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Table 22.6 Odd bivariate harmonic polynomials: POLY ξ ( ODD : q,l )
Basiselement : q r− 3 l : ξ r− 1 , 1 ( r
1) ( r
2) + 3
2 ξ r− 3 , 3 =0
Basiselement : q r− 5 l 3 : ξ r− 3 , 3 ( r
3) ( r
4) + 5
4 ξ r− 5 , 5 =0
.
Basiselement : q 3 l r− 5 : ξ 5 ,r− 5 5
4+ ξ 3 ,r− 3 ( r
3) ( r
4) = 0
Basiselement : ql r− 3 : ξ 3 ,r− 3 3
2+ ξ 1 ,r− 1 ( r
1)( r
2) = 0
Corollary (recurrence relation for odd bivariate harmonic polynomials).
k ( k
1) ξ k,r−k +( r
k +2)( r
k +1) ξ k− 2 ,r−k +2 = 0
(22.63)
k ( k
1)
ξ k− 2 ,r−k +2 =
k +1) ξ k,r−k
(22.64)
( r
k +2)( r
( r
k +2)( r
k +1)
ξ k,r−k =
ξ k− 2 ,r−k +2
(22.65)
k ( k
1)
( k +1)( k +2)
( r − k )( r − k − 1) ξ r−k− 2 ,k +2
ξ r−k,k =
(22.66)
( kodd )
End of Corollary.
Table 22.7 Bivariate harmonic polynomials factorial representation
ξ r, 0 = η r, 0 := 1 = r
r− 1 , 1 = η r− 1 , 1 := r = r
(22.67)
0
1
r
2
r ( r
1)
ξ r− 2 , 2 = η r− 2 , 2 =
=
(22.68)
2
r
3
r ( r
1) ( r
2)
ξ r− 3 , 3 = η r− 3 , 3 =
=
(22.69)
3
2
= r
4
r ( r
1) ( r
2) ( r
3)
ξ r− 4 , 4 = η r− 4 , 4 =
(22.70)
4
3
2
= r
5
r ( r
1) ( r
2) ( r
3)( r
4)
ξ r− 5 , 5 = η r− 5 , 5 =
(22.71)
5
4
3
2
Corollary (factorial representation of the coecients of bivariate harmonic polynomials).
The factorial representation of the coecients of bivariate harmonic polynomials is
k even
 
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