The halves of a 3-dimensional Riordan array

Abstract

For a Rirodan array, its vertical half and horizontal half are studied separately before. In this paper, we propose the (m1,s1,r,m2,s2,l)-halves of a 3-dimensional Riordan array, which are natural generalizations of the skew halves of a usual Riordan array in [21]. Let G=[gi,j,k]i,j,k≥0 be a 3-dimensional Riordan array, its (m1,s1,r,m2,s2,l)-halves H=[hi,j,k]i,j,k≥0 are defined byhi,j,k=g(m1+1)i+(s1−2)j+r,m1i+(s1−1)j+r,m2i+(s2−1)j+lk. We obtain that the (m1,s1,r,m2,s2,l)-halves of a 3-dimensional Riordan array are all 3-dimensional Riordan arrays. We also consider the (m1,s1,r,m2,s2,l)-antecedents of a 3-dimensional Riordan array.