Jack died Thursday, surrounded by his family ... including wife Heather Ann and son Matthew Hucko, plus his cousins and some friends ... according to his band.

Russell family spokesperson K.L. Doty tells TMZ … Jack died as a result of Lewy body dementia and multiple system atrophy.

Jack announced his diagnosis last month when he retired from touring ... and both conditions affect motor skills.

Jack's band is remembering him for his "sense of humor, exceptional zest for life, and unshakeable contribution to rock and roll where his legacy will forever thrive."

Jack sang for Great White from 1981 until 1996, when he tried his hand at a solo career. After his first solo album came out, he went back to Great White until 2001, when the brand broke up.

From there, Jack changed the band name to Jack Russell's Great White ... the band involved in the 2003 Rhode Island nightclub fire that killed 100 people.

Jack was 63.