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(
r
−
2)(
r
−
1)r
P
r,
2
(
q,p
)=
rq
r−
1
p
q
r−
3
p
3
+
−
···
,
(D.23)
3
∗
2
1)
s
+1
r
q
r−
(2
s−
1)
p
2
s−
1
.
[
r
+1
2
]
P
r,
2
(
q,p
)=
(
−
(D.24)
2
s −
1
s
=1
[
2
] denotes the largest natural number
≤
r
r
+1
2
,
respectively. In summarizing, the general solution of the first Laplace-Beltrami equation is given
by (
D.25
).
2
,
[(
r
+1)
/
2] the largest natural number
≤
1)
s
r
q
r−
2
s
p
2
s
+
[
2
[
r
+1
2
]
]
N
N
1)
s
+1
x
=
α
0
+
α
1
q
+
β
1
p
+
α
r
(
−
β
r
(
−
2
s
r
=2
s
=0
r
=2
s
=1
r
2
s
q
r−
2
s
+1
p
2
s−
1
.
(D.25)
−
1
Next, we implement the terms
x
=
P
r,
1
(
q,p
)and
x
=
P
r,
2
(
q,p
) in the d'Alembert-Euler equa-
tions (Cauchy-Riemann equations) (
D.26
). Obviously, there hold the polynomial relations (
D.27
).
(
−
1)
s
(
r −
2
s
)
r
q
r−
2
s−
1
p
2
s
=
(
−
1)
s
(
r −
2
s
)
r
[
2
[
r
+
2
−
1]
]
∂P
r,
1
∂q
=
2
s
2
s
s
=0
s
=0
q
r−
2
s−
1
p
2
s
,
1)
s
2
s
r
q
r−
2
s
p
2
s−
1
=
1)
s
2
s
r
q
r−
2
s
p
2
s−
1
,
[
2
[
2
]
]
∂P
r,
1
∂p
=
(
−
(
−
2
s
2
s
s
=0
s
=1
(
−
1)
s
+1
(
r −
2
s
+1)
r
q
r−
2
s
p
2
s−
1
=
(
−
1)
s
+1
2
s
r
[
r
+1
2
[
2
]
]
∂P
r,
2
∂q
=
2
s
−
1
2
s
s
=1
s
=1
q
r−
2
s
p
2
s−
1
,
(D.26)
1)
r
q
r−
2
s
+1
p
2
s−
2
=
2
s
)
r
[
r
+1
2
[
r
+1
2
]
]
−
1
∂P
r,
2
∂p
1)
s
(
r
1)
s
+1
(2
s
=
(
−
−
(
−
−
2
s −
1
2
s
s
=1
s
=0
q
r−
2
s
−
1
p
2
s
,
∂P
r,
1
∂q
=
∂P
r,
2
∂P
r,
1
∂p
∂P
r,
2
∂q
∂p
,
=
−
.
(D.27)
The general solution of the d'Alembert-Euler equations (Cauchy-Riemann equations) subject
to the integrability conditions of the harmonicity type thus can be represented by
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