Where is the source of the belief that Elijah will come specifically to the Passover Seder? I know Malachi states he will come, but not when.
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From the source given above (https://www.ou.org/jewish_action/03/2013/whats-the-truth-about-eliyahu-hanavi-at-the-seder/): there is a question whether or not to drink a fifth cup of wine at the seder. Elijah is supposed to come resolve unanswered questions (a common phrase in monetary law when in doubt is, "Leave it until Elijah comes" - see Bava Metzia 1:8, and many other places). When the Gemara cannot answer a doubt, it ends the discussion with תיקו, literally "let it stand" but often explained as an acronym for תשבי יתרץ קושיות ובעיות, "the Tishbi will answer difficulties and questions," (http://www.hebrewbooks.org/pdfpager.aspx?req=2590&pgnum=270). Therefore the fifth cup is referred to as "Elijah's Cup," not because Elijah will drink it, but because he will tell us whether to drink it. (the Maharal says to drink the fifth cup; the Chabad minhag is to pour the cup back into the bottle and specifically use it for kiddush the next morning)
There is another aspect of the seder which appears connected to Elijah: immediately after birkat hamazon, the door is opened, and the verse שפוך חמתך על הגוים ("Pour Your wrath upon the nations," Psalms 69:6) is said aloud. I have always heard this referred to as "opening the door for Elijah," however the Hagadah of the Baal HaTanya does not refer to it in that way, and the straightforward explanation is that we are showing we are not afraid of gentiles on the night of Pesach, which is called ליל שמורים ("Guarded night", see Rema OC 481:2).