His tradition might seem eccentric, but for the 67-year-old obstetrician, who has been delivering babies for over 30 years, his unique welcome is a way of celebrating the individual.
"I've delivered thousands and thousands of babies," he says. "When I'm singing to those babies, I think: 'I'm singing to a future important person'."
Dr Andrew-Jaja, who is originally from Nigeria, says he picked up the custom from an older doctor while he was still a young resident in the 1970s.
"When he was about to retire, he asked me: 'Andy, do you sing to your babies?'," Dr Andrew-Jaja recalls. "So I took it over, he passed on the baton."
Since that day, the doctor has been every newborn he delivers at the Magee-Womens Hospital in Pittsburgh, Pennsylvania with a song to celebrate the miracle of life.
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In fact, he's been doing it for so long that he recently delivered a little boy named Ethan whose father was also born under Dr Andrew-Jaja's care and received his own delivery room ditty.
Babies are guaranteed an impressive performance, too. Dr Andrew-Jaja, who has three adult children himself, is on the board of Pittsburgh's Mendelssohn Choir and sings solos at his local church.
"I love Dr. Jaja. We love to hear him sing," nurse Jaime Ford told the Pittsburgh Tribune-Review, whose own children were delivered by Dr Andrew-Jaja. "They love it - every patient I know."
His favourite song choice in the maternity ward? 'Happy Birthday', of course.
But his repertoire is varied: new parents and nurses might be invited to join the Singing Doctor for a rendition of What A Wonderful World, for instance, a song which has a particular resonance for Dr Andrew-Jaja.
"It's a beautiful world we live in. You forget about all the crises in the world for a moment, when you see that miracle of life in front of you."