tolkiens legendarium - Was Gandalf's resurrection ever explained in the books?

After Gandalf died from the injuries he sustained whilst fighting the Balrog, he was resurrected by Eru:

Gandalf's body lay on the peak for nineteen days, until he was resurrected by Eru and sent back to Middle-earth alive

In the Two Towers movie, we don't get much of an explanation:

I've been sent back, until my task is done.

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In the books, do we get a better explanation of who sent Gandalf back and why?

Is Eru mentioned specifically?

Answer

Here's what Gandalf says about his resurrection:

"Then darkness took me, and I strayed out of thought and time, and I wandered far on roads that I will not tell.

"Naked I was sent back - for a brief time, until my task is done. And naked I lay upon the mountain-top. [...] I was alone, forgotten, without escape upon the hard horn of the world. There I lay staring upward, while the stars wheeled over, and each day was as long as a life-age of the earth. Faint to my ears came the gathered rumour of all lands: the springing and the dying, the song and the weeping, and the slow everlasting groan of overburdened snow."

_{-- The Two Towers, chapter 5: The White Rider}

Eru Iluvatar is not explicitly mentioned in the narrative of Lord of the Rings.

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Not in the books, but still Word of God:

He was sent by a mere prudent plan of the angelic Valar or governors; but Authority had taken up this plan and enlarged it, at the moment of its failure. 'Naked I was sent back - for a time, until my task is done'. Sent back by whom, and whence? Not by the 'gods' whose business is only with this embodied world and its time; for he passed 'out of thought and time'.
_{-- Tolkien's Letters, Letter 156}