – Deepti
    Commented Oct 19 at 6:10
  • $\begingroup$ @Conrad Thank you for the remark. There was a misprint (product was missing), sorry for that. $\endgroup$
    – Deepti
    Commented Oct 19 at 6:11
  • $\begingroup$ Since initially there were obvious misprints, I'd like you to confirm that the coefficients $c_k$ are, indeed, exactly the same for $f$ and $g$. $\endgroup$
    – fedja
    Commented Oct 19 at 12:39

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