Lost Purchase Receipt

Hi All,

My Macbook air laptop that my very generous mother bought for me in 2013 for school has recently broke. When starting up, a black screen appears with a folder/question mark that blinks. I have found a way to get around it by pressing the command and R keys which takes me to a screen where I have to enter my firmware password. However , I have FORGOTTON my firmware password 😟... The apple store is saying I need my original purchase receipt in order for them to fix it. Is there a way I can retrieve this receipt digitally? Or somehow show them my bank records where I purchased it?? Because right now it seems like there is no other option, and I have this very expensive mac laptop that is useless. And yes, I have lost my receipt. Please help!!!!!