Abstract. We prove that the ordinary differentiation operator on the finite-dimensional polynomial space \[ P_n := \{p(x)\in \mathbb{C}[x] : \deg p \le n\} \] cannot serve as an internal model for classical fractional calculus. Here, by an internal model we mean a family of linear endomorphisms acting on the same space \(P_n\), indexed by nonnegative orders, satisfying the semigroup law and extending the first derivative at order \(1\). Two independent obstructions are established. First, the classical Riemann–Liouville and Caputo fractional derivatives do not act internally on \(P_n\): the former sends even the constant polynomial \(1\) to a nonpolynomial function, while the latter either leaves \(P_n\) or collapses to the zero operator at sufficiently high orders. Second, the differentiation operator \(D_n:=\frac{d}{dx}\big|_{P_n}\) is a single nilpotent Jordan block. We show that such an operator admits no nontrivial \(q\)-th root for any integer \(q\ge 2\). Consequently, no semigroup \((T_\alpha)_{\alpha\ge 0}\subset \mathrm{End}(P_n)\) with \(T_1=D_n\) can exist. The negative conclusion is therefore structural: the failure lies not in the operator-theoretic idea of fractional powers itself, but in the choice of a finite-dimensional ordinary polynomial state space.
Keywords. fractional calculus; Riemann–Liouville derivative; Caputo derivative; matrix functions; nilpotent Jordan block; polynomial space.
1. Introduction
A natural question in fractional calculus is whether differentiation of noninteger order can be realized purely algebraically as a fractional power of the ordinary differentiation operator. In finite dimensions this question becomes especially tempting: once a linear operator is represented by a matrix, one may try to define its fractional powers through matrix-function techniques and thereby recover a new version of fractional differentiation. The purpose of this note is to show that this program fails in the most basic finite-dimensional polynomial setting.
Fix \(n\ge 1\), and consider the space \[ P_n = \{p(x)\in \mathbb{C}[x]: \deg p\le n\}. \] Since \(P_n\) is invariant under ordinary differentiation, the restriction \[ D_n := \frac{d}{dx}\Big|_{P_n} \] is a well-defined nilpotent endomorphism of \(P_n\). One may then ask whether there exists a family \((T_\alpha)_{\alpha\ge 0}\subset \mathrm{End}(P_n)\) such that \(T_1=D_n\) and \[ T_{\alpha+\beta}=T_\alpha T_\beta \qquad (\alpha,\beta\ge 0), \] with \(T_\alpha\) playing the role of a derivative of order \(\alpha\).
The answer is negative for two separate reasons. The first is analytic: classical fractional derivatives do not preserve the class of ordinary polynomials. The second is algebraic: \(D_n\) is a single nilpotent Jordan block, and such an operator cannot be obtained as a nontrivial integer power of another endomorphism. Together these facts rule out any genuine internal realization of classical fractional differentiation on \(P_n\).
The phrase internal model is used here in a purely descriptive sense: the entire family of operators acts on the same finite-dimensional state space \(P_n\). No connection with the control-theoretic internal model principle is intended.
All vector spaces are taken over \(\mathbb{C}\). This causes no loss of generality for the nonexistence statements: if a real \(q\)-th root existed, it would also be a complex \(q\)-th root. We fix the base point \(0\) for the fractional operators; the same obstruction appears at any other finite base point after the translation \(x\mapsto x-a\).
2. Preliminaries
Let \(\mu>0\). The left-sided Riemann–Liouville fractional integral of order \(\mu\) based at \(0\) is defined by \[ ({}_0 I_x^\mu f)(x) := \frac{1}{\Gamma(\mu)} \int_0^x (x-t)^{\mu-1} f(t)\,dt, \qquad x>0. \] If \(m-1 < \alpha < m\), where \(m=\lceil \alpha\rceil\), the left-sided Riemann–Liouville and Caputo fractional derivatives are given by \[ {}_0^{\mathrm{RL}}D_x^\alpha f := \frac{d^m}{dx^m}({}_0 I_x^{m-\alpha}f), \quad {}_0^{\mathrm{C}}D_x^\alpha f := {}_0 I_x^{m-\alpha}(f^{(m)}). \]
We shall also use the nilpotent Jordan block \[ J_s := \begin{pmatrix} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & 0 & 1 \\ 0 & \cdots & \cdots & 0 & 0 \end{pmatrix} \in M_s(\mathbb{C}). \] Thus \(J_s\) is the Jordan block of size \(s\) associated with the eigenvalue \(0\).
3. Classical fractional derivatives do not act internally on \(P_n\)
Lemma 3.1. Let \(r\in \mathbb{N}_0\) and \(\mu>0\). Then \[ {}_0 I_x^\mu(x^r) = \frac{\Gamma(r+1)}{\Gamma(r+\mu+1)}x^{r+\mu}. \]
Proof.
By definition, \[ ({}_0 I_x^\mu x^r)(x) = \frac{1}{\Gamma(\mu)} \int_0^x (x-t)^{\mu-1} t^r\,dt. \] Substitute \(t=xu\). Then \[ ({}_0 I_x^\mu x^r)(x) = \frac{x^{r+\mu}}{\Gamma(\mu)} \int_0^1 (1-u)^{\mu-1}u^r\,du = \frac{x^{r+\mu}}{\Gamma(\mu)}B(r+1,\mu), \] where \(B\) is the beta function. Since \[ B(r+1,\mu)=\frac{\Gamma(r+1)\Gamma(\mu)}{\Gamma(r+\mu+1)}, \] the claim follows.
Proposition 3.2. Let \(m-1 < \alpha < m\), with \(m=\lceil \alpha\rceil\), and let \(r\in \mathbb{N}_0\). Then \[ {}_0^{\mathrm{RL}}D_x^\alpha (x^r) = \frac{\Gamma(r+1)}{\Gamma(r+1-\alpha)}x^{r-\alpha}, \] and \[ {}_0^{\mathrm{C}}D_x^\alpha (x^r) = \begin{cases} 0, & 0\le r \le m-1, \\[2mm] \dfrac{\Gamma(r+1)}{\Gamma(r+1-\alpha)}x^{r-\alpha}, & r\ge m. \end{cases} \]
Proof.
For the Riemann–Liouville derivative, Lemma 3.1 with \(\mu=m-\alpha\) gives \[ {}_0 I_x^{m-\alpha}(x^r) = \frac{\Gamma(r+1)}{\Gamma(r+m-\alpha+1)}x^{r+m-\alpha}. \] Differentiating \(m\) times yields \[ {}_0^{\mathrm{RL}}D_x^\alpha(x^r) = \frac{\Gamma(r+1)}{\Gamma(r+m-\alpha+1)} \cdot \frac{\Gamma(r+m-\alpha+1)}{\Gamma(r+1-\alpha)} x^{r-\alpha}, \] which simplifies to the stated formula.
For the Caputo derivative, if \(0\le r\le m-1\), then \(d^m(x^r)/dx^m=0\), so \[ {}_0^{\mathrm{C}}D_x^\alpha(x^r)=0. \] If \(r\ge m\), then \[ \frac{d^m}{dx^m}(x^r) = \frac{\Gamma(r+1)}{\Gamma(r+1-m)}x^{r-m}. \] Applying Lemma 3.1 again, now with exponent \(r-m\) and order \(m-\alpha\), we obtain \[ {}_0^{\mathrm{C}}D_x^\alpha(x^r) = \frac{\Gamma(r+1)}{\Gamma(r+1-m)} \cdot \frac{\Gamma(r-m+1)}{\Gamma(r+1-\alpha)} x^{r-\alpha}. \] Since \(\Gamma(r-m+1)=\Gamma(r+1-m)\), the desired formula follows.
Theorem 3.3. Let \(n\ge 1\).
- For every noninteger \(\alpha>0\), the Riemann–Liouville fractional derivative does not preserve \(P_n\): \[ {}_0^{\mathrm{RL}}D_x^\alpha(P_n)\not\subset P_n. \]
- For every noninteger \(\alpha\) with \(0 < \alpha < n\), the Caputo fractional derivative does not preserve \(P_n\): \[ {}_0^{\mathrm{C}}D_x^\alpha(P_n)\not\subset P_n. \]
- For every \(\alpha>n\), one has \[ {}_0^{\mathrm{C}}D_x^\alpha\big|_{P_n}=0. \]
Proof.
- Apply Proposition 3.2 to the constant polynomial \(1=x^0\). For every noninteger \(\alpha>0\), \[ {}_0^{\mathrm{RL}}D_x^\alpha(1) = \frac{1}{\Gamma(1-\alpha)}x^{-\alpha}. \] This is not an ordinary polynomial. Hence the image of \(P_n\) is not contained in \(P_n\).
- Let \(0 < \alpha < n\) be noninteger, and set \(m=\lceil \alpha\rceil\). Then \(m\le n\). Applying Proposition 3.2 to \(x^n\in P_n\), we obtain \[ {}_0^{\mathrm{C}}D_x^\alpha(x^n) = \frac{\Gamma(n+1)}{\Gamma(n+1-\alpha)}x^{n-\alpha}. \] Since \(\alpha\notin \mathbb{N}\), the exponent \(n-\alpha\) is not an integer, so the result is not an element of \(P_n\).
- If \(\alpha>n\), then \(m=\lceil \alpha\rceil\) satisfies \(m>n\). Every \(p\in P_n\) has \(p^{(m)}=0\), and therefore \[ {}_0^{\mathrm{C}}D_x^\alpha p = {}_0 I_x^{m-\alpha}(p^{(m)})=0. \] This proves the claim.
Remark 3.4. Theorem 3.3 already rules out the most naive finite-dimensional program. Classical fractional calculus does not remain inside the category of ordinary polynomials. In particular, no linear endomorphism of \(P_n\) can coincide with the Riemann–Liouville derivative of noninteger order, since every endomorphism of \(P_n\) necessarily outputs an ordinary polynomial.
4. The Jordan structure of \(D_n\) and the failure of matrix roots
Proposition 4.1. Let \[ e_k(x):=\frac{x^k}{k!}, \qquad k=0,1,\dots,n. \] Then \((e_0,e_1,\dots,e_n)\) is a basis of \(P_n\), and relative to this basis the operator \(D_n\) is represented by the nilpotent Jordan block \(J_{n+1}\): \[ [D_n]_{\mathcal E} = \begin{pmatrix} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & 0 & 1 \\ 0 & \cdots & \cdots & 0 & 0 \end{pmatrix}. \] In particular, \(D_n\) is a single nilpotent Jordan block of size \(n+1\), and \[ D_n^{\,n+1}=0, \qquad D_n^{\,n}\neq 0. \]
Proof.
Since \[ D_n e_0=0, \qquad D_n e_k = e_{k-1}\quad (1\le k\le n), \] the displayed matrix is immediate. The nilpotency statements follow at once.
Remark 4.2. The matrix-function viewpoint already encounters a local obstruction at the eigenvalue \(0\). In the standard Hermite-interpolation definition of a primary matrix function \(f(A)\), one needs the derivatives \(f^{(j)}(\lambda)\) up to order one less than the size of the largest Jordan block at the eigenvalue \(\lambda\) [3]. Since \(D_n\) has a Jordan block of size \(n+1>1\) at \(\lambda=0\), the function \(f(z)=z^\alpha\) with \(0 < \alpha < 1\) cannot be applied to \(D_n\) as a primary matrix function: already \(f'(0)\) does not exist. Thus the most naive matrix-power construction breaks down before any comparison with classical fractional calculus is made.
Lemma 4.3. Let \(s\ge 1\) and \(q\ge 2\). Then \[ J_s^q \sim \bigoplus_{r=1}^{\min\{q,s\}} J_{\ell_r}, \qquad \ell_r:=\left\lfloor \frac{s-r}{q}\right\rfloor+1 \quad (1\le r\le \min\{q,s\}). \] In particular, if \(s\ge 2\), then \(J_s^q\) has at least two Jordan blocks.
Proof.
Let \(u_1,\dots,u_s\) be the standard basis of \(\mathbb{C}^s\), so that \[ J_su_j=u_{j-1} \qquad (u_0:=0). \] Hence \[ J_s^q u_j = u_{j-q}. \] For each \(r=1,\dots,\min\{q,s\}\), define \[ V_r := \mathrm{span}\{u_r,u_{r+q},u_{r+2q},\dots\}. \] These subspaces are \(J_s^q\)-invariant, pairwise independent, and their direct sum is \(\mathbb{C}^s\). On the ordered basis \[ u_r,\ u_{r+q},\ u_{r+2q},\ \dots, \] the restriction \(J_s^q|_{V_r}\) acts by a one-step backward shift, hence is a nilpotent Jordan block. The number of vectors in this chain is precisely \[ \ell_r=\left\lfloor \frac{s-r}{q}\right\rfloor+1. \] Therefore \[ J_s^q \sim \bigoplus_{r=1}^{\min\{q,s\}} J_{\ell_r}. \] If \(s\ge 2\), then either \(q\ge s\), in which case \(J_s^q=0\) and there are \(s\) Jordan blocks of size \(1\), or \(q < s\), in which case both \(V_1\) and \(V_2\) are nonzero. In either case there are at least two Jordan blocks.
Theorem 4.4. For every integer \(q\ge 2\), the operator \(D_n\) has no \(q\)-th root in \(\mathrm{End}(P_n)\). Equivalently, there is no \(B\in \mathrm{End}(P_n)\) such that \[ B^q = D_n. \]
Proof.
Suppose that \(B^q=D_n\). Since every eigenvalue \(\lambda\) of \(B\) satisfies \(\lambda^q\in \sigma(D_n)=\{0\}\), it follows that \(\lambda=0\). Hence \(B\) is nilpotent. Over \(\mathbb{C}\), \(B\) is therefore similar to a direct sum of nilpotent Jordan blocks: \[ B \sim J_{s_1}\oplus \cdots \oplus J_{s_m}, \qquad s_1+\cdots+s_m=n+1. \] Consequently, \[ B^q \sim J_{s_1}^q \oplus \cdots \oplus J_{s_m}^q. \]
Because \(B^q=D_n\neq 0\), at least one \(s_i\) is greater than \(1\). By Lemma 4.3, the corresponding block \(J_{s_i}^q\) contributes at least two Jordan blocks. Every remaining summand contributes at least one Jordan block. Therefore \(B^q\) has at least two Jordan blocks in its Jordan normal form.
On the other hand, Proposition 4.1 shows that \(D_n\) itself is a single Jordan block. This contradiction proves that no such \(B\) exists.
Remark 4.5. The obstruction is not nilpotency alone. Some nilpotent matrices do admit roots; what is decisive here is that \(D_n\) is a single nilpotent Jordan block. Theorem 4.4 is therefore a statement about Jordan type, not merely about the vanishing of a sufficiently high power.
5. Nonexistence of internal fractional models
Definition 5.1. An internal fractional model for \(D_n\) on \(P_n\) is a family of operators \[ (T_\alpha)_{\alpha\ge 0}\subset \mathrm{End}(P_n) \] such that \[ T_0 = I, \qquad T_{\alpha+\beta}=T_\alpha T_\beta \quad (\alpha,\beta\ge 0), \qquad T_1 = D_n. \] No continuity in \(\alpha\) is assumed.
Theorem 5.2. For every \(n\ge 1\), the operator \(D_n\) admits no internal fractional model on \(P_n\).
Proof.
Assume that \((T_\alpha)_{\alpha\ge 0}\subset \mathrm{End}(P_n)\) is such a family. Let \(q\ge 2\) be any integer. By the semigroup law, \[ T_{1/q}^q = T_1 = D_n. \] Thus \(T_{1/q}\) would be a \(q\)-th root of \(D_n\), contradicting Theorem 4.4. Therefore no such family exists.
Corollary 5.3. The differentiation operator \(D_n\) on \(P_n\) cannot serve as an internal model for classical fractional calculus in the Riemann–Liouville or Caputo sense.
Proof.
By Theorem 3.3, the classical fractional derivatives do not act as a nontrivial family of endomorphisms of \(P_n\). By Theorem 5.2, even abstractly—without requiring agreement with any analytic formula—there is no semigroup of endomorphisms on \(P_n\) whose value at order \(1\) is \(D_n\). Hence \(D_n\) cannot be an internal realization of classical fractional differentiation.
6. Concluding remarks
The preceding argument isolates the precise point of failure. The problem is not that fractional calculus resists operator-theoretic interpretation; on the contrary, that interpretation is well established in suitable infinite-dimensional settings. Rather, the failure occurs because the state space \(P_n\) is too small and too rigid. Classical fractional differentiation naturally generates powers such as \(x^{r-\alpha}\), while \(P_n\) contains only integer monomials. At the same time, its internal differentiation operator is a single nilpotent Jordan block and therefore admits no nontrivial roots.
This negative result should therefore be read as a guide for model selection. If one seeks an algebraic realization of fractional differentiation, the relevant space cannot be a finite-dimensional truncation of ordinary polynomials. One must enlarge the state space, alter the basis, or pass to an infinite-dimensional function space where the operator-theoretic machinery of fractional powers and semigroups is genuinely available; see, for example, [5, 6].
References
- I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, 198, Academic Press, San Diego, 1999.
- F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010.
- N. J. Higham, Functions of Matrices: Theory and Computation, SIAM, Philadelphia, 2008.
- S. Öztürk, “On \(m\)-th roots of nilpotent matrices,” Electronic Journal of Linear Algebra 37 (2021), 718–733.
- N. Jacob and A. M. Krägeloh, “The Caputo derivative, Feller semigroups, and the fractional power of the first order derivative on \(C_\infty(\mathbb{R}_0^+)\),” Fractional Calculus and Applied Analysis 5 (2002), no. 4, 395–410.
- N. D. Cong, “Semigroup property of fractional differential operators and its applications,” Discrete and Continuous Dynamical Systems - B 28 (2023), no. 1, 1–19.