Just worked on my ipad for an hour, switched it off, switched it on 30 seconds later to check something and now get "connect to itunes". It happened on the way to a meeting and I had my keynote slides on the ipad.
When you purchase any application on an Apple mobile device, you're prompted to enter your account info (username and password). The apps purchased are stored in your account forever so if you re-download the app in the future - you'll be prompted with a message stating: "You have already purchased the application ____ Would you like to re-download for free?" I forget exact wording but it's something like that.
Just login to your iTunes acct and repurchase them. It will prompt you that the download is free 'after' you've agreed to buy it. I wish it would change the button to 'redownload' in the store. Sometimes I forgot if I've purchased an item and unsync'd it from the device or if I had only seen it on a friends device or only had the free/trial or full versions.
Apple doesn't have a 'purchase' inventory or history to review other then searching back email receipts.
But you certainly should be able to 'redownload' apps without paying twice on the same iTunes account. That's NOT the same with music purchases FYI.