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arxiv: 2605.01079 · v1 · submitted 2026-05-01 · 🧮 math.AP · math-ph· math.MP

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Convolution-to-sum identities for Mittag-Leffler type functions

William Cvetko , Elena Cherkaev
Authors on Pith no claims yet

Pith reviewed 2026-05-09 18:21 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MP
keywords Mittag-Leffler functionsconvolution-to-sum identitiesfractional derivativesLaplace transformpartial fractionsEuler identitysubdiffusionviscoelastic waves
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The pith

Mittag-Leffler type functions obey convolution-to-sum identities that turn integration into finite addition for equal or rationally related orders.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops identities that convert the convolution of certain Mittag-Leffler type functions into a sum of the same type of functions. For functions with identical fractional order, the convolution reduces exactly to a sum of two such functions by applying partial-fraction decomposition to their Laplace transforms. A generalization of Euler's addition formula for these functions further allows the reduction when the orders stand in a rational ratio n/m, producing a sum with n plus m terms. This framework directly aids the solution of fractional differential equations that model subdiffusion processes and attenuated waves.

Core claim

We identify a family of Mittag-Leffler type functions, parameterized as R_{α,v} and P_{α,w}, that serve as eigenfunctions for the Riemann-Liouville and Caputo fractional derivatives. Through Laplace domain analysis, the convolution of two members of this family can be expressed as a series, but collapses to a sum of two when the orders coincide. The functions obey a generalized Euler identity that extends the sum representation to convolutions where the orders differ by a rational factor, with the number of terms equal to the sum of the integers in the ratio. The identities are applied to forced subdiffusion and the Caputo-Wismer-Kelvin wave equation.

What carries the argument

The R_{α,v} and P_{α,w} Mittag-Leffler type functions together with partial-fraction decomposition of their Laplace transform products.

If this is right

  • The convolution integral between two such functions becomes a finite sum rather than requiring numerical integration.
  • Closed-form solutions become available for linear fractional differential equations driven by these functions.
  • For orders in ratio n/m the resulting expression involves exactly n + m terms.
  • Models of subdiffusion and fractional viscoelastic waves can be solved using only sums of the basis functions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar reductions might apply to other fractional special functions beyond the Mittag-Leffler family.
  • Numerical algorithms for fractional calculus could replace convolution steps with direct summation using these identities.
  • The method may extend to multi-term fractional equations or systems with multiple commensurate orders.

Load-bearing premise

The chosen R and P parameterizations completely describe the eigenfunctions of the Riemann-Liouville and Caputo operators and that partial-fraction decompositions in the Laplace domain introduce no additional singularities or restrictions within the relevant parameter ranges.

What would settle it

Compute numerically the convolution integral of two specific R_{α,v} functions with equal alpha and compare it term-by-term to the proposed sum of two P functions at several time points.

Figures

Figures reproduced from arXiv: 2605.01079 by Elena Cherkaev, William Cvetko.

Figure 1
Figure 1. Figure 1: First eigenfunction of the α-th Caputo (left) and Riemann–Liouville (right) frac￾tional derivatives, with eigenvalue −1, across several values of α. Note that the Pα,0(−1, t) all initially equal 1, and that for 0 < α < 1, Rα,0(−1, t) is singular at t = 0. Mittag-Leffler functions in all plots have been calculated using the methods in [10], implemented in MAT￾LAB [11]. 3 Convolution to sum identities This s… view at source ↗
Figure 2
Figure 2. Figure 2: Inhomogeneous solutions to Dα f + k 2 f = g together with the forcing term g. The left panel shows solution corresponding to α = 1/2, g(t) = e −1.5t , and k = 1. The right panel presents the solution for α = 3/4, g(t) = cos(2t), and k = 2. Note that on the left, though the source term decays exponentially, the inhomogeneous solution has a ’long tail’ characteristic of subdiffusion [31]. For another example… view at source ↗
Figure 3
Figure 3. Figure 3: A comparison of solutions to 4.11 in one spatial dim view at source ↗
Figure 4
Figure 4. Figure 4: The exact and approximate time-domain representa view at source ↗
Figure 5
Figure 5. Figure 5: Exact and approximate solutions to D2 t f+ηk2Dα t +c 2k 2 = cos(t), with α = 1/4, η = 0.2, c = 2, k = 2, and homogeneous initial conditions. The exact and approximate solutions are given by L −1 h 1 s 2+ηk2sα+c 2k 2 i and e −rkt sin(ωkt), respectively, convolved with cos(t). The relative error is less than 0.03. We have shown that a convolution of two Mittag-Leffler type functions of the same order can be … view at source ↗
read the original abstract

Product-to-sum identities for trigonometric functions play a fundamental role in function theory and numerous applications. In this spirit, we present convolution-to-sum identities for Mittag-Leffler type functions. Using a Laplace domain analysis of fractional operators, we identify a family of Mittag-Leffler type functions that encapsulates the eigenfunctions of Riemann-Liouville and Caputo fractional derivatives. We work with two closely-related parameterizations of this class, $R_{\alpha,v}$ and $P_{\alpha,w}$. The convolution of two such functions can be expressed as a series of them. Moreover, if the functions share the same order $\alpha$, the convolution can be reduced to a sum of two $P/R$ functions through a partial-fraction decomposition in the Laplace domain. Furthermore, $R$ and $P$ functions satisfy a generalization of Euler's identity, which expands the scope of the previous result to convolutions of $P/R$ functions whose orders $\alpha_1,\alpha_2$ are related by a rational factor. For $\frac{\alpha_1}{\alpha_2} = \frac{n}{m}$, the resulting sum has $n+m$ terms. The foundational results and methods developed here are illustrated by their application to forced subdiffusion and to a fractionally attenuated wave equation (the Caputo-Wismer-Kelvin, or the fractional Kelvin-Voigt model).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper derives convolution-to-sum identities for two parameterized families of Mittag-Leffler-type functions, R_{α,v} and P_{α,w}, which encapsulate eigenfunctions of Riemann-Liouville and Caputo fractional derivatives. Using Laplace-domain analysis, it shows that the convolution of two such functions is a series of them; when the orders α coincide, partial-fraction decomposition reduces the convolution to a sum of two P/R functions. A generalization of Euler's identity extends the reduction to cases where α1/α2 = n/m, producing a sum with n+m terms. The results are illustrated with applications to forced subdiffusion and the Caputo-Wismer-Kelvin fractional wave model.

Significance. If the central identities hold under appropriate parameter restrictions, the work would supply useful closed-form reductions for convolutions that arise in fractional differential equations, generalizing classical trigonometric product-to-sum formulas and potentially simplifying analytical treatments of subdiffusive and viscoelastic models. The explicit applications to the fractional Kelvin-Voigt equation provide concrete evidence of utility.

major comments (2)
  1. [Abstract] Abstract (and the Laplace-domain analysis of the same-order case): the reduction of the convolution to a sum of two P/R functions via partial fractions is stated without qualification, but the algebraic step A/(s^α−λ) + B/(s^α−μ) is valid only for λ ≠ μ. When poles coincide the correct decomposition contains a repeated factor whose inverse Laplace transform is t times a Mittag-Leffler function lying outside the two-parameter families R_{α,v} and P_{α,w}. This restriction is load-bearing for the claimed generality of the identity.
  2. [Abstract] The rational-order generalization (α1/α2 = n/m): the factorization of the denominator into n+m linear factors in s^α likewise presupposes distinct roots. Repeated roots alter both the number and the functional form of the resulting terms, yet the manuscript states the n+m-term sum without mentioning this condition.
minor comments (1)
  1. [Introduction] The introduction should explicitly define the precise parameter ranges for v and w in R_{α,v} and P_{α,w} to make the eigenfunction property immediate to readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting these important qualifications regarding pole distinctness. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and the Laplace-domain analysis of the same-order case): the reduction of the convolution to a sum of two P/R functions via partial fractions is stated without qualification, but the algebraic step A/(s^α−λ) + B/(s^α−μ) is valid only for λ ≠ μ. When poles coincide the correct decomposition contains a repeated factor whose inverse Laplace transform is t times a Mittag-Leffler function lying outside the two-parameter families R_{α,v} and P_{α,w}. This restriction is load-bearing for the claimed generality of the identity.

    Authors: We agree that the partial-fraction decomposition requires distinct poles (λ ≠ μ). When poles coincide, the inverse Laplace transform produces an extra term t times a Mittag-Leffler function outside the R_{α,v} and P_{α,w} families. We will revise the abstract and the Laplace-domain analysis section to state the identity under the explicit assumption of distinct poles and note the repeated-pole case for completeness. revision: yes

  2. Referee: [Abstract] The rational-order generalization (α1/α2 = n/m): the factorization of the denominator into n+m linear factors in s^α likewise presupposes distinct roots. Repeated roots alter both the number and the functional form of the resulting terms, yet the manuscript states the n+m-term sum without mentioning this condition.

    Authors: We concur that the n+m-term reduction for rationally related orders presupposes distinct roots. Repeated roots would change both the number of terms and their functional form via higher-multiplicity factors. We will revise the abstract and the statement of the rational-order result to include the distinct-roots assumption. revision: yes

Circularity Check

0 steps flagged

Derivation uses independent Laplace-domain techniques with no self-referential reduction

full rationale

The paper introduces the R and P families as parameterizations of Mittag-Leffler-type eigenfunctions, then invokes the standard Laplace transform of the Riemann-Liouville and Caputo operators together with algebraic partial-fraction decomposition of the product of two transforms. These steps rely on classical transform tables and the assumption of distinct poles; the resulting identities are obtained by term-by-term inversion rather than by any definition that already encodes the convolution result. No load-bearing premise is justified solely by a self-citation, no fitted parameter is relabeled as a prediction, and the rational-order generalization follows directly from factoring the denominator in the s^α variable. The derivation chain is therefore self-contained against external analytic tools.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The claims rest on standard Laplace transform rules for fractional operators and the assumption that the chosen Mittag-Leffler parameterizations serve as eigenfunctions; no free parameters are fitted and no new physical entities are postulated.

axioms (2)
  • standard math Laplace transforms of Riemann-Liouville and Caputo fractional derivatives follow the standard algebraic rules in the s-domain.
    Invoked to convert convolutions into products and enable partial-fraction decomposition.
  • domain assumption The R_α,v and P_α,w functions encapsulate the eigenfunctions of the fractional derivative operators.
    Stated directly in the abstract as the basis for identifying the function family.
invented entities (1)
  • Parameterized Mittag-Leffler families R_α,v and P_α,w no independent evidence
    purpose: To provide a convenient basis for expressing convolution identities and eigenfunction properties.
    New parameterizations introduced to facilitate the stated reductions; no independent experimental evidence supplied.

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