Authenticity, not imitation – that’s what Sandy Hackett wants you to know about his show celebrating the legendary Las Vegas Rat Pack buddies: Frank Sinatra, Dean Martin, Sammy Davis Jr. and Joey Bishop.

There are several Rat Pack shows currently touring the country, Hackett notes, but none has the personal connection to the subject like “Sandy Hackett’s Rat Pack Show,” which opens Tuesday as the final show of Theater League’s 2015-2016 season.

Hackett is the son of the late comic Buddy Hackett, and he grew up with the hip, swingin’, ring-a-ding Rat Pack buddies dropping by to visit his father during the 1950s and 1960s.

“Joey was a close personal friend. He was ‘Uncle Joey’ to me, and I knew him my whole life. I met Frank and Dean, but I didn’t have as close a relationship with them as I did Joey,” says Hackett, who eventually worked with all of them as a stand-up comic following in his father’s footsteps.

So when he got the idea for a tribute show back in 2002, it seemed like a natural because of his own experiences and recollections. It ran successfully in Las Vegas hotel show rooms for five years.

But after Hackett married singer/music producer Lisa Dawn Miller, daughter of famed songwriter Ron Miller (“For Once in My Life,” “Touch Me in the Morning,” “I’ve Never Been to Me”), in 2005, the two decided to revise the show.

“We wanted to make it less ‘tribute-y’ and more theatrical,” says Miller, who has performed one-woman shows from her father’s book of 400-plus songs. “I feel connected because all of the members of the Rat Pack recorded my father’s song ‘For Once in My Life.’ Frank toured with it for the rest of his life. It was a huge song for him.”

The resulting collaboration, renamed “Sandy Hackett’s Rat Pack Show,” debuted in 2009 and has been touring the country since. The key, they say, is not impersonation of familiar and beloved celebrities, but capturing their essence.

“I don’t think that the hardest thing in the show is how to portray an individual character, but in capturing the chemistry and the relationships that those guys had with each other,” Miller says. “Their bond was magical. That’s why the Rat Pack endured all those years.”

While Hackett and Miller are writers and producers of the show, they also are performers. Hackett plays Joey Bishop and Miller plays a composite character called “Frank’s One Love,” modeled mainly after his tempestuous second wife, Ava Gardner, who used to joke that the only time they weren’t fighting was on the way to the bedroom.

But Miller says her character also includes elements of Frank’s first wife, Nancy, mother to Frank’s three children (Nancy, Frank Jr. and Tina), and Barbara, his fourth and final wife (after two years with Mia Farrow in the 1960s), who was with him from 1976 until his death in 1998.

As Miller describes it, Nancy stuck with Sinatra through thick and thin, Barbara tamed him and was with him to the end, but Ava was the “one who stuck in his heart, who drove him crazy.” In recognition of Sinatra’s grief over Gardner’s death, Miller says the show features her father’s original song “The Things I Should Have Said.”

The show creators have a stable of singer/actors to draw from for the other characters – Sinatra, Martin and Davis.

Hackett, who got his showbiz start at age 11 on “Rowan & Martin’s Laugh-In,” dutifully got his college degree in hotel administration from the University of Nevada Las Vegas (UNLV). But he ventured into creating a weekly comedy showcase for the Sahara Hotel in Las Vegas and then opened his own comedy club. He dabbled in movies, including appearing with Frank, Dino and Sammy in “Cannonball Run II,” and TV (“The Nanny,” “Joan Rivers and Friends Celebrate Heidi Abromowitz,” “Deadly Games”).

But the lure of live audiences brought him back to the comedy stage and eventually to the Rat Pack Show, where he met his wife while playing Joey Bishop in the first iteration. They now have two children.

Although the show is scripted, Miller says there is room for surprises.

“It’s a book show but, like the real Rat Pack, you never really know what’s going to happen because Sandy (as Bishop) is so brilliant at the improv aspect,” she says.

But the music, says Hackett, is indelible and forever.

“Frank would have been 100 years old last December. There are so many people around today who patterned themselves after him, like Michael Buble and Harry Connick Jr.,” Hackett says. “The music of Frank Sinatra will be around forever.”