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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
,ξ
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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