gleichfalls
likewise
A formal and polite response to a wish, meaning 'I wish you the same'; likewise.
„Ich wünsche Ihnen ein schönes Wochenende!“ – „Danke, gleichfalls!“
'I wish you a nice weekend!' – 'Thank you, likewise!'

likewise
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likewise
A formal and polite response to a wish, meaning 'I wish you the same'; likewise.
„Ich wünsche Ihnen ein schönes Wochenende!“ – „Danke, gleichfalls!“
'I wish you a nice weekend!' – 'Thank you, likewise!'

The word is made up of 'gleich' (the same) and 'falls' (in the case). So it literally means 'in the same case' and expresses that you wish someone the same thing they have just wished you.
„Eine gute Heimfahrt!“ – „Danke, gleichfalls!“
'Have a good trip home!' – 'Thank you, likewise!'
'Gleichfalls' (likewise) is a short, independent response. It is used almost exclusively as a direct reaction to a wish from another person.
„Ich wünsche dir einen schönen Tag!“ – „Gleichfalls!“
'I wish you a nice day!' – 'Likewise!'
This word is considered polite and somewhat more formal. It is an appropriate and respectful response in almost any situation.
„Frohe Feiertage, Herr Meier.“ – „Vielen Dank, Frau Schmidt, gleichfalls.“
'Happy holidays, Mr. Meier.' – 'Thank you very much, Ms. Schmidt, likewise.'
also / even
'auch' and 'gleichfalls' can both express that something applies in the same way to another person or thing. However, 'auch' is much more general and fits in many sentences when you are adding something. 'gleichfalls' is used mainly as a polite response to a wish, for example in greetings or good wishes. Learners may confuse the words because both can mean 'likewise' or 'also'. In everyday language, 'auch' usually sounds more natural; 'gleichfalls' sounds more polite and a little more formal.

„Ich wünsche Ihnen ein schönes Wochenende.“ – „Danke, gleichfalls.“
'I wish you a nice weekend.' – 'Thank you, likewise.'
Nach der Arbeit kaufe ich auch noch Brot für morgen.
After work, I also buy some bread for tomorrow.