Was John a reincarnation?
John is not a reincarnation of Elijah. We know this because the Holy Spirit tells us so in John 1:21. When asked directly if he was Elijah, John answered that he was not. Also, if we check the message from the angel in Luke 1:17, we find that John was to go before Jesus, "in the spirit and power of Elijah." It does not appear that the angel is predicting an equation of essence but rather an equation of ministry in its power, fervency and effect.
Jesus does refer to John the Baptist as Elijah in Matthew 17:12,13; and 11:13. But a few points clear this up. The transfiguration, which occurs after John's death, involves the appearance of Moses and Elijah. If John and Elijah were one and the same, then wouldn't John have been recognized and commented on?
This is weak on its own merits, but let's go on. In Matthew 17:11, Jesus cryptically says that Elijah is coming (future), but then He turns around and says Elijah has already come. The key to this is found in Matthew 11:14, where Jesus says, "If you are willing to receive it he is Elijah to come." Notice the conditional. Just as Israel rejected the Messiah, so John's Elijah-like ministry is rejected. John is a partial, but not complete, fulfillment. Malachi 4:5,6 ties the coming of Elijah with the great and dreadful day of the Lord.
So John's ministry parallels a ministry of Elijah that is to come. And when that happens, Israel will listen because of her turmoil.