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An Invertible Family of Hurwitz--Lerch Type Functions Associated with $k$-Augmented Centered Triangular Numbers

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Pith's one-line read This paper shows that three consecutive functions from a family weighted by k-augmented centered triangular numbers exactly recover the classical Hurwitz–Lerch transcendent and its first two Euler derivatives.

desk verdict A correct, useful Vandermonde inversion for a quadratic Hurwitz–Lerch family; the only real issue is the unproved coefficient formula cited from [9]. read the letter →

arxiv 2607.26403 v1 pith:M7GOPXBF submitted 2026-07-29 math.CO

Noel B. Lacpao , Rushel S. Acope , Marlon S. Frias , Mark Ivan P. Arcillas , Rey Carl P. Tiu , Francis Jay R. Romagos This is my paper · ORCID
classification math.CO MSC 11M3511B8305A15
keywords Hurwitz–Lerchtranscendentk-augmentedcenteredtriangularnumbersEuleroperatorVandermondeinversionBernoullipolynomialsEuleriangeneratingfunction
topics The Collatz Conjecture
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 introduces a family of Hurwitz–Lerch type functions H_k(z,s,a) whose coefficients are the k-augmented centered triangular numbers, a quadratic sequence in the summation index with geometric dependence on k through 4^k and 2^k. The central result is an exact inversion: the three consecutive functions H_k, H_{k+1}, H_{k+2} linearly reconstruct the classical Hurwitz–Lerch transcendent Φ(z,s,a) and its two Euler derivatives DΦ and D^2Φ, with coefficients independent of z, s, a, and the coefficients for Φ itself independent of k. The proof uses a Vandermonde interpolation argument over the geometric factors 1, 2, 4. The paper also derives convergence conditions, a reduction formula, an Euler-operator representation, a three-term recurrence in k, generating functions, and special values in terms of Bernoulli, Eulerian, and Euler polynomials. A sympathetic reader would care because it provides a direct, explicit bridge between a new weighted family and a classical special function.

What carries the argument

The central object is the Vandermonde inversion formula for polynomially weighted Hurwitz–Lerch functions (Theorem 5.2). For a weight W_k(r) = Σ c_j λ_j^k r^j with distinct non-zero λ_j, the functions F_k = Σ_r W_k(r) z^r/(r+a)^s satisfy D^j Φ = (1/(c_j λ_j^k)) Σ_q ℓ_{j,q} F_{k+q}, where the ℓ_{j,q} are the coefficients of the Lagrange basis polynomials L_j(x) = Π_{m≠j} (x-λ_m)/(λ_j-λ_m). Specialized to the quadratic case λ = (1,2,4) and c = (1, 3/2, 3/2), this produces the invertible family. The coefficient formula A(r+1,k) = α_k r^2 + β_k r + 1, which encodes the k-augmented centered triangular numbers, provides the concrete geometric factors.

What would settle it

Verify the closed form for the k-augmented centered triangular numbers by computing small cases directly from the array definition (e.g., n=1,2,3 and k=0,1,2) and comparing to the formula; any mismatch disproves the central claim. Alternatively, evaluate both sides of identity (5.7) numerically at a specific triple such as z=1/2, s=2, a=1 using sufficiently many terms of the defining series, and check that the difference tends to zero as the truncation grows.

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Extended reading notes

Core claim

The central claim is Corollary 5.3: for every integer k ≥ 0, Φ(z,s,a) = (8/3)H_k - 2H_{k+1} + (1/3)H_{k+2}, DΦ(z,s,a) = (-4H_k + 5H_{k+1} - H_{k+2})/(3·2^k), and D^2Φ(z,s,a) = (2H_k - 3H_{k+1} + H_{k+2})/(9·4^k). These follow from the operator identity H_k = (α_k D^2 + β_k D + 1)Φ, with α_k = (3/2)4^k and β_k = (3/2)2^k, combined with a Vandermonde inversion on the geometric factors 1, 2, 4. The paper also proves the convergence of the defining series, a reduction of H_k to shifted Hurwitz–Lerch functions, the recurrence H_{k+3} = 7H_{k+2} - 14H_{k+1} + 8H_k, a generating function in k, and special values connecting to Bernoulli, Eulerian, and Euler polynomials.

Load-bearing premise

The entire argument depends on the closed form A(n,k) = (3/2)4^k(n-1)^2 + (3/2)2^k(n-1) + 1 for the k-augmented centered triangular numbers, which is asserted on the authority of an earlier paper and not proved here; if this formula is wrong, the operator representation and every inversion result collapse.

Editorial extensions

If this is right

  • The exact inversion (5.7)–(5.9) means any combination of Φ, DΦ, and D^2Φ can be rewritten as a finite linear combination of three series from the family, with coefficients that do not depend on z, s, a (and, for Φ, not on k).
  • The three-term recurrence H_{k+3} = 7H_{k+2} - 14H_{k+1} + 8H_k determines the entire family from any three consecutive members, since the geometric factors 1, 2, 4 are the roots of the characteristic polynomial.
  • The ordinary generating function in k is rational, with denominators 1-y, 1-2y, 1-4y, yielding closed forms for finite sums such as Σ_{k=0}^N H_k(z,s,a) in terms of Φ, DΦ, D^2Φ.
  • At special parameter values the family connects to classical polynomials: H_k(1,-m,a) is a combination of Bernoulli polynomials, H_k(z,-m,1) has numerator polynomials given by Eulerian polynomials, and H_k(-1,-m,a) is expressed through Euler polynomials.

Reading between the lines

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

  • The inversion suggests a numerical recipe: compute truncated sums of H_k, H_{k+1}, H_{k+2} and take the linear combination to cancel the quadratic weight, yielding approximations to Φ, DΦ, D^2Φ; the paper does not analyze the conditioning or error of this scheme.
  • The same Vandermonde mechanism applies to any polynomial weight with distinct geometric sequences, so the paper's inversion is a special case of a broader principle; the authors note this but do not explore higher-dimensional cases or other geometric ratios.
  • Because the inversion holds for every k, one could choose k to shift the weight and potentially simplify evaluation; for instance, the denominators 2^k and 4^k could be used to rescale the higher derivatives, a possibility the paper leaves implicit.
  • The explicit rational forms in terms of Eulerian and Euler polynomials invite coefficient comparisons that might yield new identities for these classical polynomials; this direction is not pursued in the paper.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 3 minor

Summary. The paper defines a family of Hurwitz–Lerch type functions H_k(z,s,a) whose coefficient sequence is the k-augmented centered triangular numbers A(r+1,k)=α_k r^2+β_k r+1, with α_k=3/2·4^k and β_k=3/2·2^k. The main results are: absolute convergence conditions (Theorem 3.1), a reduction to the classical Hurwitz–Lerch transcendent and its Euler derivatives (Theorem 3.2, Proposition 5.1), a Vandermonde inversion theorem for polynomially weighted Hurwitz–Lerch functions (Theorem 5.2), and the consequent inversions (Corollary 5.3) showing that H_k, H_{k+1}, H_{k+2} recover Φ(z,s,a), DΦ(z,s,a), and D^2Φ(z,s,a). The paper also derives a recurrence in k (Theorem 6.1), generating functions (Theorem 6.2, Corollary 6.3), finite sums (Corollary 6.4), and special values at z=1 and z=-1 in terms of Hurwitz zeta, Bernoulli, Eulerian, and Euler polynomials (Section 7).

Significance. If the results are correct, the paper contributes a clean and rather elegant addition to the Hurwitz–Lerch literature: an exact, explicit inversion of three consecutive members of a weighted family back to the classical transcendent and its first two Euler derivatives. The Vandermonde inversion is a nice general principle, and the convergence theorem is carefully argued. The special-value formulas are natural consequences of (3.2) and standard identities for Bernoulli, Eulerian, and Euler polynomials. The principal caveat is that the coefficient formula (2.6) is imported from the separate paper [9] by the first author and is not proved or even defined in this manuscript. Because every subsequent result—the reduction (3.2), the operator form (5.1), the inversion (5.7)–(5.9), the recurrence (6.1), and the special values—depends on the exact constants α_k and β_k, this is a load-bearing point rather than a minor presentational detail. I did not find circular reasoning: the inversion follows from a genuine Vandermonde/Lagrange argument, and the special values follow from standard zeta/Bernoulli/Euler facts.

major comments (1)
  1. [§2, Eq. (2.6)] The coefficient formula A(n,k)=3/2·4^k(n−1)^2+3/2·2^k(n−1)+1 is stated to follow from the arithmetic structure of [9], but this paper neither defines the k-augmented centered triangular array nor gives a proof. This formula supplies the exact α_k and β_k used in (2.7), and through it in Theorem 3.2, Proposition 5.1, Corollary 5.3, Theorem 6.1, and all of Section 7. If the constants or the index shift in (2.6) were wrong, the central inversion (5.7)–(5.9) would not hold. I ask the authors to make this point self-contained: define A(n,k) explicitly, state the exact result from [9] with theorem or equation number, and give a short derivation of (2.6), even if only from the combinatorial construction. This is the single most important revision.
minor comments (3)
  1. [§5 (after Prop. 5.1); §7 (before Thm. 7.10)] The paper repeatedly says that identities 'extend by analytic continuation whenever the functions involved are defined,' but the precise domains are not stated. In particular, Corollary 5.3 is used at z=1 and z=−1 later, and the reader must infer which functions are continued and where the identities remain valid. Please state the domain of each continuation explicitly.
  2. [§3, Theorem 3.1] In the z=1 part of the proof, the divergence of ∑ r^{−(s−2)} for Re(s−2)≤1 is asserted without justification. The assertion is correct (for positive real leading terms it is a standard Dirichlet-series fact), but a one-sentence explanation would improve readability.
  3. [§2] The paper would be easier to follow if the definition of the k-augmented centered triangular numbers appeared in the present manuscript rather than only via reference [9]. This is related to the major comment on (2.6), but even a short definition of the array would help the reader see why (2.6) is natural.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; central inversion is a direct Vandermonde identity once the cited coefficient formula is granted.

full rationale

None of the claimed results reduce to their inputs by construction. The family H_k is defined by (1.2) with A(r+1,k)=alpha_k r^2+beta_k r+1 (2.7). Theorem 3.2 and Proposition 5.1 are direct algebra using r=(r+a)-a and r^2=(r+a)^2-2a(r+a)+a^2; no parameter is fitted. Theorem 5.2 is a Vandermonde/Lagrange inversion applied to the identity F_k=sum c_j lambda_j^k D^j Phi (5.5); L_j(lambda_m)=delta_{jm} gives (5.4) exactly. Corollary 5.3 is the d=2 specialization lambda=(1,2,4), c=(1,3/2,3/2), and the displayed Lagrange polynomials; the arithmetic is correct. Recurrences, generating functions, and special values all follow from the same operator representation plus standard zeta/Bernoulli/Euler identities. The only external input is Eq. (2.6), attributed to [9] by the first author; although it is unproved here and load-bearing, this is a borrowed premise/correctness risk, not circular reasoning. The paper does not call a fitted value a prediction, and no self-citation supplies the inversion argument. Therefore no circular step is present.

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

The central claim depends on the imported coefficient formula for A(n,k) and on standard Hurwitz-Lerch/zeta background. There are no fitted free parameters and no ad hoc invented objects; the new family H_k is a definition, not a postulated entity used to force the result.

assumptions (4)
  • domain assumption A(n,k) = 3/2 4^k (n-1)^2 + 3/2 2^k (n-1) + 1 for n≥1, k≥0 (Eq. 2.6).
    Quoted from [9] (first author's prior paper) without derivation; it fixes alpha_k and beta_k and therefore determines H_k and all later formulas.
  • standard math The Hurwitz-Lerch transcendent Φ has the stated convergence, analytic continuation, and Euler-derivative identities DΦ=Φ(s-1)-aΦ(s,a), D²Φ=Φ(s-2,a)-2aΦ(s-1,a)+a²Φ(s,a).
    Invoked in Sections 2-3 and used throughout; standard background cited to [10,11,16,18].
  • standard math Termwise application of the Euler operator D and of parameter shifts is valid in the common domain of absolute convergence, and identities extend by analytic continuation.
    Used repeatedly in Sections 3,5,7; a standard analytic-continuation assumption for Dirichlet-type series.
  • standard math The rational form of negative-order polylogarithms Li_{-n}(z)=z A_n(z)/(1-z)^{n+1} and the Bernoulli/Euler polynomial identities (2.8), (7.15)-(7.17).
    Used in Section 7 to express special values; cited to [16] and derived from standard zeta identities.

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Pith. "Pith review of An Invertible Family of Hurwitz--Lerch Type Functions Associated with $k$-Augmented Centered Triangular Numbers." pith.science (2026). https://pith.science/paper/M7GOPXBF

@misc{pith2026260726403,
  author       = {Pith},
  title        = {Pith review of: An Invertible Family of Hurwitz--Lerch Type Functions Associated with $k$-Augmented Centered Triangular Numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M7GOPXBF}},
  note         = {Machine review of arXiv:2607.26403}
}
abstract

This paper defines a family of Hurwitz--Lerch type functions whose coefficients are the \(k\)-augmented centered triangular numbers. For this family, we obtain the convergence conditions, a reduction formula, and an Euler-operator form. A Vandermonde-based inversion formula is derived for a class of polynomially weighted Hurwitz--Lerch functions. The family considered here is the quadratic case with geometric factors \(1\), \(2\), and \(4\). The resulting formulas show that three consecutive functions recover the classical Hurwitz--Lerch transcendent and its first two Euler derivatives. We also derive recurrence formulas, ordinary generating functions, finite sums, and special values. The values at \(z=1\) are expressed through Hurwitz zeta functions and Bernoulli polynomials. When \(a=1\), the numerator polynomials of the rational values \(H_k(z,-m,1)\) are written in terms of Eulerian polynomials, while the alternating values \(H_k(-1,-m,a)\) are expressed through Euler polynomials.

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