Structural properties of Riordan matrices and extending the matrices

We consider an infinite lower triangular matrix L=[ℓn,k]n,k∈N0L=[ℓn,k]n,k∈N0 and a sequence Ω=(ωn)n∈N0Ω=(ωn)n∈N0 called the (a,b  )-sequence such that every element ℓn+1,k+1ℓn+1,k+1 except lying in column 0 can be expressed asℓn+1,k+1=∑i=0⌊(n-k)/m⌋ωiℓn-ai,k+bi,ω0∉0where a and b   are integers with a+b=m>0a+b=m>0 and b⩾0b⩾0. This concept generalizes the A-sequence of a Riordan matrix. As a result, we explore several structural properties of Riordan matrices by means of (a,b)-sequences. In particular, if b < 0 then this leads to an extended Riordan matrix which is a bilaterally infinite matrix.