To specify the matters a bit more, my problem is not at all related to filtering, but the mathematical function I am looking can be written in the form above. So I am hoping to apply some theorems derived in linear systems theory to my problem.
Also, the poles and zeros are all real and negative, i.e., Q(z) is a ratio of two strictly increasing positive polynomials.
Now, what I am after is some facts about the variation / oscillations of this function. Ideally, I wanted it to have no local maximums for z>0, but unfortunately this is not true. I am thinking something along the lines of that since all the poles and zeros are for z0. Also, only real z's are of interest.
Looking it as a transfer function, I then would be interested in results on the magnitude, since everything is positive, i.e. |Q| = Q.
Yes, of course (I saw N+1 instead of N-1). So they are improper in this case. About the rest I give up. I don't understand your question. If z is complex so how could it be that modulus (real value) equals complex value |Q(z)|=3DQ(z) for all z? If z is not complex then how can you even think of modulus? Or maybe it's an absolute value ||?