On this occasion, they are joined by “the first lady of jazz”, Ella Fitzgerald (the serene and mellifluous Nicola Emmanuelle).
It’s thrilling to hear so many of their signature songs, including You Make Me Feel So Young, My Kinda Town, Mr Bojangles, That’s amore and the inevitable My Way, complete with a big band accompaniment.
All three actors offer impeccable impersonations, whether of Sinatra’s supreme cool, Davis’s hyperactive jiggling or Martin’s laidback charm.
Phillips, the undoubted star of the evening, echoes Sinatra’s vocal style so perfectly that my companion was convinced he was miming (he wasn’t).
But, for all the musical delights of Mitch Sebastian’s production, the linking material is excruciating.
The racist, sexist and homophobic banter may be authentic but it should be cut.
Theatre Royal, Haymarket, London until February 3, then touring.
Tickets: 0844 871 7615/atgtickets.com; £22-£60.50