Some Power Series Expansions for Monogenic Functions in

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270 D. Peña Peña and F. Sommen CMFT An interesting particular case of the toroidal expansion is the case where the coefficients do not depend on the variable x 0 and satisfy the symmetric relation A k,l (x) = A l,k (x). With this assumption we can calculate explicitly using (3) the coefficients of series (2). Indeed, they are determined by the formula A k,l (x) = (−1)k+l 2 k+l k! l! ∂k+l x A 0,0 (x). Substituting the above in (2) gives ∞∑ k∑ f(x 0 , x) = Z k−l Z l A k−l,l (x) = = = k=0 ∞∑ k=0 k=0 l=0 (−1) k 2 k k! k∑ l=0 k! (k − l)! l! Zk−l Z l ∂ k xA 0,0 (x) ∞∑ (−1) k 2 k k! (Z + Z)k ∂xA k 0,0 (x) ∞∑ (−x 0 ) k ∂ k k! xA 0,0 (x) k=0 which is the classical Cauchy-Kowalewski extension (1). We can also solve the recurrence formula (3) if the coefficients satisfy the relation A k,l (x 0 , x) = (−1) k+l A l,k (x 0 , x). In this case we obtain that ⎧ (−1) ⎪⎨ l 2 A k,l (x 0 , x) = k+l k! l! D x(P x D x ) (k+l−1)/2 A 0,0 (x 0 , x) if k + l is odd, ⎪⎩ (−1) l 2 k+l k! l! (P xD x ) (k+l)/2 A 0,0 (x 0 , x) if k + l is even, where the differential operator P x is defined by P x g = ∂ x0 g + ω∂ r g + 1 r Γ x(ωg). Then we obtain the following expansion ∞∑ 1 f(x 0 , x) = 2 2k (2k)! (Z − Z)2k (P x D x ) k A 0,0 (x 0 , x) k=0 ∞∑ 1 + 2 2k+1 (2k + 1)! (Z − Z)2k+1 D x (P x D x ) k A 0,0 (x 0 , x) k=0 ∞∑ k (r − 1)2k = (−1) (P x D x ) k A 0,0 (x 0 , x) (2k)! k=0 ∞∑ + (−1) k (r − 1)2k+1 ω D x (P x D x ) k A 0,0 (x 0 , x). (2k + 1)! k=0
7 (2007), No. 1 Some Power Series Expansions for Monogenic Functions 271 This expansion may be considered as a kind of Cauchy-Kowalewski extension for the cylinder r = 1. Now we investigate the generalization of the previous theorem to the biaxially symmetric case. To that end we split up R m as R m = R p 1 ⊕ R p 2 , p 1 + p 2 = m, yielding p 1 p ∑ ∑ 2 x = x 1 + x 2 , x 1 = e j x j , x 2 = and accordingly j=1 j=1 e p1 +jx p1 +j p 1 p ∑ ∑ 2 ∂ x = ∂ x1 + ∂ x2 , ∂ x1 = e j ∂ xj , ∂ x2 = e p1 +j∂ . xp1 +j Introducing spherical coordinates on R p 1 and R p 2 respectively, i.e. we thus have that where j=1 j=1 x 1 = r 1 ω 1 , r 1 = |x 1 |, ω 1 ∈ S p 1−1 x 2 = r 2 ω 2 , r 2 = |x 2 |, ω 2 ∈ S p 2−1 ∂ x = ω 1 ( ∂ r1 + 1 r 1 Γ x1 ) + ω 2 ( ∂ r2 + 1 r 2 Γ x2 ) Γ x1 = −x 1 ∧ ∂ x1 and Γ x2 = −x 2 ∧ ∂ x2 . We call a toroidal expansion of biaxial type a convergent series of biaxial type around S p1−1 × S p2−1 of the form ∞∑ ∞∑ f(x 1 , x 2 ) = Z k Z l A k,l (x 1 , x 2 ), Z = (r 1 − 1) + (r 2 − 1)ω 1 ω 2 . k=0 l=0 We thus obtain the following generalization of Theorem 1. Theorem 2 (Toroidal expansion of biaxial type). A sufficient condition for f to be monogenic is given by ( ω A k+1,l (x 1 , x 2 ) = 1 (ω 2(k + 1) 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 ) (4) ( 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 )) , k, l ≥ 0.
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