An Algebraic-Operator Construction of the Half-Derivative on a Graded Monomial Space. Part I: Recurrence, Double Factorials, the Wallis Product, and Normalization

Abstract

This paper constructs a half-order differentiation operator on the graded monomial space spanned by non-negative integer and half-integer powers of x, with x > 0. The algebraic stage uses neither an integral kernel, a limiting process, nor the Gamma function. The operator is assumed to act in the form $D^{1/2}x^\beta=c(\beta)x^{\beta-1/2}$. Requiring two successive applications of the operator to reproduce the ordinary first derivative yields the fundamental recurrence relation $c(\beta)c(\beta-1/2)=\beta$. Expanding the recurrence produces two linked coefficient families, one for integer powers and one for half-integer powers. Their formulas contain a free normalization constant $c_0=c(0)$, which cancels under composition. Thus, the compositional requirement fixes the relative coefficients but not the scale of each individual half-step. Ratios of even and odd double factorials lead naturally to a partial Wallis product; however, the Wallis product alone does not determine $c_0$. Imposing compatibility with the monomial formula for the left Riemann-Liouville half-derivative with lower limit 0 selects the value $c_0=1/\sqrt{\pi}$. With this normalization, $D^{1/2}x^\beta=\frac{\Gamma(\beta+1)}{\Gamma(\beta+1/2)}x^{\beta-1/2}$ holds for every $\beta\in\{0,1/2,1,3/2,\ldots\}$. The resulting monomial formula agrees with the corresponding Riemann-Liouville formula. For positive integer powers it also agrees with the Caputo formula, whereas for the constant function it agrees only with the Riemann-Liouville rule. The resulting linear compositional operator lowers the exponent by exactly one half and is fully specified on the stated graded monomial space. Extension to broader function spaces, construction of an integral or convolution kernel, and the analysis of non-locality remain open problems.