ABSTRACT
In the previous sections we made use of x0,…,xn and L0,…,Ln to define x n * , L n * https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003066811/2eb71492-b21d-46e6-a523-e21d0fbd218b/content/inequ4_1.tif"/> , Rn,Mn and Kn. Of course similar definitions can be given by starting with Li instead of L0 (that is using Li,…,Li+n) and with xj instead of x0 (that is using xj,…,xj+n). The corresponding elements will be respectively denote by x n ( i,j ) , L n ( i,j ) , R n ( i,j ) , M n ( i,j ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003066811/2eb71492-b21d-46e6-a523-e21d0fbd218b/content/inequ4_2a.tif"/> and K n ( i,j ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003066811/2eb71492-b21d-46e6-a523-e21d0fbd218b/content/inequ4_2b.tif"/> . The case i = j = 0 corresponds to what was done above. The various families { L n ( i,j ) , x m ( i,j ) } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003066811/2eb71492-b21d-46e6-a523-e21d0fbd218b/content/inequ4_2c.tif"/> obtained for various values of i and j are called adjacent biorthogonal families. The aim of this section is to provide recurrence relations between adjacent x n ( i,j ) , L n ( i,j ) , R n ( i,j ) , M n ( i,j ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003066811/2eb71492-b21d-46e6-a523-e21d0fbd218b/content/inequ4_2d.tif"/> and K n ( i,j ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003066811/2eb71492-b21d-46e6-a523-e21d0fbd218b/content/inequ4_2e.tif"/> . Such relations will be useful in applications for their practical computation. For this purpose we shall make use of two determinantal identities named after Sylvester and Schweins. They are classical identities which have been recently proved to hold for determinants whose first (or last) row (or column) contains elements of a vector space, all the other entries being scalars [23]. They are given in appendix 3.
