Gamma series associated to elements satisfying regularized double shuffle relations

Let K be a field of characteristic 0, and R be a commutative K-algebra. Let Φ(x0,x1) be an element in R《x0,x1》 with regularized double shuffle relations. We define a gamma series ΓΦ(s)∈1+s2R〚s〛 associated to Φ. We prove that the associated beta series is just the image of ΦY(x0,x1) in the commutative formal power series ring R〚x0,x1〛, where if Φ=1+Φ0x0+Φ1x1, then ΦY=1+Φ1x1. We also give some equivalent conditions for the reflection formula of the gamma series ΓΦ(s).