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I am writing a short formal acknowledgement in a paper, and I would like to thank two people at the same time. The phrase I'm using is essentially:

Coffee was paid for by John and Anna, to both of whom I express my sincere gratitude.

I am not sure if the expression "to both of whom" is correct. Should it be "to whom both"? Maybe a small variation of the phrasing would work, but I can't think of any that is short and formal.

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Coffee was paid for by John and Anna, to both of whom I express my sincere gratitude.

is the correct of the two sentences you pose.

both OED

With a plural pronoun (denoting the members of a pair). Immediately following (or in early use preceding) a subject or object pronoun. Also in both which. Cf. sense C. 3, and both of us, both of whom, etc., at sense

To use to whom both in your sentence:

Coffee was paid for by John and Anna, to whom both I and my companion express sincere gratitude.

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It could not be “to whom both”. If the clause were not a relative clause, you would write something like “I express my sincere gratitude to both of them.” You can turn it into a relative clause by replacing “them” with the relative pronoun “whom”, which needs to be “fronted”—placed at the start of the clause. But if you wrote *”whom I express my sincere gratitude to both of,” the preposition “of” would be “stranded”—something that is possible in some sentences, but not I think here. So other things move with the relative pronoun (this is called “pied-piping”). If we just moved the “of”, we would get “of whom I express my sincere gratitude to both.” This sounds a bit better to me, but I think that it still sounds worse than the option you mentioned with pier-piping of the phrase “to both of whom”—“to both of whom I express my sincere gratitude.”

Pied-piping is fairly formal, but you said that the context of the sentence is supposed to be formal.

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