Paperfolding infinite products and the gamma function

Taking the product of (2n+1)/(2n+2)(2n+1)/(2n+2) raised to the power +1 or −1 according to the n-th term of the Thue–Morse sequence gives rise to an infinite product P   while replacing (2n+1)/(2n+2)(2n+1)/(2n+2) with (2n)/(2n+1)(2n)/(2n+1) yields an infinite product Q, whereP=(12)+1(34)−1(56)−1(78)+1⋯ andQ=(23)+1(45)−1(67)−1(89)+1⋯ Though it is known that P=2−1/2P=2−1/2, nothing is known about Q. Looking at the corresponding question when the Thue–Morse sequence is replaced by the regular paperfolding sequence, we obtain two infinite products A and B, whereA=(12)+1(34)+1(56)−1(78)+1⋯ andB=(23)+1(45)+1(67)−1(89)+1⋯ Here nothing is known for A, but we give a closed form for B   that involves the value of the gamma function at 1/4. We then prove general results where (2n+1)/(2n+2)(2n+1)/(2n+2) or (2n)/(2n+1)(2n)/(2n+1) are replaced by specific rational functions. The corresponding infinite products have a closed form involving gamma values. In some cases there is no explicit gamma value occurring in the closed-form formula, but only trigonometric functions.