You can easily concatenate two lists in O(1) time using either a single linked list or a doubly linked list, provided that you have a pointer to the last node in at least one of the lists. (And, of course, pointers to the list heads.)
You can't do it with an array implementation, because you end up having to allocate more memory and copy the new resulting list to it. Even if the array already has memory allocated, you still have to copy all of the new items to it. So it's either O(m+n) or O(n) (where m and n are the lengths of the individual lists, respectively).
With a circularly linked list, you can easily concatenate them in O(1) time. It's just a matter of breaking a link in both lists, and then hooking them together. This assumes, of course, that the order of items isn't especially important.