Last Shabbes, I noticed that in two different nusach Ari siddurim (Torah Or and Tehillas Hashem), Modim in the Shemoneh Esrei either possesses or lacks a waw depending on whether it is a Shabbes or a weekday davening.
In the weekday davening, Modim ends with the phrase, "הטוב כי לא כלו רחמיך המרחם כי לא תמו חסדיך".
In the Shabbes davening, Modim ends with the phrase, "הטוב כי לא כלו רחמיך והמרחם כי לא תמו חסדיך".
In other words, the difference between Shabbes and weekday davening hinges on whether המרחם has a waw attached to it or not. I at first assumed that this might be a typo, but since I have seen it now in two different siddurim, I can only assume that it is deliberate.
What is the reason for the waw on Shabbes?