irrtümlich
mistakenly
Based on an error or mistake; happening by accident.
Ich habe Ihnen irrtümlich die falsche Rechnung geschickt.
I mistakenly sent you the wrong invoice.

mistaken
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mistakenly
Based on an error or mistake; happening by accident.
Ich habe Ihnen irrtümlich die falsche Rechnung geschickt.
I mistakenly sent you the wrong invoice.

'Irrtümlich' (mistakenly) is derived from the noun 'der Irrtum' (the mistake, the error). When something happens 'irrtümlich' (mistakenly), it is based on a mistake.
Seine Annahme basierte auf einem Irrtum.
His assumption was based on a mistake.
'Irrtümlich' (mistaken) can stand as an adjective before a noun or as an adverb that describes a verb. The adverbial use is very common.
Das war eine irrtümliche Überweisung.
That was a mistaken transfer.
The word 'irrtümlich' (mistakenly) is often used in written language and in more formal contexts, for example in business correspondence or with public authorities.
Der Betrag wurde irrtümlich von Ihrem Konto abgebucht.
The amount was mistakenly debited from your account.
wrong / false
'falsch' and 'irrtümlich' can both express that something is not right. 'falsch' is more general: an answer, a number, or a decision can be wrong. It can also mean that something is not suitable. 'Irrtümlich' emphasizes the reason: something happened by accident or because of a mistake. Learners may confuse the words because both have to do with errors. If you only want to say that something is not correct, 'falsch' fits. If you want to say that someone was mistaken or that something was done by accident, 'irrtümlich' is more precise.

Der Kunde hat irrtümlich zweimal dieselbe Rechnung bezahlt.
The customer mistakenly paid the same invoice twice.
Für das Training morgen haben wir die falschen Schuhe eingepackt.
We packed the wrong shoes for tomorrow's training session.