General question: if you are very sick I think you have no choice but to call in sick even if it's your first week. How bad does this look to your new employer? Are there things you can do to avoid eroding their trust?
-- My specific context: I've started an executive position at a new company. When I signed, they were eager for me to get started right away but I asked for 6 weeks to wrap up with my old company and make sure I left them in a good place.
Now it's my first week and on day 1 I wake up with a fever and flu. I call in sick and it's looking like I will be sick all week at least. Because of an autoimmune condition I have, it can take me longer to recover from flu than the average person. Doctor says that I need to take days off to recuperate or it could turn into something worse. However I'm quite worried my new employer will lose patience with me and start questioning whether I'm serious or will flake out, and not sure what else I can do other than recover quickly and then be a model employee once I start.
Update after receiving all your input: I called my employer and offered to begin with some light work remotely. They recommended I not work, instead recover fully and start next week. So I will take them up on it. I will also offer to push my start date by a week and miss the pay as a sign of good faith, as suggested. Thank you all very much for the advice, I really appreciate it.