Abstract. Let \(0<\alpha<1\). We prove that the algebraic direct sum \[ \mathcal{G}_{\alpha}^{\mathrm{alg}} := \bigoplus_{n=0}^{\infty}\mathbb{C}e_n, \quad e_n(x):=\frac{x^{n\alpha}}{\Gamma(n\alpha+1)}, \] is the canonical graded monomial space on which the order-\(\alpha\) Riemann–Liouville integral and the order-\(\alpha\) Caputo derivative realize a unilateral shift algebra. More precisely, if \[ J_\alpha:={}_0 I_x^\alpha, \quad C_\alpha:={}_0^{\mathrm{C}}D_x^\alpha, \] then \[ J_\alpha e_n=e_{n+1}\quad(n\ge 0), \quad C_\alpha e_0=0, \quad C_\alpha e_n=e_{n-1}\quad(n\ge 1). \] Consequently, \[ C_\alpha J_\alpha=I, \quad J_\alpha C_\alpha=I-\Pi_0, \quad [C_\alpha,J_\alpha]=\Pi_0, \] where \(\Pi_0\) is the projection onto the vacuum component. We further prove a uniqueness theorem: among graded monomial chains with one-dimensional homogeneous components, this vacuum-anchored \(\alpha\)-chain is, up to a global scalar normalization, the unique chain on which \(J_\alpha\) and \(C_\alpha\) act as forward and backward shifts. Finally, for every \(m\in\mathbb{N}\) we show that \[ J_\alpha^m = {}_0 I_x^{m\alpha} \] on all of \(\mathcal{G}_{\alpha}^{\mathrm{alg}}\), while \[ C_\alpha^m = {}_0^{\mathrm{C}}D_x^{m\alpha} \] on the tail subspace \[ \mathcal{G}_{\alpha}^{(\ge m)} := \bigoplus_{n=m}^{\infty}\mathbb{C}e_n. \] Thus the partial semigroup behavior of the Caputo derivative is encoded exactly by the finite-dimensional low-grade defect sector.
Keywords. fractional calculus; Caputo derivative; Riemann–Liouville integral; graded space; shift algebra; Mittag-Leffler basis
1. Introduction
A central intuition in fractional calculus is that noninteger operators should act as natural analogues of ordinary shifts of degree. In finite-dimensional spaces built from ordinary polynomials this principle fails, because the integer lattice of exponents is too rigid. The present paper shows that once one replaces the ordinary grading \[ 0,1,2,3,\dots \] by the \(\alpha\)-grading \[ 0,\alpha,2\alpha,3\alpha,\dots, \] the basic order-\(\alpha\) fractional operators acquire an exact shift-algebra form.
More precisely, we study the order-\(\alpha\) Riemann–Liouville integral \[ J_\alpha={}_0 I_x^\alpha \] and the order-\(\alpha\) Caputo derivative \[ C_\alpha={}_0^{\mathrm{C}}D_x^\alpha \] on the half-line \((0,\infty)\), where \(0<\alpha<1\). We exhibit a distinguished graded monomial basis \[ e_n(x)=\frac{x^{n\alpha}}{\Gamma(n\alpha+1)}, \qquad n\in\mathbb{N}_0, \] for which \(J_\alpha\) raises the grade by one and \(C_\alpha\) lowers the grade by one, except at the vacuum vector \(e_0=1\), which is annihilated.
The resulting space \[ \mathcal{G}_{\alpha}^{\mathrm{alg}} := \bigoplus_{n=0}^{\infty}\mathbb{C}e_n \] should be understood as an algebraic core. No topology is imposed, and every vector is by definition a finite linear combination of the basis vectors. This avoids convergence issues and isolates the exact algebraic skeleton of the fractional operators.
The paper has three principal aims. First, we prove the existence of the shift realization. Second, we prove its uniqueness among graded monomial chains with one-dimensional homogeneous components. Third, we show that higher powers of the basic shifts recover the higher-order Riemann–Liouville integral globally and the higher-order Caputo derivative on natural tail subspaces, thereby making the low-grade obstruction entirely explicit.
Throughout the paper, all identities are understood pointwise for \(x>0\).
2. Preliminaries
Definition 2.1. Let \(\mu>0\). The left-sided Riemann–Liouville fractional integral of order \(\mu\) is defined by \[ ({}_0 I_x^\mu f)(x) := \frac{1}{\Gamma(\mu)} \int_0^x (x-t)^{\mu-1}f(t)\,dt, \qquad x>0, \] whenever the integral exists.
If \(\mu\notin\mathbb{N}\) and \(p:=\lceil \mu\rceil\), the left-sided Caputo derivative of order \(\mu\) is defined by \[ ({}_0^{\mathrm{C}}D_x^\mu f)(x) := \frac{1}{\Gamma(p-\mu)} \int_0^x (x-t)^{p-\mu-1}f^{(p)}(t)\,dt, \qquad x>0, \] whenever the integral exists. If \(\mu=p\in\mathbb{N}\), we set \[ {}_0^{\mathrm{C}}D_x^p f := f^{(p)}. \]
Lemma 2.2. Let \(\mu>0\) and \(\beta>-1\). Then \[ {}_0 I_x^\mu\!\left(\frac{x^\beta}{\Gamma(\beta+1)}\right) = \frac{x^{\beta+\mu}}{\Gamma(\beta+\mu+1)}. \]
Proof.
By Definition 2.1, \[ {}_0 I_x^\mu\!\left(\frac{x^\beta}{\Gamma(\beta+1)}\right) = \frac{1}{\Gamma(\mu)\Gamma(\beta+1)} \int_0^x (x-t)^{\mu-1}t^\beta\,dt. \] Substituting \(t=xu\), we obtain \[ \frac{x^{\beta+\mu}}{\Gamma(\mu)\Gamma(\beta+1)} \int_0^1 (1-u)^{\mu-1}u^\beta\,du = \frac{x^{\beta+\mu}}{\Gamma(\mu)\Gamma(\beta+1)} B(\beta+1,\mu), \] where \(B\) is the beta function. Since \[ B(\beta+1,\mu) = \frac{\Gamma(\beta+1)\Gamma(\mu)}{\Gamma(\beta+\mu+1)}, \] the result follows.
Lemma 2.3. Let \(\mu>0\), let \(p:=\lceil \mu\rceil\), and assume \(\beta>p-1\). Then \[ {}_0^{\mathrm{C}}D_x^\mu\!\left(\frac{x^\beta}{\Gamma(\beta+1)}\right) = \frac{x^{\beta-\mu}}{\Gamma(\beta+1-\mu)}. \] In addition, if \(0<\mu<1\), then \[ {}_0^{\mathrm{C}}D_x^\mu(1)=0. \]
Proof.
First consider the case \(\mu\notin\mathbb{N}\). Since \(\beta>p-1\), the \(p\)-th ordinary derivative exists on \((0,\infty)\) and equals \[ \frac{d^p}{dt^p}\left(\frac{t^\beta}{\Gamma(\beta+1)}\right) = \frac{t^{\beta-p}}{\Gamma(\beta+1-p)}. \] Hence \[ {}_0^{\mathrm{C}}D_x^\mu\!\left(\frac{x^\beta}{\Gamma(\beta+1)}\right) = \frac{1}{\Gamma(p-\mu)\Gamma(\beta+1-p)} \int_0^x (x-t)^{p-\mu-1}t^{\beta-p}\,dt. \] The substitution \(t=xu\) yields \[ \frac{x^{\beta-\mu}}{\Gamma(p-\mu)\Gamma(\beta+1-p)} B(\beta+1-p,p-\mu), \] and the beta–gamma identity gives \[ B(\beta+1-p,p-\mu) = \frac{\Gamma(\beta+1-p)\Gamma(p-\mu)}{\Gamma(\beta+1-\mu)}. \] Therefore \[ {}_0^{\mathrm{C}}D_x^\mu\!\left(\frac{x^\beta}{\Gamma(\beta+1)}\right) = \frac{x^{\beta-\mu}}{\Gamma(\beta+1-\mu)}. \]
If \(\mu=p\in\mathbb{N}\), then the Caputo derivative is the ordinary derivative of order \(p\), and the same formula follows immediately.
Finally, if \(0<\mu<1\), then the Caputo derivative depends on the first ordinary derivative, and since the derivative of the constant function \(1\) is zero, one has \[ {}_0^{\mathrm{C}}D_x^\mu(1)=0.\tag*{\(\square\)} \]
3. Shift-stable graded monomial chains
We begin with a one-parameter family of candidate graded chains and then isolate the unique one that is stable under both basic operators.
Definition 3.1. Let \(\rho\ge 0\). For \(n\in\mathbb{N}_0\), define \[ e_n^{(\rho)}(x) := \frac{x^{\rho+n\alpha}}{\Gamma(\rho+n\alpha+1)}. \] The associated graded monomial chain is \[ \mathcal{G}_{\alpha,\rho}^{\mathrm{alg}} := \bigoplus_{n=0}^{\infty}\mathbb{C}e_n^{(\rho)}. \]
Proposition 3.2. Let \(\rho\ge 0\). Then:
- For every \(n\in\mathbb{N}_0\), \[ J_\alpha e_n^{(\rho)}=e_{n+1}^{(\rho)}. \]
- For every \(n\ge 1\), \[ C_\alpha e_n^{(\rho)}=e_{n-1}^{(\rho)}. \]
- For \(n=0\), \[ C_\alpha e_0^{(\rho)} = \begin{cases} 0, & \rho=0, \\[2mm] \dfrac{x^{\rho-\alpha}}{\Gamma(\rho+1-\alpha)}, & \rho>0. \end{cases} \]
Proof.
For (i), apply Lemma 2.2 with \(\mu=\alpha\) and \(\beta=\rho+n\alpha\): \[ J_\alpha e_n^{(\rho)} = {}_0 I_x^\alpha\!\left(\frac{x^{\rho+n\alpha}}{\Gamma(\rho+n\alpha+1)}\right) = \frac{x^{\rho+(n+1)\alpha}}{\Gamma(\rho+(n+1)\alpha+1)} = e_{n+1}^{(\rho)}. \]
For (ii), note that \(\rho+n\alpha>0\) for \(n\ge 1\). Since \(0<\alpha<1\), Lemma 2.3 with \(\mu=\alpha\) and \(\beta=\rho+n\alpha\) yields \[ C_\alpha e_n^{(\rho)} = \frac{x^{\rho+(n-1)\alpha}}{\Gamma(\rho+(n-1)\alpha+1)} = e_{n-1}^{(\rho)}. \]
For (iii), if \(\rho=0\), then \(e_0^{(0)}=1\), so \(C_\alpha e_0^{(0)}=0\) by Lemma 2.3. If \(\rho>0\), then Lemma 2.3 gives \[ C_\alpha e_0^{(\rho)} = \frac{x^{\rho-\alpha}}{\Gamma(\rho+1-\alpha)}.\tag*{\(\square\)} \]
Corollary 3.3. The chain \(\mathcal{G}_{\alpha,\rho}^{\mathrm{alg}}\) is invariant under both \(J_\alpha\) and \(C_\alpha\) if and only if \(\rho=0\).
Proof.
By Proposition 3.2(i), every \(\mathcal{G}_{\alpha,\rho}^{\mathrm{alg}}\) is invariant under \(J_\alpha\). If \(\rho=0\), then Proposition 3.2(ii)–(iii) shows that \(C_\alpha\) also preserves the space. If \(\rho>0\), then Proposition 3.2(iii) gives \[ C_\alpha e_0^{(\rho)} = \frac{x^{\rho-\alpha}}{\Gamma(\rho+1-\alpha)}, \] whose exponent \(\rho-\alpha\) is not of the form \(\rho+n\alpha\) with \(n\in\mathbb{N}_0\). Hence \[ C_\alpha e_0^{(\rho)}\notin \mathcal{G}_{\alpha,\rho}^{\mathrm{alg}}. \] Therefore invariance under \(C_\alpha\) fails whenever \(\rho>0\).
Definition 3.4. The canonical \(\alpha\)-graded monomial space is \[ \mathcal{G}_{\alpha}^{\mathrm{alg}} := \mathcal{G}_{\alpha,0}^{\mathrm{alg}} = \bigoplus_{n=0}^{\infty}\mathbb{C}e_n, \qquad e_n(x):=\frac{x^{n\alpha}}{\Gamma(n\alpha+1)}. \] We call \(e_0=1\) the vacuum vector.
4. The shift algebra on the canonical chain
We now show that \(\mathcal{G}_{\alpha}^{\mathrm{alg}}\) is precisely the desired shift space.
Theorem 4.1. For every \(n\in\mathbb{N}_0\), \[ J_\alpha e_n=e_{n+1}, \] \[ C_\alpha e_0=0, \qquad C_\alpha e_n=e_{n-1}\quad(n\ge 1). \] In particular, \(J_\alpha\) and \(C_\alpha\) define linear endomorphisms of \(\mathcal{G}_{\alpha}^{\mathrm{alg}}\).
Proof.
This is the specialization of Proposition 3.2 to the case \(\rho=0\). Since \(\mathcal{G}_{\alpha}^{\mathrm{alg}}\) is an algebraic direct sum, every element is a finite sum of basis vectors, and the displayed formulas define linear endomorphisms by linear extension.
Corollary 4.2. For every \[ f=\sum_{n=0}^{N}a_n e_n\in \mathcal{G}_{\alpha}^{\mathrm{alg}}, \] one has \[ J_\alpha f=\sum_{n=0}^{N}a_n e_{n+1}, \qquad C_\alpha f=\sum_{n=1}^{N}a_n e_{n-1}. \]
Proof.
Immediate from Theorem 4.1 by linearity.
Definition 4.3. Let \(c_{00}\) denote the vector space of finitely supported complex sequences. Let \(\{u_n\}_{n\ge 0}\) be its standard basis, and define linear operators \[ S_+u_n:=u_{n+1}\quad(n\ge 0), \] \[ S_-u_0:=0, \qquad S_-u_n:=u_{n-1}\quad(n\ge 1). \]
Corollary 4.4. The map \[ U:\mathcal{G}_{\alpha}^{\mathrm{alg}}\to c_{00}, \qquad U(e_n):=u_n, \] is a vector space isomorphism, and \[ UJ_\alpha U^{-1}=S_+, \qquad UC_\alpha U^{-1}=S_-. \] Thus \((J_\alpha,C_\alpha)\) is exactly the unilateral shift pair transported to \(\mathcal{G}_{\alpha}^{\mathrm{alg}}\).
Proof.
Since \(\{e_n\}_{n\ge 0}\) and \(\{u_n\}_{n\ge 0}\) are bases, \(U\) is a vector space isomorphism. The intertwining relations follow from Theorem 4.1: \[ UJ_\alpha U^{-1}u_n = UJ_\alpha e_n = Ue_{n+1} = u_{n+1} = S_+u_n, \] and similarly \[ UC_\alpha U^{-1}u_0 = UC_\alpha e_0 = 0 = S_-u_0, \] while for \(n\ge 1\), \[ UC_\alpha U^{-1}u_n = UC_\alpha e_n = Ue_{n-1} = u_{n-1} = S_-u_n.\tag*{\(\square\)} \]
Definition 4.5. For \(n\in\mathbb{N}_0\), let \[ \Pi_n:\mathcal{G}_{\alpha}^{\mathrm{alg}}\to \mathcal{G}_{\alpha}^{\mathrm{alg}} \] denote the projection onto the \(n\)-th homogeneous component: \[ \Pi_n\!\left(\sum_{k=0}^{N}a_k e_k\right):=a_n e_n. \] In particular, \(\Pi_0\) is the vacuum projection.
Theorem 4.6. On \(\mathcal{G}_{\alpha}^{\mathrm{alg}}\), \[ C_\alpha J_\alpha=I, \qquad J_\alpha C_\alpha=I-\Pi_0, \qquad [C_\alpha,J_\alpha]=\Pi_0. \]
Proof.
It suffices to verify the identities on the basis \(\{e_n\}_{n\ge 0}\).
First, for every \(n\ge 0\), \[ C_\alpha J_\alpha e_n = C_\alpha e_{n+1} = e_n. \] Hence \(C_\alpha J_\alpha=I\).
Next, if \(n\ge 1\), then \[ J_\alpha C_\alpha e_n = J_\alpha e_{n-1} = e_n, \] while \[ J_\alpha C_\alpha e_0 = J_\alpha 0 = 0. \] Therefore \(J_\alpha C_\alpha\) acts as the identity on all positive grades and kills the vacuum component, so \[ J_\alpha C_\alpha=I-\Pi_0. \] The commutator identity is now immediate: \[ [C_\alpha,J_\alpha] = C_\alpha J_\alpha-J_\alpha C_\alpha = I-(I-\Pi_0) = \Pi_0.\tag*{\(\square\)} \]
5. Uniqueness of the canonical chain
We now show that the graded monomial chain constructed above is not accidental. Under a natural monomial ansatz, it is forced.
Definition 5.1. A graded monomial chain is a graded vector space \[ \mathcal{M} = \bigoplus_{n=0}^{\infty}\mathbb{C}f_n, \] where \[ f_n(x)=c_n x^{\beta_n}, \qquad c_n\in\mathbb{C}\setminus\{0\}, \quad \beta_n\ge 0. \]
Lemma 5.2. Let \(\mathcal{M}=\bigoplus_{n\ge 0}\mathbb{C}f_n\) be a graded monomial chain, with \[ f_n(x)=c_n x^{\beta_n}. \] Assume that \[ J_\alpha f_n=f_{n+1} \qquad(n\ge 0). \] Then \[ \beta_{n+1}=\beta_n+\alpha \qquad(n\ge 0), \] and \[ c_{n+1} = c_n\frac{\Gamma(\beta_n+1)}{\Gamma(\beta_n+\alpha+1)} \qquad(n\ge 0). \]
Proof.
By Lemma 2.2, \[ J_\alpha f_n = c_n\,{}_0 I_x^\alpha(x^{\beta_n}) = c_n\frac{\Gamma(\beta_n+1)}{\Gamma(\beta_n+\alpha+1)}x^{\beta_n+\alpha}. \] Since \(J_\alpha f_n=f_{n+1}=c_{n+1}x^{\beta_{n+1}}\) and two nonzero monomials on \((0,\infty)\) are equal if and only if both their exponents and coefficients agree, the two stated identities follow.
Lemma 5.3. Let \(\mathcal{M}=\bigoplus_{n\ge 0}\mathbb{C}f_n\) be a graded monomial chain, with \[ f_n(x)=c_n x^{\beta_n}, \qquad \beta_n\ge 0. \] Assume that \[ C_\alpha f_0=0. \] Then necessarily \[ \beta_0=0. \]
Proof.
Suppose, to the contrary, that \(\beta_0>0\). Since \(0<\alpha<1\), Lemma 2.3 applies and gives \[ C_\alpha f_0 = c_0\frac{x^{\beta_0-\alpha}}{\Gamma(\beta_0+1-\alpha)}. \] This is nonzero on \((0,\infty)\), contradicting the assumption \(C_\alpha f_0=0\). Hence \(\beta_0=0\).
Theorem 5.4. Let \(\mathcal{M}=\bigoplus_{n\ge 0}\mathbb{C}f_n\) be a graded monomial chain, with \[ f_n(x)=c_n x^{\beta_n}, \qquad c_n\neq 0, \quad \beta_n\ge 0. \] Assume that \[ J_\alpha f_n=f_{n+1} \qquad(n\ge 0), \] and \[ C_\alpha f_0=0, \qquad C_\alpha f_n=f_{n-1} \qquad(n\ge 1). \] Then there exists a unique nonzero scalar \(c\in\mathbb{C}\) such that \[ f_n(x)=c\,\frac{x^{n\alpha}}{\Gamma(n\alpha+1)} \qquad(n\ge 0). \] In particular, up to a global scalar normalization, \[ \mathcal{M}=\mathcal{G}_{\alpha}^{\mathrm{alg}}. \]
Proof.
By Lemma 5.2, \[ \beta_n=\beta_0+n\alpha \qquad(n\ge 0). \] By Lemma 5.3, \(\beta_0=0\). Therefore \[ \beta_n=n\alpha \qquad(n\ge 0). \]
Again by Lemma 5.2, the coefficients satisfy \[ c_{n+1} = c_n\frac{\Gamma(n\alpha+1)}{\Gamma((n+1)\alpha+1)}. \] Iterating this recurrence yields \[ c_n=\frac{c_0}{\Gamma(n\alpha+1)} \qquad(n\ge 0). \] Writing \(c:=c_0\), we obtain \[ f_n(x)=c\,\frac{x^{n\alpha}}{\Gamma(n\alpha+1)}, \] as claimed. The scalar \(c\) is uniquely determined by \(f_0\).
Corollary 5.5. Among graded monomial chains with one-dimensional homogeneous components, the canonical chain \(\mathcal{G}_{\alpha}^{\mathrm{alg}}\) is, up to a global scalar normalization of the basis, the unique chain on which \(J_\alpha\) acts as a forward shift and \(C_\alpha\) acts as a backward shift with vacuum annihilation.
Proof.
Immediate from Theorem 5.4.
6. Higher powers and tail spaces
We now analyze higher powers of the basic shift operators and compare them with fractional operators of order \(m\alpha\).
Definition 6.1. For \(m\in\mathbb{N}\), define the tail subspace \[ \mathcal{G}_{\alpha}^{(\ge m)} := \bigoplus_{n=m}^{\infty}\mathbb{C}e_n. \]
Lemma 6.2. Let \(m\in\mathbb{N}\). Then, for every \(n\in\mathbb{N}_0\), \[ J_\alpha^m e_n=e_{n+m}, \] and \[ C_\alpha^m e_n= \begin{cases} e_{n-m}, & n\ge m, \\[2mm] 0, & 0\le n < m. \end{cases} \]
Proof.
The first identity follows by repeated application of \(J_\alpha e_n=e_{n+1}\). The second follows by repeated application of \(C_\alpha e_0=0\) and \(C_\alpha e_n=e_{n-1}\) for \(n\ge 1\).
Theorem 6.3. For every \(m\in\mathbb{N}\), \[ J_\alpha^m = {}_0 I_x^{m\alpha} \] on all of \(\mathcal{G}_{\alpha}^{\mathrm{alg}}\).
Proof.
By Lemma 6.2, \[ J_\alpha^m e_n=e_{n+m}. \] On the other hand, Lemma 2.2 with \(\mu=m\alpha\) and \(\beta=n\alpha\) yields \[ {}_0 I_x^{m\alpha}e_n = \frac{x^{(n+m)\alpha}}{\Gamma((n+m)\alpha+1)} = e_{n+m}. \] Since the basis \(\{e_n\}_{n\ge 0}\) spans \(\mathcal{G}_{\alpha}^{\mathrm{alg}}\), the two operators agree on all of \(\mathcal{G}_{\alpha}^{\mathrm{alg}}\).
Theorem 6.4. Let \(m\in\mathbb{N}\). Then \[ C_\alpha^m = {}_0^{\mathrm{C}}D_x^{m\alpha} \] on the tail subspace \(\mathcal{G}_{\alpha}^{(\ge m)}\).
Proof.
Let \(n\ge m\), and set \[ \mu:=m\alpha. \] By Lemma 6.2, \[ C_\alpha^m e_n=e_{n-m}. \] We compare this with the Caputo derivative of order \(\mu\). Let \(p:=\lceil \mu\rceil\). Since \(n\ge m\), we have \[ n\alpha\ge m\alpha=\mu>p-1. \] Therefore Lemma 2.3 applies to \(e_n\), and we obtain \[ {}_0^{\mathrm{C}}D_x^{m\alpha}e_n = \frac{x^{n\alpha-m\alpha}}{\Gamma(n\alpha+1-m\alpha)} = \frac{x^{(n-m)\alpha}}{\Gamma((n-m)\alpha+1)} = e_{n-m}. \] Hence \[ C_\alpha^m e_n = {}_0^{\mathrm{C}}D_x^{m\alpha}e_n \qquad(n\ge m). \] By linearity, the identity holds on all of \(\mathcal{G}_{\alpha}^{(\ge m)}\).
Corollary 6.5. For every \(m\in\mathbb{N}\), \[ C_\alpha^m J_\alpha^m=I \] on \(\mathcal{G}_{\alpha}^{\mathrm{alg}}\), and \[ J_\alpha^m C_\alpha^m = I-\sum_{k=0}^{m-1}\Pi_k. \]
Proof.
By Lemma 6.2, \[ C_\alpha^m J_\alpha^m e_n = C_\alpha^m e_{n+m} = e_n \qquad(n\ge 0), \] so \(C_\alpha^m J_\alpha^m=I\). Likewise, \[ J_\alpha^m C_\alpha^m e_n = \begin{cases} e_n, & n\ge m, \\[2mm] 0, & 0\le n < m, \end{cases} \] which is exactly the action of \[ I-\sum_{k=0}^{m-1}\Pi_k.\tag*{\(\square\)} \]
Remark 6.6. Theorems 6.3 and 6.4 make the semigroup phenomenon completely explicit. The Riemann–Liouville integral closes globally on the graded space: \[ J_\alpha^m = {}_0 I_x^{m\alpha}. \] By contrast, the Caputo derivative closes as a semigroup only after removing the first \(m\) grades. Thus the failure of a full semigroup law for Caputo derivatives is encoded exactly in the finite-dimensional defect sector \[ \operatorname{span}\{e_0,\dots,e_{m-1}\}. \]
7. Mittag-Leffler generating series
The basis \(\{e_n\}_{n\ge 0}\) is precisely the coefficient chain of the Mittag-Leffler function \[ E_\alpha(z) := \sum_{n=0}^{\infty}\frac{z^n}{\Gamma(\alpha n+1)}. \] Indeed, \[ E_\alpha(\lambda x^\alpha) = \sum_{n=0}^{\infty}\lambda^n e_n(x). \]
Remark 7.1. The space \(\mathcal{G}_{\alpha}^{\mathrm{alg}}\) is an algebraic direct sum, so the full Mittag-Leffler series is not itself an element of \(\mathcal{G}_{\alpha}^{\mathrm{alg}}\). Nevertheless, the series identity above shows that \(\{e_n\}\) is the natural algebraic coefficient chain of the classical Mittag-Leffler eigenfunction. In any completion where the series converges and termwise application of \(C_\alpha\) is justified, one formally obtains \[ C_\alpha E_\alpha(\lambda x^\alpha) = \lambda E_\alpha(\lambda x^\alpha). \]
8. Conclusion
We have identified a canonical infinite-dimensional graded monomial space on which the order-\(\alpha\) Riemann–Liouville integral and the order-\(\alpha\) Caputo derivative form a concrete shift algebra. The central object is \[ \mathcal{G}_{\alpha}^{\mathrm{alg}} = \bigoplus_{n=0}^{\infty}\mathbb{C}\frac{x^{n\alpha}}{\Gamma(n\alpha+1)}, \] and its structure is determined by three facts:
- the grading follows the \(\alpha\)-lattice of exponents;
- \(J_\alpha\) is the forward shift;
- \(C_\alpha\) is the backward shift with vacuum annihilation.
The construction is not merely convenient. Among graded monomial chains with one-dimensional homogeneous components, it is unique up to a global scalar normalization. Moreover, the higher-order identities show that the gap between the global semigroup law for Riemann–Liouville integrals and the partial semigroup behavior of Caputo derivatives is exactly the presence of a finite-dimensional low-grade defect space.
In this sense, \(\mathcal{G}_{\alpha}^{\mathrm{alg}}\) is the natural algebraic internal model for order-\(\alpha\) fractional integration and Caputo differentiation.
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