Analytical calculation of the inverse nabla Laplace transform

YangQuan Chen; Yiheng Wei; Yong Wang; Yuquan Chen

REVIEW 2 major objections 5 minor 19 references

Analytical calculation of the inverse nabla Laplace transform

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper derives two exact analytic routes for inverting the nabla Laplace transform: residue summation and partial-fraction table lookup, with a sign rule tied to the clockwise contour.

desk verdict Useful table and correct inside-pole formula, but Eq. (7) is false as stated (missing residue at infinity) and the novelty claim is overstated. read the letter →

arxiv 1909.02655 v2 pith:M2MU6SNU submitted 2019-08-23 math.GM

Yiheng Wei , YangQuan Chen , Yuquan Chen , Yong Wang This is my paper · ORCID

classification math.GM MSC 39A1244A1026A33

keywords nablaLaplacetransforminverseresiduemethodpartialfractionexpansiondiscretefractionalcalculusMittag-Lefflerfunctiontransform-pairtable

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes that the inverse nabla Laplace transform—the operation that recovers a discrete-time causal sequence $f(k)$ from its transform $F(s)$—can be computed exactly without evaluating the defining contour integral. The first route is a residue method: $f(k)$ is the negative of the sum of residues of $F(s)(1-s)^{-k+a}$ at poles inside a clockwise contour, or the plain sum at poles outside it. The second route is partial-fraction expansion: once $F(s)$ is decomposed into simple rational pieces, each piece maps to a power of $(1-s_i)^{-1}$, and a table of sixteen transform pairs supplies exact inverses for common sequences, including fractional-order Mittag-Leffler forms. The methods give exact inverses for rational $F(s)$ and for fractional-order functions that can be rearranged into table entries, and the paper states explicitly where they stop working, namely for branch-cut or multi-valued cases such as $1/(s^{\alpha}-\lambda)$ with irrational $\alpha$.

What carries the argument

The load-bearing object is the kernel $s \leftrightarrow (1-s)^{-k+a}$ built into the inversion integral (3). In the residue route, the residue theorem applied to the meromorphic integrand $F(s)(1-s)^{-k+a}$ separates poles inside the clockwise contour (giving a minus sign) from poles outside (giving no sign), with higher-order poles handled by the standard derivative formula. In the partial-fraction route, the same kernel becomes the elementary inverse $N_a^{-1}\{1/(s-\lambda)\}=1/(1-\lambda)^{k-a}$, and multiple poles produce polynomial prefactors $(k-a)^{i-1}$. The table of sixteen transform pairs is the practical machine: it packages the algebra so that a user only decomposes $F(s)$ and reads off the sequence, including fractional-order cases via the discrete Mittag-Leffler function.

What would settle it

Take a rational $F(s)$ whose inverse is known independently, for example $F(s)=1/(s-2)$, whose inverse should be $(-1)^{k-a}$; evaluate the defining contour integral (3) numerically on a small clockwise circle around $s=1$ for $k-a=1,2,3$ and compare signs—any sign slip in the residue formulas appears immediately. For a fractional-order case, compute the series definition of the claimed Mittag-Leffler output for the first few $k$ and compare against direct numerical evaluation of the inverse transform; a wrong table pair produces a finite discrepancy.

Watch

Extended reading notes

Core claim

The central claim is that the inversion formula $N_a^{-1}\{F(s)\} = \frac{1}{2\pi j}\oint_c F(s)(1-s)^{-k+a}\,ds$ can be evaluated analytically by residue calculus. Because the contour $c$ winds clockwise about $s=1$, the inside-pole formula carries a minus sign, $f(k)=-\sum_m \operatorname{Res}[F(s)(1-s)^{-k+a},s_m]$, while the outside-pole formula carries a plus sign, $f(k)=\sum_n \operatorname{Res}[\cdots,s_n]$. Equivalently, when $F(s)=\sum_i r_i/(s-s_i)$, with multiple-pole terms included, the inverse is $f(k)=\sum_i r_i/(1-s_i)^{k-a}$ plus related multiple-pole terms involving $(k-a)^{i-1}$. This turns inversion into algebra plus table lookup: the sixteen transform pairs in Table 1 cover powers, exponentials, sinusoids, and discrete Mittag-Leffler functions, and the examples show a rational case and a fractional-order case solved exactly.

Load-bearing premise

The load-bearing premise is that the inverse contour integral (3) is a valid inversion formula for the chosen $F(s)$ and that all singularities of $F(s)(1-s)^{-k+a}$ are isolated poles cleanly separated inside or outside the contour; branch-point or multi-valued cases such as $1/(s^{\alpha}-\lambda)$ with irrational $\alpha$ violate this and are acknowledged as not covered.

Editorial extensions

If this is right

For rational $F(s)$ with known poles, exact inversion becomes a residue count plus table lookup, removing the approximation error of numerical inversion.
The clockwise contour flips the usual inverse-Z-transform sign pattern: inside-pole residues enter with a minus sign and outside-pole residues with a plus sign.
The sixteen transform pairs give closed-form inverses for common causal sequences and for fractional-order Mittag-Leffler entries, usable in solving nabla fractional difference equations.
Multiple poles are handled by explicit polynomial factors, so repeated-root transfer functions remain exactly invertible by the same formulas.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because $z^{-1}=1-s$ links the nabla and Z transforms, the table approach can likely port any finite-order rational Z-domain inversion table into nabla form; the residue identities here should coincide with inverse-Z-transform results up to orientation.
For rational $\alpha=p/q$, the function $1/(s^{p/q}-\lambda)$ has finitely many pole branches on the appropriate Riemann surface, so collecting residues over all branches may yield a closed-form discrete Mittag-Leffler inverse even though the irrational case is left open.
A direct numerical test of the branch-cut case $1/(\sqrt{s}-\lambda)$ by numerical integration of the inversion integral around $s=1$ would reveal whether purely polar residue formulas miss a branch-cut contribution.
The paper's own construction suggests a generative recipe: applying convolution, time-scaling, and frequency-differentiation properties to the existing sixteen pairs produces new pairs, so the table is expandable rather than closed.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear Colleague,

The short version: the paper has a useful table and a correct inside-pole residue formula, but the outside-pole formula (7) is false as stated because it drops the residue at infinity. That is not a nitpick: take F(s)=1, which is Table 1 row 1; for k=a+1, the integrand is 1/(1-s), there are no finite poles outside the clockwise contour, yet the inverse transform is 1. Equation (7) gives 0. The missing term is Res∞, and the formula works only when that residue vanishes (e.g., F(s)=O(1/s) as s→∞).

What the paper does well: the inside-pole formula (4) is correct, and Table 1's 16 transform pairs look right—they are standard Z-transform pairs under z^{-1}=1-s, which the authors explicitly state in Section 3.2. Example 1 is worked out correctly by both methods. The partial-fraction extension (11) is standard but cleanly presented. The discussion of fractional-order limitations in Section 3.3 is honest.

The soft spots: aside from (7), the claimed novelty is inflated. Because of the explicit mapping g(k)=f(k+1), z^{-1}=1-s, the inverse nabla transform is just the inverse Z transform; residue and partial-fraction methods for that are classical. The 'first time' statement in the conclusions won't hold up. Some table entries are asserted without derivation, though they are verifiable. The paper also leans heavily on self-cited prior work [14], [16] for the contour integral; that is acceptable background but worth noting.

In sum: a methods/pairs paper with a real error in one of its two headline formulas. The error is fixable—add the residue at infinity or state the hypothesis—and the table alone might justify a revised version. As it stands, I would not cite it for Eq. (7), but I would send it to a referee because the rest is correct and the table is practical. After revision, it could be a serviceable reference.

Hope this helps.

Referee Report

2 major / 5 minor

Summary. The paper develops two analytical methods for inverting the nabla Laplace transform defined in Eq. (1) and (3). The first method is a residue calculation: Eq. (4) expresses f(k) as the negative sum of residues of F(s)(1-s)^{-k+a} at finite poles inside the integration contour, and Eq. (7) claims to express f(k) as the sum of residues at finite poles outside the contour. The second method is partial fraction expansion: Eq. (9) and Eq. (11) give the inverse transform for rational F(s), and Eq. (12)-(13) extend this to certain fractional-order terms. A table of 16 transform pairs is provided, and two examples (one rational, one fractional-order) are worked out. Section 3.3 discusses limitations for fractional-order functions.

Significance. If corrected, the paper would offer a convenient exact inversion toolkit for nabla Laplace transforms, with the table and worked examples being useful for practitioners. The formula in Eq. (4) and the partial fraction formulas in Eq. (9) and Eq. (11) are consistent with the defining contour integral for the proper rational examples in Section 4, and the paper explicitly checks both residue formulas and the partial fraction method against the same result in Example 1. However, Eq. (7) is false as stated because it omits the residue at infinity, and this is an internal inconsistency with Eq. (3), not merely a fractional-order limitation. The fractional-order discussion in Section 3.3 is candid, but the residue-at-infinity gap is not acknowledged there. The novelty claim is modest: the methods are classical residue and partial fraction techniques adapted to a transform that is closely related to the Z-transform.

major comments (2)

[Section 3.1, Eq. (7)] The formula f(k)=sum_n Res[F(s)(1-s)^{-k+a}, s_n] is not correct as an unconditional statement because it omits the residue at infinity of the integrand H(s)=F(s)(1-s)^{-k+a}. By the residue theorem on the Riemann sphere, the contour integral in Eq. (3) equals 2*pi*j*(sum of residues outside c plus the residue at infinity), so Eq. (7) needs an additional term Res[H(s), infinity] on the right-hand side, or a decay hypothesis on F(s) that makes that residue vanish. A concrete counterexample is F(s)=1, which is Table 1, row 1 and corresponds to the delta sequence f(a+1)=1, f(k)=0 for k>a+1. For k=a+1, H(s)=1/(1-s), whose only finite pole is s=1 inside the contour; there are no finite poles outside, so Eq. (7) predicts f(a+1)=0, whereas direct evaluation of Eq. (3) gives 1. The paper's note that s=1 cannot be a pole of F(s) for finite f(k) does not fix the problem, because the pole of H(s) at s=1 comes from the factor (1-s)^{-k+a}, not from F(s).
[Section 3.3] The limitations discussion correctly identifies difficulties with multi-valued fractional-order functions such as F(s)=1/(s^alpha-lambda) for irrational alpha, but it does not mention the residue-at-infinity problem in Eq. (7). This problem occurs even for elementary rational transforms, so it is an internal inconsistency with the defining integral (3) rather than a limitation of fractional calculus. The section should state explicitly that Eq. (7) is valid only when the residue at infinity of F(s)(1-s)^{-k+a} is zero, or it should be amended to include the missing residue-at-infinity term.

minor comments (5)

[General] The phrase 'residual calculation method' should be 'residue calculation method' throughout the paper, including the title, abstract, and Section 3.1.
[Table 1] The 16 transform pairs in Table 1 are asserted without derivation; many can be verified directly from the definition in Eq. (1), and a short explanation or a reference to the property list in [14] would make the table more self-contained and easier to check.
[Section 3.3] There are several incomplete or garbled sentences in Section 3.3, including 'some inverse transform of fractional order polynomial' and 'The proposed two methods In other words, it is an opportunity and challenge to handle with such complicated irrational F(s)'; these need to be rewritten into complete sentences.
[Example 2] The expression 's2 = 0.3^{10/7} e^{j20*pi*i/7}' is ambiguous because the index i is not defined and the statement that the number of such poles is infinite is not obviously consistent with the displayed formula, which has finite periodicity in i; please clarify the branch structure.
[Section 2, Eq. (3)] The inversion formula in Eq. (3) should explicitly state the required analyticity assumptions on F(s) on the contour c and the precise sense in which the contour 'locates in the convergent region', since these conditions are inherited by all subsequent formulas.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the residual and partial-fraction inversion formulas are derived from the residue theorem applied to the standard contour integral, and the Table 1 pairs are directly verifiable from the definition; self-citations are background, not load-bearing.

full rationale

The paper's derivation chain is self-contained with respect to circularity. The inverse nabla Laplace transform is defined in Eq. (1), and the contour integral in Eq. (3) is taken from the authors' prior work [16], but it is a standard Cauchy-integral/power-series inversion representation and is not equivalent to the formulas the paper claims to derive. Equations (4) and (7) follow from applying the residue theorem to that contour integral, with the sign convention dictated by the clockwise contour; no fitted parameter is introduced and no target result is assumed. Equation (9) and the multiple-pole extension (11) are consequences of the residue formulas applied to a partial-fraction expansion, and each Table 1 entry can be checked directly by substituting the candidate sequence into definition (1) and summing the resulting geometric, binomial, or Mittag-Leffler series. The Examples then evaluate these formulas rather than fitting them to the answers. The citations to [14] and [16] supply background transform properties and the inversion integral, but those results are not used as unverified self-support for the central derivation. The noted mathematical issue that Eq. (7) omits a residue-at-infinity term is a correctness concern, not a circularity concern. Accordingly, no circular step is present and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper fits no free parameters and introduces no invented entities. The central results rest on the standard residue theorem, the assumed inverse formula from [16], the correctness of the transform-pair table, and the stated Z/N transform equivalence. These are background assumptions, not fitted values.

assumptions (4)

standard math The residue theorem of complex analysis applies to the contour integral in Eq. (3).
Invoked at the start of Section 3.1 to convert the integral into sums of residues; requires the integrand to be meromorphic inside and on the contour.
domain assumption The inverse nabla Laplace transform formula (3), N_a^{-1}{F(s)} = 1/(2 pi j) ∮_c F(s)(1-s)^{-k+a} ds with c clockwise around s=1, is valid.
Taken from the authors' prior work [16]; the paper does not re-derive it, and all subsequent formulas depend on it.
domain assumption The 16 transform pairs in Table 1 are correct, including the discrete Mittag-Leffler pairs #9 and #10.
Assembled using properties from [14]; no derivation is given in this paper, but the pairs are used as look-up entries in the partial fraction method.
domain assumption The relation Z_a{g(k)} = N_a{f(k)} through g(k)=f(k+1) and z^{-1}=1-s holds.
Stated in Section 3.2 before Table 3; used implicitly to justify applying Z-transform style methods to the nabla transform.

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Cite this review

Pith. "Pith review of Analytical calculation of the inverse nabla Laplace transform." pith.science (2026). https://pith.science/paper/M2MU6SNU

@misc{pith2026190902655,
  author       = {Pith},
  title        = {Pith review of: Analytical calculation of the inverse nabla Laplace transform},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M2MU6SNU}},
  note         = {Machine review of arXiv:1909.02655}
}

read the original abstract

The inversion of nabla Laplace transform, corresponding to a causal sequence, is considered. Two classical methods, i.e., residual calculation method and partial fraction method are developed to perform the inverse nabla Laplace transform. For the first method, two alternative formulae are proposed when adopting the poles inside or outside of the contour, respectively. For the second method, a table on the transform pairs of those popular functions is carefully established. Besides illustrating the effectiveness of the developed methods with two illustrative examples, the applicability are further discussed in the fractional order case.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [14]

Y . H. Wei, Y . Q. Chen, Y . Wang, and Y . Q. Chen, Some fundamental properties on the sampling free nabla Laplace transform, ASME 2019 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference (IDETC/CIE 2019), August 18-21, 2019, Anaheim, USA, No. DETC2019-97351

work page 2019

[2] [16]

Y . H. Wei, J. C. Wang, P . W . Tse, and Y . Wang, Mod- elling and simulation of nabla fractional order systems with nonzero initial conditions, Asian Journal of Control, 2019, doi: 10.1002/asjc.2232

work page doi:10.1002/asjc.2232 2019

[3] [1]

J. Hein, Z. McCarthy, N. Gaswick, B. McKain, and K. Speer, Laplace transforms for the nabla-difference oper- ator, PanAmerican Mathematical Journal, vol. 21, no. 3, pp. 79-96, 2011

work page 2011

[4] [2]

Jarad, B

F. Jarad, B. Kaymakc ¸alan, and K. Tas ¸, A new transform method in nabla discrete fractional calculus, Advances in Difference equations, vol. 2012, no. 1, id. 190, 2012

work page 2012

[5] [3]

M. D. Ortigueira, F. J. V . Coito, and J. J. Trujillo, A new look into the discrete-time fractional calculus: transfor m and linear systems, IFAC Proceedings V olumes, vol. 46, no. 1, pp. 635-640, 2013

work page 2013

[6] [4]

M. D. Ortigueira, D. F. M. Torres, and J. J. Trujillo, Ex- ponentials and Laplace transforms on nonuniform time scales, Communications in Nonlinear Science and Nu- merical Simulation, vol. 39, pp. 252-270, 2016

work page 2016

[7] [5]

Abdeljawad, F

T. Abdeljawad, F. Jarad, and D. Baleanu, A semigroup- like property for discrete Mittag-Lefﬂer functions, Ad- vances in Difference Equations, vol. 2012, no. 1, id. 72, 2012

work page 2012

[8] [6]

J. M. Jonnalagadda, D. Purnima, and G. V . S. R. Deekshi- tulu, Discrete Control Systems of Fractional Order, Inter- national Journal of Nonlinear Science, vol. 21, no. 1, pp. 37-46, 2016

work page 2016