Some Power Series Expansions for Monogenic Functions in

272 D. Peña Peña and F. Sommen CMFT
Proof. Let z = (r 1 − 1) + (r 2 − 1)i and put B k,l (x 1 , x 2 ) = Z k Z l A k,l (x 1 , x 2 ), then
we have that
B k,l (x 1 , x 2 ) = z k¯z l (1 − iω 1ω 2 )
2
Therefore
∂ x B k,l (x 1 , x 2 ) = 2kZ k−1 Z l ω 1 A k,l (x 1 , x 2 )
A k,l (x 1 , x 2 ) + ¯z k z l (1 + iω 1ω 2 )
A k,l (x
2
1 , x 2 ).
+Z k Z l (ω 1 ∂ r1 A k,l (x 1 , x 2 ) + ω 2 ∂ r2 A k,l (x 1 , x 2 )
( (p1 − 1)ω
+
1
+ (p ) )
2 − 1)ω 2
A k,l (x
2r 1 2r 1 , x 2 )
2
(
+Z k Z l ω 1
Γ x1 A k,l (x
r 1 , x 2 ) + ω 2
Γ x2 A k,l (x
1 r 1 , x 2 )
2
( (p1 − 1)ω
−
1
+ (p ) )
2 − 1)ω 2
A k,l (x
2r 1 2r 1 , x 2 )
2
and the action of the operator ∂ x on f is given by
∂ x f(x 1 , x 2 ) =
∞∑
k=0
∞∑
Z k Z
(2(k l + 1)ω 1 A k+1,l (x 1 , x 2 )
l=0
+ω 1
(∂ r1 A k,l (x 1 , x 2 ) + 1 )
Γ x1 A l,k (x
r 1 , x 2 )
1
+ω 2
(∂ r2 A k,l (x 1 , x 2 ) + 1 )
Γ x2 A l,k (x
r 1 , x 2 )
2
( (p1 − 1)ω
+
1
+ (p )
2 − 1)ω 2
2r 1 2r 2
)
× (A k,l (x 1 , x 2 ) − A l,k (x 1 , x 2 ))
.
We thus have that the recurrence relation (4) is sufficient for function f to be
monogenic.
Note that for the toroidal expansion of biaxial type the sequence of functions
{A 0,l (x 1 , x 2 )} l≥0 is the initial condition.