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 July 26 at his home in Chesapeake, Va. He was 82 and had complications from kidney failure.

Mr. Foster joined the Basie big band as a tenor saxophonist in 1953, just as the group was returning to its glory established two decades before. He was a spirited soloist whose bold playing brought a fresh 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,” Mr. 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,” Mr. Foster said of Basie’s musical conception. “It always had to swing. That was the one basic ingredient that always had to be there: to swing.”

In addition to countless performances by the Basie band, “Shiny Stockings” has been recorded more than 300 times and has been called “essential for any jazz collection” by critic Scott Yanow.

Mr. Foster had several other compositions that became staples of the Basie repertoire, including “Blues in Hoss’ Flat,” “Blues Backstage” and “Down for the Count.” The band’s 1960 album “Easin’ It” consisted of tunes written and arranged by Mr. Foster.

He and fellow tenor saxophonist Frank Wess made a compelling team in the Basie band, with their contrasting styles. Wess excelled in ballads and hewed to a lyrical prototype established by Lester Young in the 1930s. Mr. Foster had a more vigorous and harmonically adventurous approach, adopting many of the musical advances of Charlie Parker and the bebop generation.

The saxophonists became known as the Two Franks, and composer Neal Hefti wrote a tune by that name. Wess and Mr. Foster continued to perform and record together for more than 40 years.

In 1965, Mr. Foster left the Basie orchestra to work with drummer Elvin Jones and other musicians, taught at several colleges and formed a big band called the Loud Minority. He composed a jazz suite for the 1980 Winter Olympics and arranged music for Sarah Vaughan, Frank Sinatra, Dizzy Gillespie, Tony Bennett and other entertainers.

After Basie died in 1984, Mr. Foster returned to the orchestra two years later as director. He led the group until 1995, typically spending 250 nights a year on the road. His big-band arrangements won Grammy Awards in 1987 and 1988.

Mr. Foster played the band’s traditional favorites, but he also introduced many new and more thorny works that were not as popular.

“The old school says they’re tired of hearing the new stuff,” he told the Los Angeles Times in 1995. “They don’t understand that musicians have to be constantly challenged or you lose them. . . . I’m on the side of those who like newer things.”

Frank Benjamin Foster III was born Sept. 23, 1928, in Cincinnati. He often went to operas as a child with his mother — once seeing Paul Robeson in the title role in Verdi’s “Otello” — and began playing the clarinet at age 11. He later turned to the saxophone and, in high school, led a 12-piece jazz band.

After attending Wilberforce University in Ohio, Mr. Foster spent two years in Detroit’s lively jazz scene before serving in the Army during the Korean War. On the recommendation of singers Vaughan and Billy Eckstine, he joined the Basie band soon after his military discharge in 1953.

That year, Mr. Foster recorded the first of more than 25 albums under his own name. He had offers to join groups led by Thelonious Monk and Miles Davis, but he chose to stay with Basie.

After moving to Chesapeake, Mr. Foster suffered a severe stroke in 2001 that left him unable to play the saxophone, but he remained active as a composer and arranger.

His first marriage, to Vivian Wilson, ended in divorce. Survivors include his wife of 45 years, Cecilia Jones Foster of Chesapeake; two children from his first marriage; two children from his second marriage; and six grandchildren.

In 2002, Mr. Foster received the country’s highest honor for jazz musicians when he was named a National Endowment for the Arts Jazz Master.

“I’ll always be about one thing,” Mr. Foster said in 1995. “I’m a big-band fanatic. The Basie band is about total swing with a bluesy concept. Its key is its simplicity.”