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2.5. A closed-form expression for f d 57Whence the following definition.Definition 2.5.12. For i ≥ 0, let us denote by A i the quantityA i =∑∑ i ij=0 A 0(i, j)4 −j−1 j=0(3/4) i+1 =The first few values for A i are given in Table 2.3.〈 ij〉4 −j−1(3/4) i+1 .Table 2.3: Values of A i for 0 ≤ i ≤ 7i = 0 1 2 3 4 5 6 7A i = 1/3 4/9 20/27 44/27 380/81 4108/243 17780/243 269348/729Then, the following corollary of Lemmas 2.5.10 and 2.5.11 gives a simple expression of thesum.Corollary 2.5.13. For k ≥ 1,β−1∑e k 4 −e = A k −e=1So, for k ≥ 1, the sum becomes( k−1 ∑( )kk−i βi)A i 4 −β − 4A 0 β k 4 −β .i=0β−1∑k−1∑( k(β − e) k (4 e − 1) = A k 4 β − A k−i βi)i − 4A 0 β k − 1k + 1e=1i=0k∑( k= A k 4 β − A k βi)k−i − 4A 0 β ki=1− 1k + 1 βk+1 + 1 2 βk −i=2= A k (4 β − 1) − 1k + 1 βk+1 − 5 6 βk −k∑( ) k + 1(−1) 1i=1 B i β k+1−iii=0k∑( ) k + 1B i β k+1−iik−1∑i=1( ( kA i +i) B )i+1β k−i .i + 1According to the above discussion about the different sums on e i , Π d X can be expressed as∑ j0Π d X = 4 − +j 1 ∏β i4 β j− 1 − 3βi=j 0 +1 j∏ (Akj (4 β j− 1) − ΘX,j)d3where=∏{j 0 +1≤j≤j 0 +j 1 |k j =0}{j 0 +1≤j≤j 0 +j 1 |k j =0}1 − 4 −β j− 3β j4 −β jΘ d X,j = 1k j + 1 βkj+1 j + 5 6 βkj j +3{j 0 +1≤j≤j 0 +j 1 |k j ≠0}∏{j 0 +1≤j≤j 0 +j 1 |k j ≠0}k∑ j−1i=1(Akj (1 − 4 −β j) − Θ d X,j4 −β j ) .( ) (kjA i + B )i+1β kj−i .i i + 1Hence, Π d X , Σd X and T X d are all as stated in Proposition 2.5.1. The values of the degrees of themultivariate polynomials follow from the above expressions.