Axial monogenic functions \(f_ k\) are considered. These are solutions of the equation \((\partial_{\mathbf x} + \partial_{x_ 0}) f_ k = 0\) where \(\partial_{\mathbf x}\) is the Dirac operator of the Euclidean \(m\)- space of the form \[ f_ k(x) = \bigl[ A_ k (x_ 0, \rho) + e_ \rho B_ k (x_ 0, \rho) \bigr] P_ k (e_ \rho) \] where \(x = x_ 0 e_ 0 + {\mathbf x},x = \rho e_ \rho\), and \(P_ k\) is a spherical monogenic. The central result is that, given a holomorphic function \(f\), then \[ \int_{-1}^ 1(1 - t^ 2)^{k + (m - 3)/2} f(x_ 0 + i \rho t) (1 - it e_ \rho) dt P_ k ({\mathbf x}) \] is an axial monogenic function, where the integration path can be deformed in order to avoid singularities of \(f\). Notice that for \(\rho \sim 0\), \(f_ k(x) \sim f(x_ 0) P_ k ({\mathbf x})\). First this construction is compared with a construction using plane waves (monogenic functions depending only on the projection of \(x\) on a fixed plane), then it is applied to several specific cases, such as homogeneous axial monogenics. Further the axial exponential function, where \(f(x) = e^ z\) is treated, where an estimate for the modulus of the function is given. A second application is the theory of generalised orthogonal (Hermite and Gegenbauer) polynomials. Since these polynomials arise in the series expansion of axial monogenic functions (respectively the monogenic extensions of \(e^{ - \rho^ 2/2} P_ k ({\mathbf x})\) and \((1 - {\mathbf x}^ 2)^{ - \beta}\) for some \(\beta)\) the previous representation of axial monogenics can be used to obtain explicit expressions for these polynomials. Notice however that this explicit form was already given by the reviewer [the reviewer, Bull. Soc. Math. Belg., Sér. B 43, 1-17 (1991; Zbl 0721.33003)].