Multiplicity of zeros and discrete orthogonal polynomials

We consider a problem of bounding the maximal possible multiplicity of a zero at of some expansions $sum a_i F_i(x)$, at a certain point $c,$ depending on the chosen family ${F_i }$. The most important example is a polynomial with $c=1.$ It is shown that this question naturally leads to discrete orthogonal polynomials. Using this connection we derive some new bounds, in particular on the multiplicity of the zero at one of a polynomial with a prescribed norm.