Abstract. Let \(d\in\mathbb{N}\) and let \(\boldsymbol{\alpha}=(\alpha_1,\dots,\alpha_d)\in(0,1)^d\). We prove that the algebraic direct sum \[ \mathcal{G}_{\boldsymbol{\alpha}}^{\mathrm{alg}} := \bigoplus_{\mathbf{k}\in\mathbb{N}_0^d}\mathbb{C}e_{\mathbf{k}}, \quad e_{\mathbf{k}}(x) := \prod_{j=1}^d \frac{x_j^{k_j\alpha_j}}{\Gamma(k_j\alpha_j+1)}, \] is the canonical multi-graded monomial space on which the partial Riemann–Liouville integrals \( J_j:={}_0 I_{x_j}^{\alpha_j} \text{ for } 1\le j\le d \) and the partial Caputo derivatives \( C_j:={}_0^{\mathrm{C}}D_{x_j}^{\alpha_j} \text{ for } 1\le j\le d \) realize a commuting shift algebra. More precisely, for every \(\mathbf{k}\in\mathbb{N}_0^d\) and every \(j\in\{1,\dots,d\}\), \( J_j e_{\mathbf{k}} = e_{\mathbf{k}+\mathbf{e}_j}, \) and \[ C_j e_{\mathbf{k}} = \begin{cases} 0, & k_j=0,\\[1mm] e_{\mathbf{k}-\mathbf{e}_j}, & k_j\ge 1. \end{cases} \] Consequently, \( C_jJ_j=I,\quad J_jC_j=I-\Pi_j,\quad [C_j,J_j]=\Pi_j, \) where \(\Pi_j\) is the projection onto the coordinate-vacuum hyperplane \(\{\,\mathbf{k}\in\mathbb{N}_0^d : k_j=0\,\}\). We further prove that the tuples \((J_1,\dots,J_d)\) and \((C_1,\dots,C_d)\) commute on \(\mathcal{G}_{\boldsymbol{\alpha}}^{\mathrm{alg}}\), and that the model is unique up to a global scalar normalization among multi-indexed graded monomial lattices with one-dimensional homogeneous components. Finally, for every \(\mathbf{m}\in\mathbb{N}_0^d\), the operator \( J^{\mathbf{m}} := J_1^{m_1}\cdots J_d^{m_d} \) coincides with the partial Riemann–Liouville integral of order \(\mathbf{m}\odot\boldsymbol{\alpha}\) on all of \(\mathcal{G}_{\boldsymbol{\alpha}}^{\mathrm{alg}}\), whereas \( C^{\mathbf{m}} := C_1^{m_1}\cdots C_d^{m_d} \) coincides with the partial Caputo derivative of order \(\mathbf{m}\odot\boldsymbol{\alpha}\) on the natural tail subspace. Thus the multidimensional Caputo defect appears as a family of coordinate boundary layers rather than a single vacuum sector.
Keywords. multidimensional fractional calculus; partial Caputo derivative; partial Riemann–Liouville integral; multi-graded space; commuting shift algebra; Mittag-Leffler basis.
1. Introduction
Classical fractional calculus extends integration and differentiation to noninteger orders; see, for example, [3, 4, 5]. In one variable, a particularly simple algebraic model appears when one replaces the ordinary monomial chain \[ 1,x,x^2,x^3,\dots \] by the \(\alpha\)-graded chain \[ 1,x^\alpha,x^{2\alpha},x^{3\alpha},\dots. \] The resulting one-dimensional shift-algebra model was established in [2], while the obstruction to finite-dimensional models built from ordinary polynomial spaces was analyzed in [1].
The present paper develops the corresponding multidimensional picture. Our starting point is the coordinatewise framework for multidimensional fractional calculus studied by Kostić in [7], where partial Riemann–Liouville and Caputo operators are defined by applying the one-variable operators to the individual coordinates. Motivated by that viewpoint, we ask the following question.
Is there a canonical multi-graded monomial space on which the partial Riemann–Liouville integrals and the partial Caputo derivatives act as a commuting shift algebra?
We show that the answer is yes. Let \(d\in\mathbb{N}\) and let \[ \boldsymbol{\alpha}=(\alpha_1,\dots,\alpha_d)\in(0,1)^d. \] We define \[ e_{\mathbf{k}}(x) := \prod_{j=1}^d \frac{x_j^{k_j\alpha_j}}{\Gamma(k_j\alpha_j+1)}, \qquad \mathbf{k}=(k_1,\dots,k_d)\in\mathbb{N}_0^d, \] and consider the algebraic direct sum \[ \mathcal{G}_{\boldsymbol{\alpha}}^{\mathrm{alg}} := \bigoplus_{\mathbf{k}\in\mathbb{N}_0^d}\mathbb{C}e_{\mathbf{k}}. \] On this space, the partial Riemann–Liouville integral in the \(j\)-th coordinate acts as a forward shift in the \(j\)-th index, while the partial Caputo derivative acts as the corresponding backward shift, annihilating the hyperplane on which the \(j\)-th index is zero.
The passage from one variable to several variables is not merely a tensor-product repetition. In one variable there is a single vacuum vector and a finite-rank low-grade defect for higher Caputo powers. In several variables the single vacuum becomes a family of coordinate-vacuum hyperplanes, and the higher-order defects become boundary layers indexed by multi-indices. This is the essential new structural feature of the multidimensional model.
It is important to emphasize what this paper does not claim. We do not assert that the full ambient multidimensional Riemann–Liouville or Caputo calculus is commuting. In fact, mixed partial fractional derivatives need not commute in general; see [7, Example 2]. Rather, we isolate a canonical invariant sector—the multi-graded monomial lattice—on which the coordinatewise operators do commute and admit an exact shift-algebra realization.
The paper is organized as follows. In Section 2 we recall the coordinatewise partial Riemann–Liouville and Caputo operators and prove the required power-function formulas. In Section 3 we define the canonical multi-graded monomial space. Section 4 establishes the commuting shift realization and the fundamental operator identities. Section 5 treats higher powers, tail spaces, and the multidimensional defect projections. Section 6 proves a uniqueness theorem showing that the canonical basis is forced under a natural monomial ansatz. Section 7 records the product Mittag–Leffler generating series and explains the relation of the model to the ambient multidimensional calculus. We conclude in Section 8.
Throughout, all identities are understood pointwise on \((0,\infty)^d\), and \(\mathbb{N}_0:=\{0,1,2,\dots\}\).
2. Preliminaries on partial fractional operators
Fix \(d\in\mathbb{N}\). If \[ x=(x_1,\dots,x_d)\in(0,\infty)^d \] and \(j\in\{1,\dots,d\}\), we write \[ \widehat{x}_j := (x_1,\dots,x_{j-1},x_{j+1},\dots,x_d) \] for the \((d-1)\)-tuple obtained by removing the \(j\)-th coordinate.
If \[ \mathbf{k}=(k_1,\dots,k_d),\ \mathbf{m}=(m_1,\dots,m_d)\in\mathbb{N}_0^d, \] we write \[ \mathbf{k}\le \mathbf{m} \quad \text{if and only if} \quad k_j\le m_j \ \text{for all } j, \] and \(\mathbf{e}_j\) denotes the \(j\)-th standard basis vector in \(\mathbb{N}_0^d\). If \[ \boldsymbol{\mu}=(\mu_1,\dots,\mu_d)\in(0,\infty)^d, \] and \[ \mathbf{m}=(m_1,\dots,m_d)\in\mathbb{N}_0^d, \] then \[ \mathbf{m}\odot\boldsymbol{\mu} := (m_1\mu_1,\dots,m_d\mu_d). \]
Definition 2.1. Let \(j\in\{1,\dots,d\}\) and let \(\mu>0\).
- The left-sided partial Riemann–Liouville integral of order \(\mu\) in the \(j\)-th coordinate is defined by \[ ({}_0 I_{x_j}^{\mu}f)(x) := \frac{1}{\Gamma(\mu)} \int_0^{x_j} (x_j-t)^{\mu-1} f(x_1,\dots,x_{j-1},t,x_{j+1},\dots,x_d)\,dt, \] whenever the integral exists.
- If \(\mu\notin\mathbb{N}\) and \(p:=\lceil \mu\rceil\), the left-sided partial Caputo derivative of order \(\mu\) in the \(j\)-th coordinate is defined by \[ ({}_0^{\mathrm{C}}D_{x_j}^{\mu}f)(x) := \frac{1}{\Gamma(p-\mu)} \int_0^{x_j} (x_j-t)^{p-\mu-1} \partial_j^p f(x_1,\dots,x_{j-1},t,x_{j+1},\dots,x_d)\,dt, \] whenever the integral exists. If \(\mu=p\in\mathbb{N}\), we set \[ {}_0^{\mathrm{C}}D_{x_j}^{p}f := \partial_j^{p}f. \]
Lemma 2.2. Let \(j\in\{1,\dots,d\}\), let \(\mu>0\), and let \(\phi=\phi(\widehat{x}_j)\) be independent of the variable \(x_j\).
- If \(\beta>-1\), then \[ {}_0 I_{x_j}^{\mu}\!\big(\phi(\widehat{x}_j)\,x_j^{\beta}\big) = \phi(\widehat{x}_j)\, \frac{\Gamma(\beta+1)}{\Gamma(\beta+\mu+1)}\, x_j^{\beta+\mu}. \]
- If \(p:=\lceil \mu\rceil\) and \(\beta>p-1\), then \[ {}_0^{\mathrm{C}}D_{x_j}^{\mu}\!\big(\phi(\widehat{x}_j)\,x_j^{\beta}\big) = \phi(\widehat{x}_j)\, \frac{\Gamma(\beta+1)}{\Gamma(\beta+1-\mu)}\, x_j^{\beta-\mu}. \]
- If \(0 < \mu < 1\), then \[ {}_0^{\mathrm{C}}D_{x_j}^{\mu}\phi(\widehat{x}_j)=0. \]
Proof.
Since \(\phi(\widehat{x}_j)\) is independent of \(x_j\), it factors out of all partial integrals and partial derivatives in the \(j\)-th variable.
For (i), Definition 2.1 gives \[ {}_0 I_{x_j}^{\mu}\!\big(\phi(\widehat{x}_j)x_j^\beta\big) = \phi(\widehat{x}_j)\, \frac{1}{\Gamma(\mu)} \int_0^{x_j}(x_j-t)^{\mu-1}t^\beta\,dt. \] The remaining one-variable integral is the classical Riemann–Liouville integral of the monomial \(x^\beta\), hence \[ \frac{1}{\Gamma(\mu)} \int_0^{x_j}(x_j-t)^{\mu-1}t^\beta\,dt = \frac{\Gamma(\beta+1)}{\Gamma(\beta+\mu+1)}x_j^{\beta+\mu}. \] This proves (i).
For (ii), first assume \(\mu\notin\mathbb{N}\) and write \(p:=\lceil \mu\rceil\). Since \[ \partial_j^p\!\big(\phi(\widehat{x}_j)x_j^\beta\big) = \phi(\widehat{x}_j)\, \frac{\Gamma(\beta+1)}{\Gamma(\beta+1-p)}x_j^{\beta-p}, \] Definition 2.1 yields \[ {}_0^{\mathrm{C}}D_{x_j}^{\mu}\!\big(\phi(\widehat{x}_j)x_j^\beta\big) = \phi(\widehat{x}_j)\, \frac{\Gamma(\beta+1)}{\Gamma(\beta+1-p)} \cdot \frac{1}{\Gamma(p-\mu)} \int_0^{x_j}(x_j-t)^{p-\mu-1}t^{\beta-p}\,dt. \] Applying part (i) with exponent \(\beta-p\) and order \(p-\mu\) gives \[ \frac{1}{\Gamma(p-\mu)} \int_0^{x_j}(x_j-t)^{p-\mu-1}t^{\beta-p}\,dt = \frac{\Gamma(\beta+1-p)}{\Gamma(\beta+1-\mu)}x_j^{\beta-\mu}, \] and the desired formula follows. If \(\mu=p\in\mathbb{N}\), then \({}_0^{\mathrm{C}}D_{x_j}^{\mu}=\partial_j^p\), and the same expression follows immediately.
For (iii), if \(0 < \mu < 1\), then the Caputo derivative depends on the first partial derivative with respect to \(x_j\), and \[ \partial_j \phi(\widehat{x}_j)=0. \] Hence the Caputo derivative vanishes identically.
3. The canonical multi-graded monomial space
We now define the multi-indexed monomial lattice that will carry the commuting shift algebra.
Definition 3.1. Let \[ \boldsymbol{\alpha}=(\alpha_1,\dots,\alpha_d)\in(0,1)^d. \] For each \[ \mathbf{k}=(k_1,\dots,k_d)\in\mathbb{N}_0^d, \] define \[ e_{\mathbf{k}}(x) := \prod_{j=1}^d \frac{x_j^{k_j\alpha_j}}{\Gamma(k_j\alpha_j+1)}, \qquad x=(x_1,\dots,x_d)\in(0,\infty)^d. \] The canonical multi-graded monomial space is the algebraic direct sum \[ \mathcal{G}_{\boldsymbol{\alpha}}^{\mathrm{alg}} := \bigoplus_{\mathbf{k}\in\mathbb{N}_0^d}\mathbb{C}e_{\mathbf{k}}. \] Its \(\mathbf{k}\)-th homogeneous component is \[ \mathcal{G}_{\boldsymbol{\alpha}}^{(\mathbf{k})} := \mathbb{C}e_{\mathbf{k}}. \] The vector \[ e_{\mathbf{0}}=1 \] will be called the vacuum vector.
Remark 3.2. The basis functions \(e_{\mathbf{k}}\) are pairwise linearly independent because they have distinct exponent tuples \[ (k_1\alpha_1,\dots,k_d\alpha_d). \] Hence the direct sum in Definition 3.1 is well defined. Every element of \(\mathcal{G}_{\boldsymbol{\alpha}}^{\mathrm{alg}}\) is, by definition, a finite linear combination of the basis vectors.
4. The commuting shift realization
For the remainder of the paper, fix \[ \boldsymbol{\alpha}=(\alpha_1,\dots,\alpha_d)\in(0,1)^d, \] and define, for \(1\le j\le d\), \[ J_j:={}_0 I_{x_j}^{\alpha_j}, \qquad C_j:={}_0^{\mathrm{C}}D_{x_j}^{\alpha_j}. \]
Theorem 4.1. For every \(\mathbf{k}\in\mathbb{N}_0^d\) and every \(j\in\{1,\dots,d\}\), \[ J_j e_{\mathbf{k}} = e_{\mathbf{k}+\mathbf{e}_j}, \] and \[ C_j e_{\mathbf{k}} = \begin{cases} 0, & k_j=0,\\[1mm] e_{\mathbf{k}-\mathbf{e}_j}, & k_j\ge 1. \end{cases} \] In particular, each \(J_j\) and each \(C_j\) defines a linear endomorphism of \(\mathcal{G}_{\boldsymbol{\alpha}}^{\mathrm{alg}}\).
Proof.
Fix \(\mathbf{k}=(k_1,\dots,k_d)\) and \(j\in\{1,\dots,d\}\). Write \[ e_{\mathbf{k}}(x) = \phi(\widehat{x}_j)\, \frac{x_j^{k_j\alpha_j}}{\Gamma(k_j\alpha_j+1)}, \] where \[ \phi(\widehat{x}_j) := \prod_{\substack{1\le i\le d\\ i\neq j}} \frac{x_i^{k_i\alpha_i}}{\Gamma(k_i\alpha_i+1)}. \] Since \(\phi\) is independent of \(x_j\), Lemma 2.2(i) with \(\mu=\alpha_j\) and \(\beta=k_j\alpha_j\) yields \[ J_j e_{\mathbf{k}} = \phi(\widehat{x}_j)\, \frac{x_j^{(k_j+1)\alpha_j}}{\Gamma((k_j+1)\alpha_j+1)} = e_{\mathbf{k}+\mathbf{e}_j}. \]
If \(k_j=0\), then \[ e_{\mathbf{k}}(x)=\phi(\widehat{x}_j), \] so Lemma 2.2(iii) gives \[ C_j e_{\mathbf{k}}=0. \] If \(k_j\ge 1\), then \(k_j\alpha_j>0\), so Lemma 2.2(ii) with \(\mu=\alpha_j\) and \(\beta=k_j\alpha_j\) yields \[ C_j e_{\mathbf{k}} = \phi(\widehat{x}_j)\, \frac{x_j^{(k_j-1)\alpha_j}}{\Gamma((k_j-1)\alpha_j+1)} = e_{\mathbf{k}-\mathbf{e}_j}. \] The final statement follows by linearity because every vector in \(\mathcal{G}_{\boldsymbol{\alpha}}^{\mathrm{alg}}\) is a finite linear combination of basis vectors.
Corollary 4.2. For all \(i,j\in\{1,\dots,d\}\) with \(i\neq j\), \[ J_iJ_j=J_jJ_i, \qquad C_iC_j=C_jC_i, \qquad C_iJ_j=J_jC_i. \] Hence the tuples \[ (J_1,\dots,J_d) \qquad \text{and} \qquad (C_1,\dots,C_d) \] are commuting tuples on \(\mathcal{G}_{\boldsymbol{\alpha}}^{\mathrm{alg}}\).
Proof.
It suffices to verify the identities on the basis vectors.
For the integrals, \[ J_iJ_j e_{\mathbf{k}} = J_i e_{\mathbf{k}+\mathbf{e}_j} = e_{\mathbf{k}+\mathbf{e}_j+\mathbf{e}_i} = J_j e_{\mathbf{k}+\mathbf{e}_i} = J_jJ_i e_{\mathbf{k}}. \]
For the Caputo derivatives, one has \[ C_iC_j e_{\mathbf{k}} = \begin{cases} e_{\mathbf{k}-\mathbf{e}_i-\mathbf{e}_j}, & k_i\ge 1 \text{ and } k_j\ge 1,\\[1mm] 0, & \text{otherwise}, \end{cases} \] which is symmetric in \(i\) and \(j\), so \(C_iC_j=C_jC_i\).
Finally, if \(i\neq j\), then \[ C_iJ_j e_{\mathbf{k}} = \begin{cases} e_{\mathbf{k}+\mathbf{e}_j-\mathbf{e}_i}, & k_i\ge 1,\\[1mm] 0, & k_i=0, \end{cases} \] and the same formula is obtained for \(J_jC_i e_{\mathbf{k}}\). Therefore \(C_iJ_j=J_jC_i\).
Definition 4.3. For \(j\in\{1,\dots,d\}\), define a linear projection \[ \Pi_j:\mathcal{G}_{\boldsymbol{\alpha}}^{\mathrm{alg}} \longrightarrow \mathcal{G}_{\boldsymbol{\alpha}}^{\mathrm{alg}} \] by \[ \Pi_j e_{\mathbf{k}} := \begin{cases} e_{\mathbf{k}}, & k_j=0,\\ 0, & k_j\ge 1. \end{cases} \] Thus \(\Pi_j\) is the projection onto the coordinate-vacuum hyperplane \[ \mathcal{V}_j := \operatorname{span}\{\,e_{\mathbf{k}} : k_j=0\,\}. \]
Theorem 4.4. For every \(j\in\{1,\dots,d\}\), \[ C_jJ_j=I, \qquad J_jC_j=I-\Pi_j, \qquad [C_j,J_j]=\Pi_j. \]
Proof.
It suffices to verify the formulas on the basis vectors.
For every \(\mathbf{k}\in\mathbb{N}_0^d\), \[ C_jJ_j e_{\mathbf{k}} = C_j e_{\mathbf{k}+\mathbf{e}_j} = e_{\mathbf{k}}, \] so \(C_jJ_j=I\).
If \(k_j\ge 1\), then \[ J_jC_j e_{\mathbf{k}} = J_j e_{\mathbf{k}-\mathbf{e}_j} = e_{\mathbf{k}}, \] whereas if \(k_j=0\), then \[ J_jC_j e_{\mathbf{k}} = J_j 0 = 0. \] Hence \[ J_jC_j e_{\mathbf{k}} = (I-\Pi_j)e_{\mathbf{k}} \] for every \(\mathbf{k}\), proving \[ J_jC_j=I-\Pi_j. \] The commutator identity follows immediately: \[ [C_j,J_j] = C_jJ_j-J_jC_j = I-(I-\Pi_j) = \Pi_j.\tag*{\(\square\)} \]
Definition 4.5. Let \(c_{00}(\mathbb{N}_0^d)\) denote the vector space of finitely supported complex-valued functions on \(\mathbb{N}_0^d\). Let \(\{u_{\mathbf{k}}\}_{\mathbf{k}\in\mathbb{N}_0^d}\) be its standard basis. For each \(j\in\{1,\dots,d\}\), define linear operators \[ S_j^+u_{\mathbf{k}}:=u_{\mathbf{k}+\mathbf{e}_j}, \] and \[ S_j^-u_{\mathbf{k}} := \begin{cases} 0, & k_j=0,\\ u_{\mathbf{k}-\mathbf{e}_j}, & k_j\ge 1. \end{cases} \]
Corollary 4.6. The map \[ U:\mathcal{G}_{\boldsymbol{\alpha}}^{\mathrm{alg}} \longrightarrow c_{00}(\mathbb{N}_0^d), \qquad U(e_{\mathbf{k}}):=u_{\mathbf{k}}, \] is a vector space isomorphism, and \[ UJ_jU^{-1}=S_j^+, \qquad UC_jU^{-1}=S_j^-, \qquad (1\le j\le d). \] Thus the family \[ \{J_1,\dots,J_d,C_1,\dots,C_d\} \] is exactly the standard commuting multidimensional shift algebra transported to \(\mathcal{G}_{\boldsymbol{\alpha}}^{\mathrm{alg}}\).
Proof.
Since \(\{e_{\mathbf{k}}\}\) and \(\{u_{\mathbf{k}}\}\) are bases, \(U\) is a vector space isomorphism. By Theorem 4.1, \[ UJ_jU^{-1}u_{\mathbf{k}} = UJ_j e_{\mathbf{k}} = Ue_{\mathbf{k}+\mathbf{e}_j} = u_{\mathbf{k}+\mathbf{e}_j} = S_j^+u_{\mathbf{k}}, \] and similarly \[ UC_jU^{-1}u_{\mathbf{k}} = \begin{cases} 0, & k_j=0,\\ u_{\mathbf{k}-\mathbf{e}_j}, & k_j\ge 1, \end{cases} = S_j^-u_{\mathbf{k}}.\tag*{\(\square\)} \]
Remark 4.7. When \(d=1\), Theorem 4.4 and Corollary 4.6 recover exactly the one-dimensional shift algebra of [2]. For \(d\ge 2\), the essential new feature is that each \(\Pi_j\) has infinite rank: the one-dimensional vacuum defect becomes a family of coordinate-vacuum hyperplanes.
5. Higher powers, tail spaces, and boundary defects
We now analyze higher powers of the basic shifts and compare them with the corresponding higher-order partial fractional operators.
Definition 5.1. Let \[ \mathbf{m}=(m_1,\dots,m_d)\in\mathbb{N}_0^d. \] Define \[ J^{\mathbf{m}} := J_1^{m_1}\cdots J_d^{m_d}, \qquad C^{\mathbf{m}} := C_1^{m_1}\cdots C_d^{m_d}. \] These are well defined by Corollary 4.2.
The corresponding tail subspace is \[ \mathcal{G}_{\boldsymbol{\alpha}}^{(\ge \mathbf{m})} := \bigoplus_{\mathbf{k}\ge \mathbf{m}}\mathbb{C}e_{\mathbf{k}}. \]
If \[ \boldsymbol{\mu}=(\mu_1,\dots,\mu_d)\in(0,\infty)^d, \] we define the coordinatewise higher-order partial operators by \[ {}_0 I_x^{\boldsymbol{\mu}} := {}_0 I_{x_1}^{\mu_1}\cdots {}_0 I_{x_d}^{\mu_d}, \] and \[ {}_0^{\mathrm{C}}D_x^{\boldsymbol{\mu}} := {}_0^{\mathrm{C}}D_{x_1}^{\mu_1}\cdots {}_0^{\mathrm{C}}D_{x_d}^{\mu_d}, \] with composition taken in the natural order \(1,\dots,d\). On the canonical lattice this order is immaterial because the operators commute there.
Lemma 5.2. Let \[ \mathbf{m}\in\mathbb{N}_0^d. \] Then, for every \[ \mathbf{k}\in\mathbb{N}_0^d, \] one has \[ J^{\mathbf{m}} e_{\mathbf{k}} = e_{\mathbf{k}+\mathbf{m}}, \] and \[ C^{\mathbf{m}} e_{\mathbf{k}} = \begin{cases} e_{\mathbf{k}-\mathbf{m}}, & \mathbf{k}\ge \mathbf{m},\\[1mm] 0, & \mathbf{k}\not\ge \mathbf{m}. \end{cases} \]
Proof.
The formula for \(J^{\mathbf{m}}\) follows by repeated application of Theorem 4.1: \[ J^{\mathbf{m}} e_{\mathbf{k}} = J_1^{m_1}\cdots J_d^{m_d} e_{\mathbf{k}} = e_{\mathbf{k}+\mathbf{m}}. \]
For \(C^{\mathbf{m}}\), repeated application of Theorem 4.1 shows that each \(C_j^{m_j}\) lowers the \(j\)-th coordinate by \(m_j\) if possible and annihilates the vector otherwise. Since the \(C_j\) commute, the result is \[ C^{\mathbf{m}} e_{\mathbf{k}} = \begin{cases} e_{\mathbf{k}-\mathbf{m}}, & \mathbf{k}\ge \mathbf{m},\\ 0, & \mathbf{k}\not\ge \mathbf{m}. \end{cases} \]
Theorem 5.3. Let \[ \mathbf{m}\in\mathbb{N}_0^d. \] Then \[ J^{\mathbf{m}} = {}_0 I_x^{\mathbf{m}\odot\boldsymbol{\alpha}} \] on all of \(\mathcal{G}_{\boldsymbol{\alpha}}^{\mathrm{alg}}\).
Proof.
It suffices to verify the identity on the basis vectors. Let \[ \mathbf{k}=(k_1,\dots,k_d)\in\mathbb{N}_0^d. \] By Lemma 5.2, \[ J^{\mathbf{m}} e_{\mathbf{k}}=e_{\mathbf{k}+\mathbf{m}}. \] On the other hand, applying Lemma 2.2(i) coordinatewise gives \[ {}_0 I_x^{\mathbf{m}\odot\boldsymbol{\alpha}} e_{\mathbf{k}} = \prod_{j=1}^d \frac{x_j^{(k_j+m_j)\alpha_j}}{\Gamma((k_j+m_j)\alpha_j+1)} = e_{\mathbf{k}+\mathbf{m}}. \] Hence the two operators agree on the basis and therefore on all of \(\mathcal{G}_{\boldsymbol{\alpha}}^{\mathrm{alg}}\).
Theorem 5.4. Let \[ \mathbf{m}\in\mathbb{N}_0^d. \] Then \[ C^{\mathbf{m}} = {}_0^{\mathrm{C}}D_x^{\mathbf{m}\odot\boldsymbol{\alpha}} \] on the tail subspace \[ \mathcal{G}_{\boldsymbol{\alpha}}^{(\ge \mathbf{m})}. \]
Proof.
It suffices to check the identity on a basis vector \[ e_{\mathbf{k}} \qquad (\mathbf{k}\ge \mathbf{m}). \] By Lemma 5.2, \[ C^{\mathbf{m}}e_{\mathbf{k}} = e_{\mathbf{k}-\mathbf{m}}. \]
We now compute \({}_0^{\mathrm{C}}D_x^{\mathbf{m}\odot\boldsymbol{\alpha}}e_{\mathbf{k}}\) coordinatewise. For each fixed \(j\), set \[ \mu_j:=m_j\alpha_j. \] Since \(\mathbf{k}\ge \mathbf{m}\), one has \[ k_j\alpha_j\ge m_j\alpha_j=\mu_j>\lceil \mu_j\rceil-1. \] Hence Lemma 2.2(ii) applies to the \(j\)-th coordinate. Applying it successively for \(j=1,\dots,d\) yields \[ {}_0^{\mathrm{C}}D_x^{\mathbf{m}\odot\boldsymbol{\alpha}}e_{\mathbf{k}} = \prod_{j=1}^d \frac{x_j^{(k_j-m_j)\alpha_j}}{\Gamma((k_j-m_j)\alpha_j+1)} = e_{\mathbf{k}-\mathbf{m}}. \] Therefore \[ C^{\mathbf{m}}e_{\mathbf{k}} = {}_0^{\mathrm{C}}D_x^{\mathbf{m}\odot\boldsymbol{\alpha}}e_{\mathbf{k}} \] for every \(\mathbf{k}\ge \mathbf{m}\), and the result follows by linearity.
Definition 5.5. Let \[ \mathbf{m}\in\mathbb{N}_0^d. \] Define the boundary-layer projection \[ \Pi_{< \mathbf{m}}:\mathcal{G}_{\boldsymbol{\alpha}}^{\mathrm{alg}} \longrightarrow \mathcal{G}_{\boldsymbol{\alpha}}^{\mathrm{alg}} \] by \[ \Pi_{< \mathbf{m}} e_{\mathbf{k}} := \begin{cases} e_{\mathbf{k}}, & \mathbf{k}\not\ge \mathbf{m},\\ 0, & \mathbf{k}\ge \mathbf{m}. \end{cases} \] Equivalently, \(\Pi_{< \mathbf{m}}\) is the projection onto the complement of the tail subspace \[ \mathcal{G}_{\boldsymbol{\alpha}}^{(\ge \mathbf{m})}. \]
Corollary 5.6. For every \[ \mathbf{m}\in\mathbb{N}_0^d, \] one has \[ C^{\mathbf{m}}J^{\mathbf{m}}=I \] on all of \(\mathcal{G}_{\boldsymbol{\alpha}}^{\mathrm{alg}}\), and \[ J^{\mathbf{m}}C^{\mathbf{m}}=I-\Pi_{< \mathbf{m}}. \]
Proof.
By Lemma 5.2, \[ C^{\mathbf{m}}J^{\mathbf{m}}e_{\mathbf{k}} = C^{\mathbf{m}}e_{\mathbf{k}+\mathbf{m}} = e_{\mathbf{k}} \] for every \(\mathbf{k}\in\mathbb{N}_0^d\), proving \[ C^{\mathbf{m}}J^{\mathbf{m}}=I. \]
Likewise, \[ J^{\mathbf{m}}C^{\mathbf{m}}e_{\mathbf{k}} = \begin{cases} e_{\mathbf{k}}, & \mathbf{k}\ge \mathbf{m},\\[1mm] 0, & \mathbf{k}\not\ge \mathbf{m}, \end{cases} \] which is exactly the action of \[ I-\Pi_{< \mathbf{m}}.\tag*{\(\square\)} \]
Remark 5.7. The multidimensional defect differs qualitatively from the one-dimensional defect studied in [2]. When \(d=1\), the projection \(\Pi_{ < m}\) has finite rank \(m\). For \(d\ge 2\), the boundary-layer projection \(\Pi_{ < \mathbf{m}}\) has infinite rank in general. Thus the higher-order Caputo defect is no longer a finite-dimensional low-grade sector but a union of coordinate boundary layers.
6. Uniqueness of the canonical multi-graded lattice
We now show that the canonical basis is not an arbitrary choice. Under a natural monomial ansatz, it is forced.
Definition 6.1. A graded monomial lattice is a graded vector space \[ \mathcal{M} = \bigoplus_{\mathbf{k}\in\mathbb{N}_0^d}\mathbb{C}f_{\mathbf{k}}, \] where \[ f_{\mathbf{k}}(x) = c_{\mathbf{k}} \prod_{j=1}^d x_j^{\beta_j(\mathbf{k})}, \qquad c_{\mathbf{k}}\in\mathbb{C}\setminus\{0\}, \qquad \beta_j(\mathbf{k})\ge 0. \]
Lemma 6.2. Let \(\mathcal{M}\) be a graded monomial lattice as in Definition 6.1. Assume that \[ J_j f_{\mathbf{k}} = f_{\mathbf{k}+\mathbf{e}_j} \qquad (\mathbf{k}\in\mathbb{N}_0^d,\ 1\le j\le d). \] Then, for every \(\mathbf{k}\in\mathbb{N}_0^d\) and every \(i,j\in\{1,\dots,d\}\), \[ \beta_i(\mathbf{k}+\mathbf{e}_j) = \begin{cases} \beta_i(\mathbf{k}), & i\neq j,\\[1mm] \beta_j(\mathbf{k})+\alpha_j, & i=j, \end{cases} \] and \[ c_{\mathbf{k}+\mathbf{e}_j} = c_{\mathbf{k}}\, \frac{\Gamma(\beta_j(\mathbf{k})+1)} {\Gamma(\beta_j(\mathbf{k})+\alpha_j+1)}. \]
Proof.
Fix \(\mathbf{k}\) and \(j\). Write \[ f_{\mathbf{k}}(x) = c_{\mathbf{k}} \Big(\prod_{\substack{1\le i\le d\\ i\neq j}} x_i^{\beta_i(\mathbf{k})}\Big) x_j^{\beta_j(\mathbf{k})}. \] Applying Lemma 2.2(i) with \[ \phi(\widehat{x}_j) = c_{\mathbf{k}} \prod_{i\neq j}x_i^{\beta_i(\mathbf{k})} \] gives \[ J_j f_{\mathbf{k}} = c_{\mathbf{k}} \frac{\Gamma(\beta_j(\mathbf{k})+1)} {\Gamma(\beta_j(\mathbf{k})+\alpha_j+1)} \Big(\prod_{i\neq j}x_i^{\beta_i(\mathbf{k})}\Big) x_j^{\beta_j(\mathbf{k})+\alpha_j}. \] By assumption, \[ J_j f_{\mathbf{k}}=f_{\mathbf{k}+\mathbf{e}_j} = c_{\mathbf{k}+\mathbf{e}_j} \prod_{i=1}^d x_i^{\beta_i(\mathbf{k}+\mathbf{e}_j)}. \] Since two nonzero monomials on \((0,\infty)^d\) are equal if and only if all corresponding exponents and coefficients agree, the stated identities follow.
Lemma 6.3. Let \(\mathcal{M}\) be a graded monomial lattice as in Definition 6.1. Assume that for some fixed \(j\in\{1,\dots,d\}\) one has \[ C_j f_{\mathbf{k}} = 0 \qquad \text{whenever } k_j=0. \] Then \[ \beta_j(\mathbf{k})=0 \qquad \text{whenever } k_j=0. \]
Proof.
Fix \(\mathbf{k}\) with \(k_j=0\). Suppose, for contradiction, that \[ \beta_j(\mathbf{k})>0. \] Write \[ f_{\mathbf{k}}(x) = c_{\mathbf{k}} \Big(\prod_{i\neq j}x_i^{\beta_i(\mathbf{k})}\Big) x_j^{\beta_j(\mathbf{k})}. \] Since \(0 < \alpha_j < 1\) and \(\beta_j(\mathbf{k})>0\), Lemma 2.2(ii) gives \[ C_j f_{\mathbf{k}} = c_{\mathbf{k}} \frac{\Gamma(\beta_j(\mathbf{k})+1)} {\Gamma(\beta_j(\mathbf{k})+1-\alpha_j)} \Big(\prod_{i\neq j}x_i^{\beta_i(\mathbf{k})}\Big) x_j^{\beta_j(\mathbf{k})-\alpha_j}, \] which is nonzero on \((0,\infty)^d\). This contradicts the assumption \(C_j f_{\mathbf{k}}=0\). Hence \[ \beta_j(\mathbf{k})=0 \] whenever \(k_j=0\).
Theorem 6.4. Let \[ \mathcal{M} = \bigoplus_{\mathbf{k}\in\mathbb{N}_0^d}\mathbb{C}f_{\mathbf{k}} \] be a graded monomial lattice, where \[ f_{\mathbf{k}}(x) = c_{\mathbf{k}} \prod_{j=1}^d x_j^{\beta_j(\mathbf{k})}, \qquad c_{\mathbf{k}}\neq 0, \qquad \beta_j(\mathbf{k})\ge 0. \] Assume that, for every \(\mathbf{k}\in\mathbb{N}_0^d\) and every \(j\in\{1,\dots,d\}\), \[ J_j f_{\mathbf{k}} = f_{\mathbf{k}+\mathbf{e}_j}, \] and \[ C_j f_{\mathbf{k}} = \begin{cases} 0, & k_j=0,\\[1mm] f_{\mathbf{k}-\mathbf{e}_j}, & k_j\ge 1. \end{cases} \] Then there exists a unique nonzero scalar \(c\in\mathbb{C}\) such that \[ f_{\mathbf{k}}(x) = c\, \prod_{j=1}^d \frac{x_j^{k_j\alpha_j}}{\Gamma(k_j\alpha_j+1)} = c\,e_{\mathbf{k}}(x) \qquad (\mathbf{k}\in\mathbb{N}_0^d). \] In particular, up to a global scalar normalization, \[ \mathcal{M} = \mathcal{G}_{\boldsymbol{\alpha}}^{\mathrm{alg}}. \]
Proof.
Fix \(i\in\{1,\dots,d\}\). By Lemma 6.2, if \(j\neq i\) then \[ \beta_i(\mathbf{k}+\mathbf{e}_j)=\beta_i(\mathbf{k}). \] Hence \(\beta_i(\mathbf{k})\) depends only on the single coordinate \(k_i\). Thus there exists a function \[ \varphi_i:\mathbb{N}_0\to [0,\infty) \] such that \[ \beta_i(\mathbf{k})=\varphi_i(k_i) \qquad (\mathbf{k}\in\mathbb{N}_0^d). \]
Again by Lemma 6.2, \[ \varphi_i(k_i+1)=\varphi_i(k_i)+\alpha_i \qquad (k_i\in\mathbb{N}_0). \] On the other hand, Lemma 6.3 implies that \[ \varphi_i(0)=0, \] because whenever \(k_i=0\) we have \(C_i f_{\mathbf{k}}=0\). Therefore \[ \varphi_i(k_i)=k_i\alpha_i \qquad (k_i\in\mathbb{N}_0). \] Since \(i\) was arbitrary, we conclude that \[ \beta_i(\mathbf{k})=k_i\alpha_i \qquad (1\le i\le d,\ \mathbf{k}\in\mathbb{N}_0^d). \]
It remains to determine the coefficients. Define \[ d_{\mathbf{k}} := c_{\mathbf{k}} \prod_{j=1}^d \Gamma(k_j\alpha_j+1). \] By Lemma 6.2, \[ c_{\mathbf{k}+\mathbf{e}_j} = c_{\mathbf{k}}\, \frac{\Gamma(k_j\alpha_j+1)} {\Gamma((k_j+1)\alpha_j+1)}. \] Multiplying both sides by \[ \prod_{i=1}^d \Gamma((k_i+\delta_{ij})\alpha_i+1), \] we obtain \[ d_{\mathbf{k}+\mathbf{e}_j}=d_{\mathbf{k}} \qquad (\mathbf{k}\in\mathbb{N}_0^d,\ 1\le j\le d). \] Since the lattice \(\mathbb{N}_0^d\) is connected by successive additions of basis vectors, it follows that \[ d_{\mathbf{k}}=c \] is constant on all of \(\mathbb{N}_0^d\) for some fixed \(c\in\mathbb{C}\setminus\{0\}\). Therefore \[ c_{\mathbf{k}} = \frac{c}{\prod_{j=1}^d \Gamma(k_j\alpha_j+1)}, \] and hence \[ f_{\mathbf{k}}(x) = c\, \prod_{j=1}^d \frac{x_j^{k_j\alpha_j}}{\Gamma(k_j\alpha_j+1)} = c\,e_{\mathbf{k}}(x). \] The scalar \(c\) is uniquely determined by \(f_{\mathbf{0}}\).
Corollary 6.5. Among graded monomial lattices with one-dimensional homogeneous components, the canonical lattice \[ \mathcal{G}_{\boldsymbol{\alpha}}^{\mathrm{alg}} \] is, up to multiplication of the entire basis by a single nonzero scalar, the unique lattice on which the partial Riemann–Liouville integrals act as forward coordinate shifts and the partial Caputo derivatives act as backward coordinate shifts with coordinate-vacuum annihilation.
Proof.
Immediate from Theorem 6.4.
7. Product Mittag-Leffler generating series and ambient remarks
The basis constructed above is naturally connected with the classical Mittag–Leffler function; see [6].
Proposition 7.1. Let \[ \boldsymbol{\lambda}=(\lambda_1,\dots,\lambda_d)\in\mathbb{C}^d. \] Then \[ \prod_{j=1}^d E_{\alpha_j}(\lambda_j x_j^{\alpha_j}) = \sum_{\mathbf{k}\in\mathbb{N}_0^d} \boldsymbol{\lambda}^{\mathbf{k}} e_{\mathbf{k}}(x), \] where \[ E_{\alpha_j}(z) := \sum_{n=0}^{\infty}\frac{z^n}{\Gamma(\alpha_j n+1)} \] is the one-variable Mittag–Leffler function and \[ \boldsymbol{\lambda}^{\mathbf{k}} := \prod_{j=1}^d \lambda_j^{k_j}. \]
Proof.
For each \(j\), \[ E_{\alpha_j}(\lambda_j x_j^{\alpha_j}) = \sum_{k_j=0}^{\infty} \frac{\lambda_j^{k_j}x_j^{k_j\alpha_j}}{\Gamma(k_j\alpha_j+1)}. \] Multiplying the \(d\) series together gives \[ \prod_{j=1}^d E_{\alpha_j}(\lambda_j x_j^{\alpha_j}) = \sum_{(k_1,\dots,k_d)\in\mathbb{N}_0^d} \prod_{j=1}^d \frac{\lambda_j^{k_j}x_j^{k_j\alpha_j}}{\Gamma(k_j\alpha_j+1)}, \] which is exactly \[ \sum_{\mathbf{k}\in\mathbb{N}_0^d} \boldsymbol{\lambda}^{\mathbf{k}} e_{\mathbf{k}}(x).\tag*{\(\square\)} \]
Remark 7.2. The algebraic direct sum \(\mathcal{G}_{\boldsymbol{\alpha}}^{\mathrm{alg}}\) consists only of finite linear combinations of the \(e_{\mathbf{k}}\), so the full series in Proposition 7.1 does not itself belong to \(\mathcal{G}_{\boldsymbol{\alpha}}^{\mathrm{alg}}\). Nevertheless, it shows that the basis \(\{e_{\mathbf{k}}\}\) is precisely the coefficient lattice of the product Mittag–Leffler function.
Remark 7.3. In any completion that allows termwise application of the operators, one formally obtains \[ C_j\!\left(\prod_{i=1}^d E_{\alpha_i}(\lambda_i x_i^{\alpha_i})\right) = \lambda_j \prod_{i=1}^d E_{\alpha_i}(\lambda_i x_i^{\alpha_i}), \qquad 1\le j\le d. \] Thus the product Mittag–Leffler function is the generating eigenfunction of the commuting Caputo shift tuple.
Remark 7.4. The commuting relations proved in this paper are specific to the canonical lattice \(\mathcal{G}_{\boldsymbol{\alpha}}^{\mathrm{alg}}\). They should not be confused with the behavior of mixed partial Riemann–Liouville or Caputo derivatives on general ambient function spaces, where commutativity may fail; see [7, Example 2]. In this sense, the present paper isolates a canonical commuting submodel rather than a full representation of generic multidimensional fractional calculus.
8. Conclusion
We have identified a canonical multi-indexed monomial lattice on which the partial Riemann–Liouville integrals and the partial Caputo derivatives form a commuting shift algebra. The central object is \[ \mathcal{G}_{\boldsymbol{\alpha}}^{\mathrm{alg}} = \bigoplus_{\mathbf{k}\in\mathbb{N}_0^d} \mathbb{C} \prod_{j=1}^d\frac{x_j^{k_j\alpha_j}}{\Gamma(k_j\alpha_j+1)}, \] and its defining properties are the following:
- the grading follows the \(\boldsymbol{\alpha}\)-lattice of multi-exponents;
- each partial Riemann–Liouville integral \(J_j\) acts as a forward shift in the \(j\)-th coordinate;
- each partial Caputo derivative \(C_j\) acts as the corresponding backward shift with coordinate-vacuum annihilation.
The model is not merely convenient but canonical: Theorem 6.4 shows that, under a natural monomial ansatz, it is forced up to a global scalar normalization. Moreover, higher powers of the shift tuple recover the expected higher-order partial fractional operators on the appropriate sectors, and the multidimensional Caputo defect is made completely explicit through the boundary-layer projections \(\Pi_{< \mathbf{m}}\).
In this sense, \(\mathcal{G}_{\boldsymbol{\alpha}}^{\mathrm{alg}}\) is the natural algebraic internal model for the coordinatewise partial Riemann–Liouville and Caputo operators of orders \(\alpha_1,\dots,\alpha_d\). It provides the multi-graded, multidimensional counterpart of the one-dimensional canonical shift algebra of [2], and it stands in deliberate contrast to the finite-dimensional obstruction described in [1].
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