##Seriously, 12 bytes
,,1WX■@│+)-1 Outputs an infinite stream, format is b(n) a(n), one pair of outputs per line.
No online link because TryItOnline doesn't do so well with infinite loops.
Explanation:
,,1WX■@│+)-1 ,,1 push a(0), push b(0), push 1 W while loop: X discard the 1 (only used to make sure the while loop always runs) ■ print all stack elements, separated by spaces, without popping @│ swap, duplicate entire stack +) push a(n) + b(n) (a(n+1)) and move it to the bottom of the stack - push a(n) - b(n) (b(n+1)) 1 push 1 to make sure the loop continues