Reincarnation is a concept about rebirth in some cultures/religions, whereby the deceased returns to the living world in one form or another. The strongest hint of this is indicated here: image, in which Roger tells Rayleigh that he will not die. I analysed the page, and I inferred two things: one, Roger was starting a new age of piracy through his death and two, he wanted to return to the living world with a new younger body.
The question is why? Well, we know that Roger and Rayleigh had discovered the true history of the world, however, before the Roger pirates could act upon their new found knowledge, Roger had become sick with an incurable disease. Perhaps Roger started the new age of piracy to encourage people to fight back against the tyranny of the World Government. He had a personal interest to return to the living world for one other reason: Ace. Roger was a father, so while he couldn't prevent his death, he probably had the power to return from death. This is not unique or impossible, as we know Brook has accomplished a form of rebirth already. Maybe Roger and Garp made some kind of deal which would allow Dragon to father the next birth of Roger. Perhaps that explains the strange tattoo's on Dragon's face. Roger returned to the living world, and despite everything Ace says about Roger doing nothing for him, maybe through Luffy, Roger gave Ace purpose to life. It may have been Rogers way of protecting Ace, and if it wasn't for Akainu, Luffy would've saved Ace successfully.
According to Rayleigh, Gol D Roger was a man who loved to do "anything and everything with flair and pomp, it was the same for partying and fighting" which you can see here: image. Luffy too loves to do anything and everything, whose personality is almost identical to Rogers.
Shanks believes in Luffy
Why would Shanks, who was a legendary pirate bet his arm on a small child with absolutely no extraordinary powers/intelligence to the point of giving Luffy a hat he had been given by his Captain? Shanks says here image, that Luffy "said the same thing as Captain Roger. Those words that the captain once said." What Shanks said truly surprised me, it wasn't what Luffy said, it was the fact that it was something Roger also said. Just look at his face, he looks extremely happy, as though Roger himself has returned to the world of the living. Indeed, if Shanks believes Luffy is the reincarnation of Roger, then it makes sense he would bet his arm on Luffy becoming a great pirate. What didn't Shanks tell Luffy? Rayleigh says here: image, that Shanks is keeping a secret from Luffy about something. It has to be something incredibly important, which could put Luffy's life in danger and it's likely a connection between Luffy and Roger.
What was Shanks doing in East Blue?
If the deal that I talked about earlier; the deal Roger and Garp made was in fact true, maybe Shanks wanted to see for himself... to see whether Roger truly was reborn. Maybe he was also asked to be there in order to become Luffy's role model. Maybe taking that devil fruit with them on to the island was intentional? I mean they could've left the devil fruit on the ship, as opposed to leaving it lying around a kid who eats just about anything.
Luffy and Roger share the same dream of conquering the grand line.