In "Inside Llewyn Davis" was Mike (Llewyn's deceased musical partner) the Gorfeins' son?
I am most interested in in-universe proof, but would also appreciate quotes of any relevant discussion by the filmmakers.
The movie is peppered with clues that point to the Gorfeins as Mike's parents:
By the way, and off the topic, all the reviewers missed the fact that the scenes where Llewyn wakes up in the morning at the beginning and end are the same footage except for one element - in the second iteration Llewyn does NOT let the cat out. So many people write about how dark this movie is but here is Llewyn back in the beginning, without a mark on his face, with a chance to go through the day again. Without the cat would he have to go to Jim and Jean's in the morning? Probably not. If he didn't know about the pregnancy he might have gotten royalties for his part in the hit record. To me this movie is highly redemptive. There is even an implication that he is remembering what happened the first time - why else would he stop to look at the poster of the Incredible Journey?
No, there is no definitive proof given that affirms that Mikey is a Gorfein.
There are plenty of moments in the movie where it would be easy to confirm that he is in fact their son, but such opportunities are never taken. For example:
When Lillian says "I don't want to be in this room" she could have said "his room" which would imply more of a personal connection, but she didn't.
When Carey says "I miss Mike," nobody else said anything like "Imagine how the Gorfeins must feel."
If Mikey was the Gorfeins' kid, that would make Llewyn a total dick for the way he treated them (the whole abortion thing, etc.)
Most grieving parents wouldn't have made Llewyn perform for dinner guests.
There are no photographs of Mikey in the Gorfeins' house. In fact, the only reminder of him is the album If We Had Wings.
While this is all circumstantial and based on what characters could have said or done, it's more compelling evidence than we're given to believe that Mike is a Gorfein.