The series of photos -- featuring Paris and Carter Reum -- include plenty of kisses and snuggling of their baby boy, who they've named Phoenix Barron Hilton Reum.

Paris Hilton shared Phoenix's name on her "This is Paris" podcast, telling fans she came up with the name years ago when she was searching cities, countries and states ... hoping to find a name to go with Paris and London -- a name she'd like to name her future daughter.

She says, "More important, it's the bird that flames out and then rises from the ashes to fly again. I want my son to grow up knowing that disaster and triumph go around and come around throughout our lives and that this should give us great hope for the future."

Mama Paris also revealed she and Carter kept Phoenix a secret from everyone until he was about a week old ... for the sake of cherishing the moment for themselves.