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Skew analysis over quaternions. I

We introduce a class of rings using which we define the concept of skew regularity for quaternion-valued functions over quaternions. It is shown that the notion of skew regularity coincides with the concept of slice regularity over symmetric slice domains. Known results regarding slice-regular functions over symmetric slice domains are generalized to skew-regular functions over general symmetric domains. Furthermore, we present new results concerning slice-regular functions.


Introduction
This paper consists of two parts: An algebraic part and an analytical part. In the algebraic part, we introduce and study a class of rings which we shall call skew-convex function rings. They can be regarded as generalizations of skew polynomial rings over skew fields. In the analytical part, we use skew-convex function rings over the skew field of quaternions to develop an analytical theory over quaternions which will be called skew analysis over quaternions. It turns that skew analysis over quaternions is closely related to the theory of slice-regular functions as introduced in [6] and later developed in a series of papers. For a detailed account of slice-regular functions, see [5] and references therein.
In Section 2, we introduce the notion of the skew product which is a binary operation on certain sets of functions with values in a fixed skew field (see Definition 2.1). The concept of the skew product is motivated by the product formula for the evaluation of skew polynomials introduced in [10, Theorem 2.7]. We show that the skew product gives rise to near-ring structures on sets of functions with values in the skew field. Restricting our attention to the class of "skew-convex" functions, we arrive at the notion of skew-convex function rings (see Definition 2.2 and Theorem 2.4). In the rest of Section 2, we study invertible elements in skew-convex function rings. A more complete treatment of the algebraic properties of skew-convex function rings will appear elsewhere. Section 3 deals with skew analysis over quaternions. In this section, the notion of skew regularity is introduced (see Definition 3.2) using which we develop an analytical theory for quaternion-valued functions over quaternions. In particular, we define the notion of the skew derivative which is very similar to the classical definition of derivatives (see Definition 3.3). It turns out that skew-regular functions are always skew-convex, a fact with far-reaching consequences. As mentioned above, the concept of skew regularity is closely related to the notion of slice regularity. More precisely, we show that every skew-regular function is slice regular (see Proposition 3.19). More importantly, we will prove that a function defined on a "symmetric slice domain" is skew regular if and only if it is slice regular (see Subsection 3.5 and Theorem 3.20). It is known that the notion of slice regularity is not well-behaved over domains that are not slice domains. The strength of our approach lies in the fact that the concept of skew-regularity is well-behaved even over domains which are not slice domains. One of our main results is that over a general symmetric domain, a function is skew regular if and only if it is symmetrically analytic (see Theorem 3.26). Furthermore, our approach simplifies and clarifies some of the results and features of slice-regular functions.

Rings of skew maps
The theory of skew polynomial rings over skew fields is rich and well-developed. For a detailed account of this theory, see for example [1,Section 1.1]. In [10], the authors defined the evaluation of skew polynomials using which skew polynomials can naturally be considered as maps on the ground skew field. It turns out that the value of the product of two skew polynomials at a given point is not necessarily equal to the product of the values of the skew polynomials at the same point. The correct formula, called the product formula, is given in [10,Theorem 2.7]. The product formula can be regarded as a binary operation which we shall call the skew product. This section will deal with the skew product and its properties. Subsection 2.1 introduces the notions of the skew product and skew-convex functions, and gives their basic properties. In Section 2.2, we study skew-invertible functions, that is, functions which are invertible with respect to the skew product. that a y ∈ Y ; By an orbit, we mean the smallest invariant subset containing some element.
We denote the set of all functions f : X → K by F(X). By abuse of notation, a constant function in F(X), whose value is a ∈ K, is simply denoted by a. Given functions f, g : X → K, let f + g denote the pointwise addition of the functions f and g.
Definition 2.1. The left skew product of the functions f and g as follows The right skew product is defined as follows In this paper, we will mostly work with the left skew product. Therefore, we usually drop the adjective "left" if there is no risk of confusion. We leave it to the reader to formulate and prove similar results for the right skew product.
It is easy to see that (a ⋄ f )(x) = af (x), for every a ∈ K, x ∈ X. We shall henceforth denote a ⋄ f by af . In the following lemma, the proof of which is straightforward, we collect some properties of the skew product.
Lemma 2.1. Let f, h, g : X → K be arbitrary functions. Then: (1) The constant function 1 : x → 1 is a unit for ⋄, that is, that is, the right distributive law (with respect to pointwise addition) holds for the skew product.
The following lemma justifies the introduction of S(X).
holds for all functions f, g : X → K if and only if h is skew convex.
Proof. The result follows from the identity As a consequence of this lemma, we have the following result.
Theorem 2.4. Equipped with the left skew product, the additive group S(X) is a ring with identity.
Proof. The result follows from Lemma 2.3 and the general fact that in any right near-ring R, the set {r ∈ R | r(s 1 + s 2 ) = rs 1 + rs 2 for all s 1 , s 2 ∈ R}, is a ring.
We call S(X), equipped with pointwise addition and the skew product, the ring of skew-convex functions on X determined by the action of K * on X. To introduce a subring of S(X), let us denote the set of all functions f : X → K which are constants on every orbit, by C(X). By Proposition 2.2, we have C(X) ⊂ S(X). Moreover, we have the following proposition whose proof is left to the reader.
Proposition 2.5. The set C(X) is a subring of S(X). Moreover, it is isomorphic to the ordinary ring of functions f : X/K * → K (equipped with pointwise addition and pointwise multiplication). Here, X/K * denotes the set of equivalence classes of the action.

Let us now give some examples.
Example 2.1. If the action of K * on X is trivial, the ring S(X) is just the ring of all functions f : X → K equipped with pointwise addition and pointwise multiplication.
Example 2.2. Let σ : K → K be an endomorphism and δ : K → K be a σ-derivation. Let K[T ; σ, δ] denote the ring of skew polynomials determined by σ and δ. Every elements of K[T ; σ, δ] can uniquely be written as n m=0 a m T m where a m ∈ K with a n = 0. Recall that the identity T a = σ(a)T + δ(a), where a ∈ K, holds in K[T ; σ, δ]. One can define the (σ, δ)-action of K * on K as follows 3) The ring of skew-convex functions determined by this action is denoted by K[σ, δ]. One can verify that there exists a unique ring homomorphism which sends each a ∈ K to the constant function a, and T to the identity function 1 K : K → K. In particular, any skew polynomial P (T ) ∈ K[T ; σ, δ] can, under this homomorphism, be considered as a skew-convex function P : K → K. The reader can verify that the value P (a) of P at a ∈ K coincides with the evaluation map introduced in [10], that is, P (a) is the unique element of K for which we have In the particular case where δ is zero and σ is the identity homomorphism, the value P (a) is computed as follows We conclude this part with a discussion regarding pullbacks. Let Y be a second set on which K * acts (on the left). For a map φ : for all a ∈ K * and x ∈ X. Proposition 2.6. Suppose that φ : X → Y is action-preserving. Then, for any function f ∈ S(Y ), we have φ * (f ) ∈ S(X). Moreover, the map φ * : S(Y ) → S(X) is a homomorphism of rings.
Proof. The fact that φ * (f ) ∈ S(X) for f ∈ S(Y ), follows from the identity The rest of the proof is straightforward.
In particular, if Z ⊂ X is an invariant subset of X, then the restriction map f ∈ S(X) → f | Z ∈ S(Z) is a well-defined homomorphism of rings.

Skew-invertible functions
As before, let K be a skew field and X be a set on which K * acts. A function f ∈ F(X) is called skew invertible if it is invertible with respect to the skew product, in which case, the inverse of f is called its skew inverse and denoted by f −1 . The notion of skew inverse with respect to the right skew product is defined in the obvious way. The skew inverse of a function f with respect to the right skew product will be denoted by f −1 r . Lemma 2.7. If f ∈ S(X) is skew invertible, then the skew inverse of f belongs to S(X).
Proof. By Lemma 2.3, we have The result follows from the following general fact whose proof is left to the reader: Let R be a right near-ring and consider the ring If r ∈ R ′ is invertible in R, then its inverse belongs to R ′ .
Next, we give a characterization of skew-invertible functions. (1) Let f ∈ F(X). There exists g ∈ F(X) such that f ⋄g = 1 if and only if for every x ∈ X, there exists some a ∈ K * such that f ( a x) = a −1 .
(2) Let g ∈ F(X). There exists f ∈ F(X) such that f ⋄ g = 1 if and only if g(X) ⊂ K * , and the map x → g(x) x is 1-1.
(3) A function f ∈ F(X) is skew invertible if and only if f (X) ⊂ K * and the assignment x → f (x) x establishes a bijection from X onto X.
(2) First, we prove the "if" direction. It follows from f ⋄ g = 1 that f ( g(x) x)g(x) = 1 for all x ∈ X. Clearly, g must be nonzero on X, i.e., g(X) ⊂ K * . If g(x) x = g(y) y = z ∈ K for some x, y ∈ X, then we have f (z)g(x) = f (z)g(y) = 1, implying that g(x) = g(y). One can easily verify that g(x) x = g(y) y and g(x) = g(y) imply that x = y. This proves the "if" direction. Conversely, let g satisfy the stated properties. We define a map f : X → K as follows: If y = g(x) x for some x ∈ X, we set f (y) = g(x) −1 .
Otherwise, we set f (y) = x 0 where x 0 ∈ X is a fixed element. It is easy to see that f is well-defined and f ⋄ g = 1.
(3) Suppose that f is skew invertible and let g be its skew inverse. By (2), f is nonzero on X and x → f (x) x is 1-1. The fact that x → f (x) x is onto follows from the following identity Conversely, suppose that f satisfies the stated properties. By Part (2), there exists g ∈ F(X) such that g ⋄ f = 1. We need only show that f ⋄ g = 1.
Given an arbitrary element x ∈ X, we can choose y ∈ X such that x = f (y) y. We have Therefore, we have g(x) x = g(x)f (y) y = y from which it follows that Since x ∈ X was arbitrary, we conclude that f ⋄ g = 1.
A skew-invertible function shares some properties with its skew inverse. In the following proposition, we give two such properties. Proposition 2.9. Let f : X → K * be a skew-invertible function. Then: Proof. By Part (3) of Lemma 2.8, the function α : X → X, defined by α(x) = f (x) x, is a bijection. Let β : X → X denote its inverse (with respect to function composition). It is easy to check that from which the results follow.
Regarding skew-invertible elements of S(X), we have the following characterization.
Proposition 2.10. A skew-convex function f : X → K satisfying f (X) ⊂ K * is skew invertible if and only if for any x ∈ X, there exists some a ∈ K * such that f ( a x) = a −1 .
Proof. The "only if" direction follows from Part 1 of Lemma 2.8. To prove the converse, we use Part (2) of Lemma 2.8. Therefore, we need only show that if f (x) x = f (y) y, then x = y. Let f (x) x = f (y) y for some x, y ∈ X. Then y and x are in the same orbit, implying that there exists a ∈ K * such that It follows that Therefore, we can write We conclude the algebraic part of the paper with an example of a skewinvertible function. Example 2.3. As in Example 2.2, let σ : K → K be an endomorphism and δ : K → K be a σ-derivation. Let A ⊂ K be an invariant set under the (σ, δ)-action. Using Lemma 2.8 and Proposition 2.10, we see that a linear polynomial The second condition is equivalent to the condition that for any a ∈ A, there exists b ∈ K * satisfying the equation b a − c = b −1 , or equivalently, the equation In the case where δ is zero and σ is the identity homomorphism, we have

Skew analysis over quaternions
A central problem of "quaternionic analysis" is to find a suitable notion of "quaternionic derivative". It is known that the most obvious definition, namely, is too restrictive. From the point of view of the skew product, one is led to the informal definition which we shall call the skew derivative (see Definition 3.3 for the precise definition). This section deals with the notion of the skew derivative and develops a version of quaternionic analysis based on the skew derivative. Throughout this section, H denotes the skew field of quaternions. Given a quaternion we denote the real part, imaginary part, conjugate and norm of q by Re(q) = r 0 , Im(q) = r 1 i + r 2 j + r 3 k, q = r 0 − r 1 i − r 2 j − r 3 k and |q| = r 2 0 + r 2 1 + r 2 2 + r 2 3 , respectively.
In this section, we fix the following action of H * on H: p q = pqp −1 . The orbit containing a quaternion q ∈ H is denoted by O(q). It is known that for all p, q ∈ H, O(p) = O(q) holds if and only if Re(p) = Re(q) and |p| = |q| hold. In particular, an orbit is either a single point, called a trivial orbit, or a 2-dimensional sphere, called a nontrivial orbit. We also consider H as a topological space where the topology is the Euclidean topology induced by the norm q → |q|. For convenience, we shall use the following terminology: A subset of H is called a region if it is both open and invariant under the action p q = pqp −1 .

Skew-convex functions on H
In this section, we study skew-convex functions on invariant subsets of H.
We begin with a characterization of skew-convex functions on orbits.
Proof. If O is a trivial orbit, there is nothing to prove, so assume that O is nontrivial. The proof of the "if" direction is straightforward. To prove the converse, fix q 0 ∈ O. We need only show that This identity can be rewritten as using which the left-hand side of Identity 3.1 can be rewritten as It is easy to check that ((q1−q0) −1 −(q 2 −q 0 ) −1 ) q 0 = q 0 form which Identity 3.1 follows.
In view of this proposition, the following definition is well-defined.
Let us give another characterization of skew-convex functions which will be needed later. In the following proposition, the function T : H → H denotes the identity function on H.
Using the elementary formula pq−qp q = q, for all p, q ∈ H with pq = qp, we obtain F ( p q) = a + E(q)( p q − q), or equivalently, In particular, F is skew convex. Note that a = F (q) and In the following proposition, we collect some properties of the orbital derivative. Proposition 3.3. Let F, G : U → H be skew-convex functions on a region U in H. Then, F + G and F ⋄ G are skew convex on U , and moreover, we have Proof. The first part follows from Theorem 2.4. The first formula is trivial.
To prove the second formula, let p ∈ U be fixed and set O = O(p). We assume that G(p) = 0. The case G(p) = 0 is left to the reader. Then, we have It easy to check that the following identity holds on U : By Proposition 3.2, we have It is easy to verify that the orbital derivative of To introduce the next class of skew-convex functions on H, we need the following result.
Proposition 3.4. Let X ⊂ H be invariant and F 1 , F 2 , ... be a sequence of skew-convex H-valued functions on X. Suppose that the sequence F 1 , F 2 , ... converges pointwise to a function F : X → H. Then, F is skew convex on X.
Proof. One can easily prove the proposition using the definition of a skewconvex function (see Definition 2.2).
As an application of this proposition, we show that (left) spherical series are skew convex. Let us review some facts regarding spherical series over H (see also [11]). Suppose that O ⊂ H is an orbit. We associate a polynomial Note that P O (T ) does not depend on the choice p ∈ O. By a (formal) spherical series centered at O, we shall mean a series of the form if the series converges. The notion of right spherical series is defined in a similar fashion. We note that a spherical series centered at a trivial orbit O = {r} is just an ordinary series of the form ∞ n=0 q n (T − r) n .
The following proposition is easily proved using the Weierstrass M -test.
Proposition 3.5. Let ∞ n=0 q n P O (T ) n be a spherical power series centered at an orbit O. Suppose, furthermore, that lim sup n→∞ |q n | 1/n = 1/R for some R > 0. Then, the series converges absolutely and uniformly on the compact subsets of the region The number R in this Proposition will be called the radius of convergence of the spherical power series. The region U (O, R) will be called the region of convergence of the spherical power series. It follows immediately from Proposition 3.4 that any spherical series which is convergent on an invariant set X is skew convex on X. It is clear that the orbital derivative of a spherical series is zero at its center.
We conclude this part with the following lemma the proof of which is left to the reader.

Skew-invertible functions on H
In this section, we study the question of skew invertibility for skew-convex functions on invariant subsets of H. We begin with the following result. Proof. The result is clear if O is a trivial orbit. So, we assume that O is nontrivial. The "only if" direction follows from Part 3 of Lemma 2.8. We prove the other direction. Since F is skew-convex, there exists constants a, b ∈ H such that F (p) = a + bp for all p ∈ O. By Lemma 2.8 and Proposition 2.10, we need only show that for any p ∈ O, there exists x ∈ H * such that F ( x p) = x −1 . For a fixed p ∈ O, we need to show that the equation has a nonzero solution. The assignment x → ax + bxp defines an R-linear map on H. Since its kernel is zero, it must be onto, and therefore, there exists x ∈ H * such that ax + bxp = 1.
As an example, we see that T −q ∈ H[T ] is skew invertible on H\O(q) (see also Example 2.3). Moreover, we have the following description of (T −q) −1 .
Proposition 3.8. For any q 0 ∈ H, the function T − q 0 is skew invertible on H \ O(q 0 ) and its skew inverse satisfies In particular, we have Moreover, (T − q 0 ) −1 : H \ O(q 0 ) → H * is 1-1 and its image is the set Proof. It is known that the equation xq − q 0 x = 1 has a (unique) solution [9]). In fact, the unique solution of xq − q 0 x = 1 is As seen in Example 2.3, it follows that T − q 0 is skew invertible on H \ O and its skew inverse (T − q 0 ) −1 is skew convex. Moreover, we have for all q ∈ H \ O. The fact that the skew inverse of T − q 0 is 1-1 follows from proposition 2.9. To prove the last assertion, we note that the equation xq−q 0 x = 1 gives q = x −1 q 0 x+x −1 . It follows from the above discussion that the image of (T − q 0 ) −1 consists of all x ∈ H * such that from which the last part of the proposition follows.
Remark 3.1. Similarly, one can show that T − q 0 is skew invertible on H \ O(q 0 ) with respect to the right skew product. Moreover, the skew inverse In [2], the function (T − q 0 ) −1 r is studied in more detail.
Now, we treat polynomials of higher degree. Let P (T ) ∈ H[T ] be a nonzero polynomial. Set It follows directly from Proposition 3.7 that P is skew invertible on the set H \ Z(P ). If P (T ) ∈ R[T ] is a polynomial with real coefficients, then it is easy to see that P (T ) −1 (q) = P (q) −1 , for all q / ∈ Z(P ).
Therefore, there is no risk of confusion in using the notation P (T ) −1 instead of P (T ) −1 when P (T ) has real coefficients. For a detailed account of polynomials over quaternions, see [4, Chapter 3].

Limits and the skew product
In this part, we investigate the behavior of the skew product with respect to limits. We begin with a simple lemma whose proof is left to the reader.  Next, we show that the skew product of continuous functions is continuous. Proof. (a) It follows from Lemma 3.9 and the fact that any orbit in H is compact. (b) Let F −1 denote the skew inverse of F . By Lemma 2.8, F is nonzero on U and the map g(x) : = F (x) x is a bijection of U onto itself. Note that g : U → U is continuous. By Brouwer's invariance of domain theorem, g is a homeomrphism, that is, its inverse (with respect to composition) h : U → U is continuous. Since we conclude that F −1 is continuous.
Regarding limits of skew products of sequences of functions, we have the following result. Proof. Since F is skew-convex, we can write Note that (F n − F ) ⋄ G n and F ⋄ (G n − G) converge uniformly to the zero function, from which the desired result follows.

The skew derivative
We begin by introducing the definition of a skew-differentiable function.

Note that the function D q (F ) introduced in this definition is unique because it is continuous on O(q) and
Remark 3.2. The function D q (F ) has also been used in the context of sliceregular functions. For more details, see [11,Section 2].
Remark 3.3. Similarly, one can define the concept of right skew-differentiability using the right skew product. We leave to the reader the analogous definitions and results concerning right skew-differentiability. Since we mainly work with the concept of left skew-differentiability, we shall often drop the adjective "left".
As a direct consequence of Proposition 3.2, we obtain the following result. We show that skew regularity implies continuity. Our definition of the (left) skew derivative is as follows. Definition 3.3. Let F : U → H be a function on a region U ⊂ H. Suppose that F is skew differentiable at q 0 ∈ U . Then, the (left) skew derivative of F , denoted by F ′ (q 0 ), is defined to be the limit In the particular case where r ∈ U is real, we have In order to derive some basic properties of the skew derivative, we need the following lemma.
Lemma 3.14. Let a function F : U → H on a region U ⊂ H be skew differentiable at q ∈ U . If F is skew convex on U , so is D q (F ), and we have Proof. Note that the function (T − q) −1 is defined (by Proposition 3.8) and continuous on U \ O(q) (by Proposition 3.10). In the light of Lemma 3.6, the first part follows from the facts that D q (F ) = (F − F (q)) ⋄ (T − q) −1 is skew convex on U \ O(q), and D q (F ) is continuous at all p ∈ O(q). The second part is a consequence of the facts that D q (F ) is skew convex and In the following proposition, some further properties of the skew derivative are summarized. (1) Let q ∈ U . If F and G are skew differentiable at q, then F + G is also skew differentiable at q and (F + G) ′ (q) = F ′ (q) + G ′ (q).
(2) If F, G are skew regular on U , then so is F ⋄ G. Moreover, we have (3) If F is skew regular and skew invertible on U , then its skew inverse is skew regular on U , and moreover, we have Proof.
(2) Fix q ∈ U . We only treat the case G(q) = 0, and leave the easier case G(q) = 0 to the reader. Set We have Since F and E 1 are skew convex (see Proposition 3.12 and Lemma 3.14), we have By Lemma 3.9, the function E 1 ⋄ G(q) + F ⋄ E 2 is continuous at all p ∈ O(q). Therefore, F ⋄ G is skew regular at q and Evaluating at q, we obtain (3) Fix p ∈ U and set q = F −1 (p) p. Note that the quaternion F (q) is the inverse of the quaternion F −1 (p). Set E = D q (F ). We can write from which the result follows.
As an application of this proposition, we show that the skew derivative is skew convex. Proof. Using Proposition 3.12, we see that F is skew convex on U . By definition, we have

Part (2) of Proposition 3.15 shows that
Therefore, F ′ is skew convex on U .
Next, we present the chain rule for skew-differentiable functions.
Proposition 3.17. Let φ : U → U ′ be a continuous action-preserving function between regions in H. If φ is skew differentiable at some q 0 ∈ U and a function F : U ′ → H is skew differentiable at φ(q 0 ), then F • φ is skew differentiable at q 0 . Moreover, we have In particular, Using the fact that φ is action preserving, one can easily check that Since (E 1 • φ) ⋄ E 2 is continuous on U , we conclude that F • φ is skew differentiable at q 0 , and The chain rule for the orbital derivative is trivial.
Regarding the inverse of a skew-differentiable function with respect to function composition, we prove the following result.
Proposition 3.18. Let φ : U → H be a continuous action-preserving function on a region U . Suppose that ψ : U ′ → H is a continuous function on a region U ′ such that φ(U ) ⊂ U ′ and ψ(φ(q)) = q for all q ∈ U . If ψ is skew differentiable at some φ(q 0 ) ∈ U ′ and the function D φ(q 0 ) (ψ) does not have a root on the orbit O(φ(q 0 )), then φ is skew differentiable at q 0 . Moreover, we have Composing with φ and using the fact that φ is action-preserving, we obtain Since φ is 1-1 and E • φ does not have a zero on the orbit O(q 0 ), one can see that E • φ is nowhere zero on U . By Proposition 3.7, E • φ is skew invertible on U . It follows that By Part (b) of Proposition 3.10, (E • φ) −1 is continuous. Therefore, φ is skew differentiable at q 0 , and As an application of this proposition, one can show that the principal quaternionic logarithm Log : H \ (−∞, 0] → H (see [7,Definition 3.4]) is skew regular, using the facts that e Log(q) = q, for all q ∈ H \ (−∞, 0], and e q is a continuous action-preserving function on H. Moreover, we have Log ′ (q) = q −1 for all q ∈ H \ (−∞, 0]. The details are left to the reader.
We conclude this part with some remarks.
Remark 3.4. In [3], the regular product of slice-regular functions is first defined for the series centered at 0, and then extended to slice-regular functions (see [3,Definition 5.3]). Proposition 5.12 in the same reference gives a formula for the regular product of regular functions over axially symmetric slice domains. This formula is just the formula for the right skew product (see Definition 2.2).
Remark 3.5. We will prove that any slice-regular function over a symmetric slice domain is skew regular (see Theorem 3.20). It follows from Proposition 3.12 that slice-regular functions over symmetric slice domains are skew-convex. The so-called representation formulas (see for example [5, Formulas 1.8,1.9,1.10]) are simply consequences of the fact that slice-regular functions over symmetric slice domains are skew-convex.
Remark 3.6. The notion of orbital derivative, defined in Definition 3.1, is also introduced in [11], where it is called spherical derivative. In [8], a Leibniz-type formula is given for the orbital product. The formula given in Proposition 3.3 is a new formula. Remark 3.8. The author has not been able to find any form of the chain rule for slice-regular functions in the literature. The only result in this direction seems to be Lemma 1.32 in [5] which, roughly speaking, states that the composition of two slice-regular functions is slice-regular if one of them is slice preserving.

Skew regularity and slice regularity
In this section, we show that skew-regular functions are slice regular, and moreover, the converse holds over symmetric slice domains. We need to review some facts regarding slice-regular functions. For more details, see [5] and references therein. Let S denote the set of all I ∈ H such that I 2 = −1.
For I ∈ S and a subset A of H, put A I = A∩ (R + RI). Note that R + RI is a commutative subfield of H. In fact, it is isomorphic to C via x + yI → x + yi, using which we shall identify R + RI with the complex plane C. If F is slice regular on U , its slice derivative F ′ s (q) at q ∈ U I is defined to be First, we show that every skew-differentiable function is slice regular.
Proposition 3.19. Let F : U → H be a function on a region. If F is skew regular on U , then it is slice regular on U , and moreover, its skew derivative is equal to its slice derivative.
Proof. Fix I ∈ S. We need to show that the restriction of F to the set U I satisfies the Cauchy-Riemann equations. We choose J ∈ S such that J is orthogonal to I, that is, IJ = −JI. Set z = x + yI, where x, y ∈ R. There exist functions F 1 , F 2 : U I → R + RI such that The fact that R + RI is a commutative field implies that from which we obtain In particular, F 1 and F 2 are holomorphic on U I . Therefore, F is slice regular. The identity F ′ = F ′ s can easily be verified.
In general, a slice-regular function may not be skew regular. In fact, there exist slice-regular functions which are not even continuous (see Example 1.10 in [5]). To exclude such pathologies, one is forced to restrict one's attention to a certain class of open sets known as slice domains. Recall that a connected open subset U of H is called a slice domain if U ∩ R is nonempty and the intersection U ∩ (R + Rq) is connected for all q ∈ U \ R. It is known that any slice-regular function on a slice domain U can uniquely be extended to a slice-regular function on the symmetric completion of U (see [5,Theorem 1.24]). We remark that the symmetric completion of a set A ⊂ H is the smallest invariant subset of H which contains A. It turns out that the concepts of slice regularity and skew regularity coincide over symmetric (meaning invariant) slice domains. Proof. The "if" direction has been proved in Proposition 3.19. Conversely, let the function F : U → H be slice regular. Fix q ∈ U . It follows from Theorem 3.17 in [5] that there exists a slice-regular function E : , as functions on U.
Since E is slice regular on a symmetric slice domain, E must be continuous on U . Therefore, F is skew differentiable at q. As the slice derivative is equal to E(q) by the product rule for the slice derivative (see [5,Proposition 1.40]), the proof is complete.
Next, we show that strongly analytic functions are skew differentiable. Recall that a function is called strongly analytic on a region if at every point p in the region, the function can locally be represented as a convergent series centered at p. For more details, see [4]. Proof. Fix p ∈ U . Since F is strongly analytic at p, we can find an open neighborhood p ∈ U 0 ⊂ U , and q 0 , q 1 , ... ∈ H such that where the series converges absolutely and uniformly on U 0 . Set Then E is a continuous function on U 0 since the convergence is uniform. Using Proposition 3.11, one can see that It follow that F is skew differentiable at p.

Spherical series representation of skew-regular functions
In this section, we show that skew-regular functions can locally be represented by spherical series. Let us begin with the following result which is an improvement of the fact that any spherical series is slice regular on its region of convergence (see [11,Proposition 1.11]).
where the series is absolutely and uniformly convergent on the compact subsets of U (O, R).
Proof. We only prove the first assertion. The proof of the second assertion is similar to that of the corresponding result for ordinary power series over real or complex numbers. If O is a trivial orbit, the result follows from Theorem 16 in [4] since every skew-regular function is slice regular by Proposition 3.19. Assume that O is a nontrivial orbit with P O (T ) = T 2 − 2x 0 T + y 2 0 . Fix q 0 ∈ U (O, R). For every n ≥ 1, there exists P n (T ) ∈ H[T ] such that It is easy to check that P 1 (T ) = T + q 0 − 2x 0 . Using induction on n, we obtain the following recursion formula Consider the series E(T ) = ∞ n=0 q n P n (T ). I claim that this series is uniformly convergent on the compact subsets of U (O, R). Let q ∈ U (O, R) and m be the maximum of the norms |P O (q)| and |P O (q 0 )|. Using the recursion formula, one can easily show that |P n (q)| ≤ n|q + q 0 − 2x 0 |m n−1 , for all n ≥ 1. Now, the claim is easily proved using the Weierstrass M -test. In particular, we see that E is continuous on U (O, R). Since S = S(q 0 ) + E ⋄ (T − q 0 ), the result follows.
We need some preparations about line integrals over quaternions. Suppose that a curve γ : [a, b] → H is piecewise smooth. For H-valued functions F, G, defined and being continuous on the image of γ, we will use the following line integral along γ: In the following proposition, we use the skew inverse of T − p with respect to the right skew product (see Remark 3.1). Proposition 3.23. Let F : U → H be skew regular on a region and O ⊂ U be a nontrivial orbit such that U contains the closure of the set For any I ∈ S, we have where γ is the boundary of the set U (O, R) ∩ (R + RI). Moreover, we have An argument similar to the proof of Lemma 6.3 in [5] shows that the desired formula holds for p = p 0 and p = p 0 , since (T − p) −1 r (q) = (q − p) −1 when pq = qp. It is easy ot verify that the assignment defines a (left) skew-convex function on O(p ′ ). Since F is skew convex on O(p ′ ), it follows immediately that the desired formula holds on O(p ′ ). This completes the proof of the first formula. The second formula is derived using the first formula and the definition of the orbital derivative.
Let us give two remarks about this propositions.
Remark 3.9. The first formula given in Proposition 3.23 can be regarded as an analogue of the Cauchy formula for holomorphic functions. Although our formula looks similar to Formula 6.4 in [5], there is a fundamental difference between the two formulas. Using our notations, we can rewrite the left-hand analogue of Formula 6.4 in [5] as This formula does not seem suitable for deriving spherical series presentations of skew-regular functions (see Proposition 3.25).
Remark 3.10. As a consequence of Proposition 3.23, we see that the orbital derivative of a skew-regular function is continuous. In fact, one can use the second formula in the proposition to show that the orbital derivative of a skew-regular function is real analytic. However, it may not be skew regular.
We also need the following lemma.
Lemma 3.24. Let p ∈ H, and O ⊂ H be a nontrivial orbit such that p / ∈ O. Then, the right skew-regular function (T − p) −1 r can be represented as the (right) series where the series is absolutely and uniformly convergent on the compact subsets of the region Proof. We have  Note that |P O (p)| < |P O (q)| = R for all p ∈ U (O, R) and q ∈ γ. Let P O (T ) = T 2 − 2x 0 T + y 2 0 . Using Lemma 3.24, we see that It is allowed to change the order of summation and integration since the (right) series converges absolutely and uniformly on the compact set γ. Setting  Proof. The "if" direction follows from Proposition 3.22. The other direction follows from Proposition 3.25 if O is a nontrivial orbit. In the case of trivial orbits, the result follows from Proposition 2.7 in [3]. Theorem 3.26 has a number of important consequences which will be studied in a sequel to this paper. Let us just mention two direct consequences. The first corollary is the generalization of the fact that the slice derivative of a slice-regular function is slice regular.
Corollary 3.27. If a function F : U → H is skew regular on a region U ⊂ H, then it is infinitely skew differentiable. In particular, the skew derivative F ′ : U → H of F is skew regular on U .
Proof. This result follows from Theorem 3.26 and the fact that spherical series can be differentiated term by term. The details are left to the reader.
In the following corollary, (T − p) −n−1 r denotes the (n + 1)-th power of (T − p) −1 r with respect to the right skew product. Proof. In view of Corollary 3.27, one can use an argument similar to the proof of Proposition 3.23 to derive the desired formula.