Hyperbolic Function Theory in the Skew-Field of Quaternions

We are studying hyperbolic function theory in the total skew-field of quaternions. Earlier the theory has been studied for quaternion valued functions depending only on three reduced variables. Our functions are depending on all four coordinates of quaternions. We consider functions, called α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-hyperbolic harmonic, that are harmonic with respect to the Riemannian metric dsα2=dx02+dx12+dx22+dx32x3α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} ds_{\alpha }^{2}=\frac{dx_{0}^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}}{x_{3}^{\alpha }} \end{aligned}$$\end{document}in the upper half space R+4={x0,x1,x2,x3∈R4:x3>0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}_{+}^{4}=\{\left( x_{0},x_{1},x_{2} ,x_{3}\right) \in {\mathbb {R}}^{4}:x_{3}>0\}$$\end{document}. If α=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =2$$\end{document}, the metric is the hyperbolic metric of the Poincaré upper half-space. Hempfling and Leutwiler started to study this case and noticed that the quaternionic power function xm(m∈Z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x^{m}\,(m\in {\mathbb {Z}})$$\end{document}, is a conjugate gradient of a 2-hyperbolic harmonic function. They researched polynomial solutions. Using fundamental α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-hyperbolic harmonic functions, depending only on the hyperbolic distance and x3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{3}$$\end{document}, we verify a Cauchy type integral formula for conjugate gradient of α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-hyperbolic harmonic functions. We also compare these results with the properties of paravector valued α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-hypermonogenic in the Clifford algebra Cℓ0,3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{{\mathcal {C}}\ell }\,}}_{0,3}$$\end{document}.


Introduction
We study quaternion valued twice continuous differentiable functions f
Earlier the theory has been studied for quaternion valued functions depending only on three reduced variables [5]. In case α = 2, this system was studied by Hempfling and Leutwiler in [11]. Recently, we verified Cauchy type formulas for these function in [6]. In this paper, we study integral formulas and operators produced by these formulas. The results are interesting, since we are building hyperbolic function theory in the full skew field of quaternions. We also develop the theory of paravector valued α-hypermonogenic functions in the Clifford algebra C 0,3 and find similar integral theorems as in the quaternionic hyperbolic function theory.

Preliminaries
The skew-field of quaternions H is four dimensional associative division algebra over reals with an identity 1. We denote by 1, i, j and k the generating elements of H satisfying the relations The elements β1 and β are identified for any β ∈ R.
Any quaternion x may be represented with respect to the base {1, i, j, k} by where x 0 , x 1 , x 2 and x 3 are real numbers. The real vector spaces R 4 and H may be identified.
We denote the upper half space by 3 ) | x m ∈ R, m = 0, 1, 2, 3 and x 3 > 0} and the lower half space by and a * = (a 0 , a 1 , a 2 , −a 3 ) , (see a proof for example in [12]). Similarly, we may compute the hyperbolic distance between the points x and a in R 4 − . The following simple calculation rules are useful. We recall that the hyperbolic ball B h (a, r h ) with the hyperbolic center a in R 4 + and the radius r h is the same as the Euclidean ball with the Euclidean center c a (r h ) = (a 0 , a 1 , a 2 , a 3 cosh r h ) and the Euclidean radius r e = a 3 sinh r h .
The inner product x, y in R 4 is defined as usual by x m y m .
If x = x 0 + x 1 i + x 2 j + x 3 k and y = y 0 + y 1 i + y 2 j + y 3 k are quaternions their inner product is defined similarly as in R 4 by x m y m .
The elements and it satisfies (xy) = x y for all quaternions x and y. The reversion ( ) * in H is the mapping x → x * defined by for a = a 0 + a 1 i + a 2 j + a 3 k ∈ H. Using the reversion, we compute the formulas We use the identities If Re a = Re b = Re c = 0, then (see [10]) Notice that the triple product is linear with respect to a, b and c. Moreover, (2.10)

Hyperregular Functions
We define the following hyperbolic generalized Cauchy-Riemann operators H l α (x) and H r α (x) for x ∈ Ω\{x 3 = 0} as follows where the parameter α ∈ R and When there is no confusion, we abbreviate D q l f by D q f and H l α by H α . and We emphasize that a function is α-hyperregular provided that it is continuous differentiable in the total open set Ω ⊂ R 4 and satisfies the preceding equation for all x with x 3 = 0.
Our operators are connected to the hyperbolic metric via the hyperbolic Laplace operator as follows.
is the Laplace-Beltrami operator (see [13]) with respect to the Riemannian metric We list a couple of simple observations. 3 (Ω) then the function ∂h ∂x3 satisfies the equation Proof. Assume that x ∈ Ω\{x 3 = 0}. We just compute as follows Real valued α-hyperbolic functions are especially important, since they produce α-hyperregular functions.
Theorem 3.7. [6] Let Ω be an open subset of R 4 . If a twice continuously differentiable function f : Ω → H is α-hyperregular then the coordinate functions f n for n = 0, 1, 2 are α-hyperbolic harmonic and f 3 satisfies the equation for any x ∈ Ω.
The following transformation property is proved in [1,3]. f (x) satisfies the equation

Cauchy Type Integral Formulas
We recall the Stokes theorem for T and S-parts proved in [6].
where dσ is the surface element and dm the usual Lebesgue volume element in R 4 .
where dσ is the surface element and dm the usual Lebesgue volume element in R 4 .
where the associated Legendre function is defined by

) and the hypergeometric function by
We remark that the fundamental α-hyperbolic harmonic function is unique up to a harmonic function. The reason why we picked the preceding function is that it leads to nice symmetry properties of a kernel, verified after the following theorem.
The function s (x, y) is the kernel computed in [2] and in [3]. Clearly, the function h α (x, y) is not symmetrical with respect to x and y. However, it has the following symmetry properties. , x) , and , y)) .

Proposition 4.5. The function h α has the properties
If m = 0, 1, 2, then The last properties follow from the tedious calculations [7].
We recall the integral formulas for S-and T -parts verified in [6].
If we combine these formulas we obtain a new formula.  x, a, ν, T f k) Proof. We combine the preceding integral formulas using the formula f (a) = Sf (a) + T f (a) k.
Using the definition of the triple product we infer , a) , ν Sf Using the symmetry properties, we obtain In order to shorten the notations, we abbreviate g = T fk. Then we simply compute a) , ν g. Symmetry properties imply that We consider the Teodorescu and Cauchy type operators in subsequent papers. Also the case a ∈ R 4 + \K involves some technical assumptions and left for later work.

Comparison of α-Hyperregular and α-Hypermonogenic Functions
The universal real Clifford algebra C 0,3 is a real associated algebra with a unit 1 and is generated by e 1 , e 2  where δ st is the usual Kronecker delta and s, t = 1, 2, 3. We denote r1 briefly by r ∈ R.
The elements x = x 0 + x 1 e 1 + x 2 e 2 + x 3 e 3 for x 0 , x 1 , x 2 , x 3 ∈ R are called paravectors. The real number x 0 is the real part of the paravector x.
The main involution in C 0,3 is the mapping a → a defined by e s = −e s for s = 1, . . . , 3 and extended to the total algebra by linearity and the product rule (ab) = a b . Similarly the reversion is the mapping a → a * defined by e * s = −e s for s = 1, . . . , 3 and extended to the total algebra by linearity and the product rule (ab) The conjugation is the mapping a → a defined by a = (a ) * = (a * ) .
Any element w in C 0,3 may be written as w = w 0 + w 1 e 1 + w 2 e 2 + w 3 e 3 + w 12 e 12 + w 13 e 13 + w 23 e 23 + w 123 e 123 , where e mn = e m e n for 1 ≤ m < n ≤ 3 and e 123 = e 1 e 2 e 3 . The element e 123 ,denoted by I, is commuting with all elements and (e 1 e 2 e 3 ) 2 = 1.
We recall that C 0,1 may be identified with the field of complex numbers. The universal Clifford algebra C 0,2 may be identified with the quaternions, by setting i = e 1 , j = e 2 and k = e 1 e 2 . This identification we used in the first section when we defined involutions.
We generalize the imaginary part of a complex number to C 0,3 by decomposing any element a ∈ C 0,3 as a = b + ce 3 for b, c ∈ C 0,2 . The mappings P : C 0,3 → C 0,2 and Q : C 0,3 → C 0,2 are defined in [9] by P a = b, Qa = c.
In order to compute the P -and Q-parts we use the involution a → a defined by e i = (−1) δs3 e i for s = 1, 2, 3 and extended to the total algebra by linearity and the product rule ab = a b. Then we obtain the formulas We consider functions f : Ω → C 0,3 , defined on an open subset Ω of R 4 , and assume that its components are continuously differentiable. The left Dirac operator (also called the Cauchy-Riemann operator) in C 0,3 is defined by