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Corollary (
POLY
ξ
(1
,l
; 3)).
Given a harmonic polynomial
POLY
ξ
(
q,l
;3) of degree 3. If
ξ
3
,
0
=1
,
2
,
1
=3then
ξ
1
,
2
=
−
3
,
ξ
0
,
3
=
−
1 such that
POLY
ξ
(
q,l
;3)=
q
3
3
q
2
l
+3
ql
2
l
3
−
−
holds.
End of Corollary.
There exists a clever factorial representation of the coecients of bivariate harmonic polyno-
mials we have outlined in Table
22.7
,Eqs.(
22.67
)-(
22.73
) and summarized in a corollary. Due to
the recurrence relations it is sucient to fix
ξ
r,
0
=
η
r,
0
:= 1 and
ξ
r−
1
,
1
=
η
r−
1
,
1
:=
r
: All the other
coecients are already determined according to Eqs.(
22.72
)and(
22.73
).
As soon as we implement “
the clever choice
” of the coecients Eq.(
22.72
) into the bivariate
harmonic polynomials we are led to the explicit representation of the homogeneous bivariate
harmonic polynomials of even and odd type, Eqs.(
22.72
)and(
22.75
), of Table
22.8
. In summary,
we have achieved the basis elements Eq.(
22.76
),
POLY
ξ
(1
,l
;3),andEq.(
22.77
),
POLY
η
(
q,l
;
r
),
of Table
22.9
of the space of
bivariate harmonic functions of degree r
(
rank r
). Table
22.10
is a
print out of base functions of type even and odd up to
degree 10
(rank 10).
Table 22.5
Even bivariate harmonic polynomials:
Δ
POLY
ξ
(
EVEN
:
q,l
)
Basiselement
:
q
r−
4
l
2
:
ξ
r−
2
,
2
(
r
−
2) (
r
−
3) + 4
∗
3
ξ
r−
4
,
4
=0
Basiselement
:
q
r−
6
l
4
:
ξ
r−
4
,
4
(
r
−
4) (
r
−
5) + 6
∗
5
ξ
r−
6
,
6
=0
.
Basiselement
:
q
2
l
r−
4
:
ξ
4
,r−
4
4
∗
3+
ξ
2
,r−
2
(
r
−
2) (
r
−
3) = 0
Basiselement
l
r
−
2
:2
ξ
2
,r−
2
+
ξ
0
,r
r
(
r
−
1) = 0
Corollary (recurrence relation for even bivariate harmonic polynomials).
k
(
k −
1)
ξ
k,r−k
+(
r − k
+2)(
r − k
+1)
ξ
k−
2
,r−k
+2
= 0
(22.59)
k
(
k
+1)
↔
ξ
k−
2
,r−k
+2
=
−
k
+1)
ξ
k,r−k
= 0
(22.60)
(
r
−
k
+2)(
r
−
(
r
−
k
+2)(
r
−
k
+1)
↔
ξ
k,r−k
=
−
ξ
k−
2
,r−k
+2
= 0
(22.61)
k
(
k
−
1)
(
k
+1)(
k
+2)
(
r − k
)(
r − k −
1)
ξ
r−k−
2
,k
+2
= 0
↔
ξ
r−k,k
=
−
(22.62)
(
k even
)
End of Corollary.
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