AI 中文总结
该论文在分次单项式空间构造半阶微分算子，通过递归关系确定系数，利用沃利斯乘积和与特定公式兼容性确定归一化常数，得到\(D^{1/2}x^\beta\)公式并与多种公式一致，此线性复合算子在该空间完全确定，还有一些相关问题待解决。
AI 中文摘要
本文在由x（x>0）的非负整数和半整数幂所张成的分次单项式空间上构造了一个半阶微分算子。代数阶段既不使用积分核、极限过程，也不使用伽马函数。假设该算子以\(D^{1/2}x^\beta=c(\beta)x^{\beta - 1/2}\)的形式作用。通过要求该算子连续应用两次以重现普通一阶导数，得到基本递归关系\(c(\beta)c(\beta - 1/2)=\beta\)。展开递归产生两个相关的系数族，一个用于整数幂，一个用于半整数幂。它们的公式包含一个自由归一化常数\(c_0 = c(0)\)，在复合运算下会消去。因此，复合要求确定了相对系数，但不是每个半步的尺度。偶数和奇数双阶乘的比值自然地引出了一个部分沃利斯乘积；然而，仅沃利斯乘积并不能确定\(c_0\)。通过要求与下限为0的左黎曼 - 刘维尔半导数的单项式公式兼容，选择\(c_0 = 1/\sqrt{\pi}\)。在此归一化下，对于每个\(\beta\in\{0,1/2,1,3/2,\ldots\}\)，\(D^{1/2}x^\beta=\frac{\Gamma(\beta + 1)}{\Gamma(\beta + 1/2)}x^{\beta - 1/2}\)成立。所得的单项式公式与相应的黎曼 - 刘维尔公式一致。对于正整数幂，它也与卡普托公式一致，而对于常数函数，它仅与黎曼 - 刘维尔规则一致。所得的线性复合算子将指数恰好降低一半，并在所述分次单项式空间上完全确定。扩展到更广泛的函数空间、构造积分或卷积核以及非局部性分析仍然是未解决的问题。
英文摘要
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 $D^{1/2}x^β=c(β)x^{β-1/2}$. Requiring two successive applications of the operator to reproduce the ordinary first derivative yields the fundamental recurrence relation $c(β)c(β-1/2)=β$. 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 $c_0=1/\sqrtπ$. With this normalization, $D^{1/2}x^β=\frac{Γ(β+1)}{Γ(β+1/2)}x^{β-1/2}$ holds for every $β\in\{0,1/2,1,3/2,\ldots\}$. 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.
Comments40 pages
