Frank Foster, a Grammy Award-winning musical arranger and jazz saxophonist who wrote one of the Count Basie Orchestra’s classic tunes of the 1950s and later became the band’s leader, died Tuesday at his home in Chesapeake, Va. He was 82 and had complications from kidney failure.

Foster joined the Basie big band as a tenor saxophonist in 1953, just as the group was returning to its glory established 20 years before. He was a spirited soloist whose bold playing brought a bebop-accented flair to the orchestra, and he became one of the ensemble’s leading writers and arrangers.

His best-known composition was “Shiny Stockings,” a relaxed tune with a propulsive sense of swing that builds to a brassy curtain of shimmering trumpets. It has been a staple of the Basie songbook since its debut on the landmark 1956 recording “April in Paris.”

The first time the band played “Shiny Stockings,” Foster recalled in a 2008 interview with the National Endowment for the Arts, “it sounded like a 43-car pileup.” The problems were ironed out, with Basie featured in a characteristically understated piano solo.

“He just didn’t want it to be too full of notes and ‘too busy,’ as he called it,” Foster said of Basie’s musical conception. “It always had to swing.”

In 1965, Foster left the orchestra to work with other musicians, taught at several colleges and formed a big band.

Basie died in 1984, and Foster returned to the orchestra two years later as director. He led the group until 1995. His big-band arrangements won Grammys in 1987 and 1988.