Monty Alexander likes to tell the story of how, as a 19-year-old pianist fresh out of Jamaica in the 1960s, he ran into Frank Sinatra in Miami – and how the relationship continued with many boozy all-night jams after Sinatra's friend Jilly Rizzo set him up at his New York club. Now 70, Alexander splits his work between vivacious jazz-reggae and the kind of canny trio-jazz that suggests a Sinatra lookalike is about to set a bourbon on the piano and start crooning.

The trio-jazz opened Alexander's week of shows at Ronnie Scott's, the virtuoso effortlessly delivering standards and originals with long-serving bassist Hassan Shakur and the UK's Steve Brown expertly filling in for the trio's flight-delayed regular drummer. Alexander's jazz-reggae sextet, the Harlem-Kingston Express, will take over from Thursday.

Monday's show began with a quiet double-bass vamp and a swishing Latin pulse from Brown, to which the leader added embroidery – downward-diving low runs, stabbed chords, trills and skipping treble melody – with plenty of space between. The followup was a mid-tempo blues with a quote-packed bass solo. And the pianist's Caribbean-inspired Renewal, although introspective at first – unfolding in soft piano-string strummings, porous chords and bowed-bass hums – soon snapped into funk, swarming arpeggios and another wry bass break in which Shakur, quoted The Pink Panther, Eleanor Rigby and These Boots Were Made for Walking. But serious business followed. Alexander silenced the room on a luxurious ballad that eased into a bossa. Then he developed How High the Moon as tugging counterpoint between bright right-hand trills and mobile low-end melodies – straying from the theme into tumbling improvisations, but periodically pulled listeners back to the hook as those great jazz entertainers Ahmad Jamal and Sonny Rollins do. Throughout his long career, Monty Alexander has never been that far out of their class.

• Until 21 June. Box office: 020-7439 0747. Venue: Ronnie Scott's, London.