How to interpret these filter coefficients? I am trying to make sense of the filter coefficients. 

The author claims that they are running a 3rd order lowpass Butterworth filter and were willing to give their design. They are not the most communicative type and it is typically weeks until I am able to get any sort of answer from them.

The coefficients are as follows:

1.22293 1.394312 1.994313
-0.3339159 -0.6572268 -0.9943293


They also provided the transfer function:

$$
H_k(z) = \frac{b_{k0}+b_{k1}z^{-1}+b_{k2}z^{-2}}{1-a_{k1}z^{-1}-a_{k2}z^{-2}},\quad k = 1, 2, 3. 
$$

Whey I plug in these coefficients into the FDA toolbox in MATLAB, I get the following frequency response (sampling rate was set to $20833\textrm{ Hz}$):



What is the correct way to interpret these numbers? I am somewhat out of my element when it comes to the Digital Filtering and could use some help.
Sorry, but that does not make any sense


A 3rd order BW lowpass can be described most simply with a single number: the cutoff off frequency
It's the not numerator a denominator since this would have 4 numbers and the numerator would have the shape of x*[1 3 3 1] and the denominator should start with 4.
It's not the poles and zeros since all the zeros are at -1 and one pole is conjugate complex 
It sure as heck isn't state space either
The transfer function is a boiler plate text book copy and also mostly wrong for a third order BW (which has one biqaud and one single real pole section).


You are not missing anything obvious and having you guessing blindly doesn't serve any useful purpose. You have to get back to the client and tell them that you need proper specification or you can't get the work done.