It is known that if z0 is a zero of entire function F(z) with multiplicity p, then z0 is a zero of their derivative but with multiplicity p-1. It is known also that is not true in general if we replace the derivative of F by the difference operator Delta(F)=F(z+1)-F(z). But, if we suppose that F and Delta(F) having the same zeros (not necessary with the same multiplicity). Can we say something about the comparison of the multiplicity of their zeros. In other words, Can we say that the order of the multiplicity of zeros of Delta(F) is less than the order of multiplicity of zeros of F.
Consider F(z) = z(z-1). Then, Delta(F) = 2z, so in z = 0 the both polynomials have a root of the same multiplicity 1.
Dear Vladimir, Thank you for your answer, but the factorial polynomial F(z) = z(z-1) does not satisfy the condition F and Delta(F) having the same zeros (not necessary with the same multiplicity) .
Here is a detailed paper on the behavior of difference operators... http://math.ucsd.edu/~jwavrik/pub/13_ew.pdf
As you can see: there is no simple answer to that question. Remember that all functions have polynomial representations, hence, the paper covers all possible functions; although, they only cover a small set of orders, because of the extreme difficulty in identifying the zeros. One thing: because the difference operator essentially shifts a function along the abscissa, then one may conclude the multiplicity would remain the same, but you would gain an additional set of zeros for the shifted portion of the polynomial, with concomitant shifts in the placement of zeros. Depending upon how many times you operate, that would delineate the number of additional zeros added to the over all field. Not an easy problem, indeed, not in the least!
Dear Latreuch,
If you want that ALL the zeros of f and F(f) are the same, the example is the following: f(z) =ez sin(2 \pi z) . The zeros of f are all integers (and only integers). Since sin(2 \pi z) is periodic with period 1, f(z+1) = ez+1 sin(2 \pi z) and F(z) is just ez(e-1)sin(2 \pi z). So, the set of zeros of F(f) is the same as for f and the multiplicities of zeros of f are the same as of F(f).
Best, Vladimir
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