Umbral Methods, Function Factorisation and Mittag–Leffler Fourier-Type Integral Transform

We propose a systematic way to construct trigonometric-like functions beyond the classical sine–cosine pair by factorising rational expressions in the umbral operator and then evaluating their action on the vacuum. The guiding idea is simple: the usual trigonometric functions may be viewed as cyclic components arising from a finite factorisation, and the same principle can be extended to an n-fold decomposition of rational umbral expressions. For each integer n≥2, the construction produces n functions which play the role of higher-order trigonometric-like components: their sum reconstructs the corresponding umbral function, while the individual components isolate the different cyclic sectors of its expansion. The construction is developed first in the formal umbral setting. The quadratic case n=2 gives the Gaussian-trigonometric functions, in which the cosine-like component is a Gaussian and the sine-like component is its natural umbral companion. The cubic case n=3 yields a three-component cyclic system and shows how the same idea extends beyond the usual even–odd decomposition. These examples suggest that trigonometric factorisation is not restricted to ordinary rotations, but belongs to a broader cyclic principle in umbral calculus. We then reinterpret the same formal identities through the recently developed analytic umbral framework. In this second step, the cyclic components are realised by Mellin–Barnes pairings, and the root-of-unity decomposition is related to the splitting of the corresponding spectral kernel. This analytic formulation provides contour representations and local expansions for the functions obtained formally, while also fixing the branch and residue conventions needed for their analytic continuation. Finally, we indicate how the same cyclic kernels act as deformations of the Fourier transform. The resulting framework presents higher-order umbral trigonometric functions as natural cyclic components of factorised rational or exponential umbral operators.