Asking the audience to hang around for a few minutes after taping last night's (Oct. 15) show, David Letterman gave the Foo Fighters a chance to perform another song on the stage of the 'Late Show.' Unlike the other three televised performances this week, though, this was not a cover (and it's exclusive to Letterman's YouTube channel) -- instead, the Foo Fighters played an original song that has extra special meaning to the iconic talk show host. Check it out in the video above.

While it may seem like an odd choice for the Foos to jam 'Miracle' (from 2005's 'In Your Honor') -- especially considering they have a new album coming out in just a matter of weeks -- Letterman explained why he wanted to hear this particular song.

“They have crossed my path in many ways, unexpected and expected," he told the audience. "As a result, each and every year they grow to me as human beings and talented music artists."

As he continued to tug at the heartstrings of his fans, Letterman opened up about his son: "Years and years ago, when I became a father, I recognized that I was older than most fathers. And so did my son." After the crowd got a few laughs in, he went on to detail how he took his son (just 4 years old at the time) to Montana so that they could both learn how to ski together.

After a few days of training, their coach took them to the top of the mountain and filmed their final day. A little while later, Letterman received the video in the mail; the footage was edited together with 'Miracle' as the soundtrack. As he said, "This is the second song of theirs that will always have great, great meaning for the rest of my life."

The Foo Fighters are on the 'Late Show With David Letterman' again tonight (Oct. 16) and tomorrow night as they prepare for the debut of their HBO series, 'Sonic Highways,' on Oct. 17 -- followed by the release of their latest studio album of the same name on Nov. 10.

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