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For parameters $\,c\in(0,1)\,$ and $\,\beta>0$, let $\,\ell_{2}(c,\beta)\,$be the Hilbert space of real functions defined on $\,\mathbb{N}\,$ (i.e., realsequences), for which $$ \| f \|_{c,\beta}^2 :=\sum_{k=0}^{\infty}\frac{(\beta)_k}{k!}\,c^k\,[f(k)]^2<\infty\,. $$ We studythe best (i.e., the smallest possible) constant $\,\gamma_n(c,\beta)\,$ in thediscrete Markov-Bernstein inequality $$ \|\Delta P\|_{c,\beta}\leq\gamma_n(c,\beta)\,\|P\|_{c,\beta}\,,\quad P\in\mathcal{P}_n\,, $$ where$\,\mathcal{P}_n\,$ is the set of real algebraic polynomials of degree at most$\,n\,$ and $\,\Delta f(x):=f(x+1)-f(x)\,$.  We prove that:  (i) $\displaystyle \gamma_n(c,1)\leq 1+\frac{1}{\sqrt{c}}\,$ for every$\,n\in \mathbb{N}\,$ and $\displaystyle \lim_{n\to\infty}\gamma_n(c,1)=1+\frac{1}{\sqrt{c}}\,$.  (ii) For every fixed $\,c\in (0,1)\,$, $\,\gamma_n(c,\beta)\,$ is amonotonically decreasing function of $\,\beta\,$ in $\,(0,\infty)\,$.  (iii) For every fixed $\,c\in (0,1)\,$ and $\,\beta>0\,$, the bestMarkov-Bernstein constants $\,\gamma_n(c,\beta)\,$ are bounded uniformly withrespect to $\,n$.  A similar Markov-Bernstein unequality is proved for sequences in$\,\ell_{2}(c,\beta)\,$. We also establish a relation between the bestMarkov-Bernstein constants $\,\gamma_n(c,\beta)\,$ and the smallest eigenvaluesof certain explicitly given Jacobi matrices.

A discrete weighted Markov--Bernstein inequality for polynomials and sequences