The elements of finite order in the Riordan group over the complex field

An element of finite order in the Riordan group over the real field must have order 1 or 2. If we extend all the entries to be complex numbers then it may have any finite order. In the present paper, we investigate the elements of finite order of the Riordan group over the complex field. This notion leads us to define an element of pseudo order k⩾2k⩾2 and k-pseudo involution, respectively. It turns out that the inverse of a k-pseudo involution only differs from it in signs. We clarify some relationship between the elements of pseudo order k and k-pseudo involutions. In particular, k  -pseudo involutions for k≡0k≡0(mod4) are characterized by a single sequence. The subgroups of the Riordan group formed by the elements having pseudo order of a prime power prpr are also introduced.