Distributional Fractional Taylor Series and Interpretation of the Fractal and Number-Theoretic Explicit Formulas

Motivated by the work of M. L. Lapidus and M. van Frankenhuisjen on fractal explicit formulas for generalized fractal strings, we develop in this paper a distributional fractal Taylor’s formula with error term and an exact distributional fractal Taylor series representation for generalized fractal strings. The explicit formulas that the aforementioned authors developed provide a way of expressing a generalized fractal string $\eta$ in terms of the underlying fractal complex dimensions, under suitable growth assumptions on its geometric zeta function, ${\zeta }_{\eta }$, known as the languidity assumptions. This involves a sum indexed by the complex dimensions of the generalized fractal string, summing over all residues of ${\zeta }_{\eta }$, multiplied by the Mellin transform of a tempered or Schwartz test function $\phi$, denoted by $\tilde{\phi }$. Under languidity assumptions, there is an error term present. Under stronger assumptions, that is, in the case of strong languidity, no error term is present and the resulting fractal explicit formula is said to be exact. In this paper, we express the residue term in the fractional distributional explicit formulas as the fractional $(-\omega {)}^{\mathrm{t}\mathrm{h}}$-ordered distributional derivative of the Dirac $\delta$ distribution, denoted by $Y_{\omega }$, applied to modified test functions of the form $\phi (x){\ln }^{k-1}(x)$, where $k$ runs from $1$ to the multiplicity of the pole $\omega$ of ${\zeta }_{\eta }$. The coefficients in this sum are written in terms of the coefficients in the principal part of the Laurent expansion at $\omega$ of the geometric zeta function ${\zeta }_{\eta }$, multiplied by the gamma function evaluated at $\omega$. The results obtained in this paper contribute to the broader program of characterizing fractals in terms of fractal Taylor series expansions involving their underlying fractal complex dimensions.