Abstract. The aim of this paper is to put the foundations of a new theory of functions, called holomorphic Cliffordian, which should play an essential role in the generalization of hoiomorphic functions to higher dimensions. Let~0.2m+l be the Clifford Here, we will study polynomial and singular solutions of this equation, we will obtain integral representation formulas and deduce the analogous of the Taylor and Laurent expansions for holomorphic Cliffordian functions.In a following paper, we will put the foundations of the Cliffordian elliptic function theory.

Abstract.-In the study of holomorphic functions of one complex variable, one well-known theory is that of elliptic functions and it is possible to take the ζ-function of Weierstrass as a building stone of this vast theory. We are working the analogue theory in the natural context of higher dimensional spaces : holomorphic and elliptic Cliffordian functions.

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SUMMARYIn this paper we introduce a real integral transform which links trigonometric and Bessel functions. This allows us to construct a monogenic pseudo-exponential in Cli ord analysis. There is a deep di erence between odd and even dimensions. It is impossible to deÿne what is a 'special function'. But these are essential tools to study deep facts in mathematics (e.g. the -Riemann function in number theory), physics (e.g. the Hermite functions for the hydrogen atom), engineering (e.g. the Bessel functions for the skin e ect in electricity) and so on. When the study of a structure gives a classical function, immediately we get many informations. When two di erent structures give the same function, immediately we get links. Hence, the subject is not at all out of fashion.Most well-known and useful special functions are one-dimensional (real or complex). To grasp the several (real)-variables case, we have to choose what to put for complex numbers: here the choice is Cli ord algebras. The aim of this work is to build an exponential in the Cli ord analysis setting. It is strange that Bessel functions pop out here.In the ÿrst part we deÿne and study an integral transform which is a kind of modiÿcation of the classical Riemann- In the second part we are studying the problem of the existence of an exponential function in Cli ord analysis restricted to the point of view of monogenic functions.Our main tool is a formula expressing the interaction between those integral transforms (we named them Bessel transforms) and the vector derivation operator. Curiously, this formula depends on the dimension of the vector space on which the Cli ord algebra is spanned. The structure of Cli ord algebras of even and odd dimensions are di erent. We expect that results re ect this: in topology see Reference [2], in analysis see References [3,4]. The exponential in Cli ord analysis was often tackled (for example, References [5][6][7][8]).Spr ossig [9] introduced related special functions. His aim was to get functions of exponential type and his idea was to take a radial variable and to use the Fueter mapping. Here, our starting point is di erent: we need an integral transform. It is quite striking that we obtain closely related results. REAL INTEGRAL TRANSFORMS: THE BESSEL TRANSFORMS Deÿnition and ÿrst propertiesIn this section we will introduce new integral transforms acting on real functions of a real variable f : R−→R, t −→ f(t), provided that f is integrable. The suggestion to call them Bessel transforms comes from formula (19), which shows explicitly how they are generated by the well-known Bessel functions.Denote by f P the even part of f, f P (t) = 1 2 [f(t) + f(−t)], for t ∈ R, and by f I the odd part of f, fRecall the notation for the Beta functionwhere a¿0 and b¿0.and (S −1=2 f)(t) = f(t), under the assumption the integral exists.Thanks to the linearity of the integral, we have that S is a linear integral transform, i.e.(i) S ( 1 f 1 + 2 f 2 ) = 1 S f 1 + 2 S f 2 for 1 ; 2 ∈ R. Obviously, because of the normalization constant, ...

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