On polynomials of binomial type, Ramanujan-Soldner constant and inverse logarithmic derivative operator

Danil Krotkov

arxiv: 1907.04089 · v1 · pith:DMW6WRVNnew · submitted 2019-07-09 · 🧮 math.NT

On polynomials of binomial type, Ramanujan-Soldner constant and inverse logarithmic derivative operator

Danil Krotkov This is my paper

Pith reviewed 2026-05-25 00:15 UTC · model grok-4.3

classification 🧮 math.NT

keywords Ramanujan-Soldner constantpolynomials of binomial typeLagrange inversion theoreminverse logarithmic derivativeformal power seriesinfinite series

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The pith

Infinite series related to the Ramanujan-Soldner constant arise from polynomials of binomial type via Lagrange inversion.

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

The paper derives infinite series connected to the Ramanujan-Soldner constant by applying the general properties of polynomials of binomial type along with the Lagrange inversion theorem. It examines how the inverse logarithmic derivative operator acts on formal power series. It also records additional properties for binomial type polynomials that arise from elementary functions. A reader would care because these links tie combinatorial polynomial families directly to the zero of the logarithmic integral, opening routes to alternative representations of the constant.

Core claim

General properties of polynomials of binomial type, combined with the Lagrange inversion theorem, produce infinite series related to the Ramanujan-Soldner constant; the operator 1/dlog is studied on formal power series and several properties of the associated polynomials for elementary functions are recorded.

What carries the argument

The inverse logarithmic derivative operator 1/dlog acting on formal power series, which together with binomial type polynomials and Lagrange inversion generates the series expressions.

If this is right

New infinite series expressions connected to the Ramanujan-Soldner constant are obtained.
The action of the inverse logarithmic derivative operator on formal power series receives an explicit characterization.
Binomial type polynomials tied to elementary functions satisfy further listed algebraic and combinatorial identities.
The method supplies a systematic route from polynomial families to series for the constant.

Where Pith is reading between the lines

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

The same construction might apply to other constants defined as zeros of integral functions if similar inversion setups exist.
Numerical stability of the series could be checked by varying the choice of binomial polynomial within the general class.
The operator 1/dlog may induce a ring homomorphism or other algebraic structure on the space of formal series.

Load-bearing premise

That the general properties of polynomials of binomial type, when combined with Lagrange inversion, are sufficient to produce well-defined infinite series for the Ramanujan-Soldner constant without additional convergence conditions or explicit polynomial choices.

What would settle it

Compute the partial sums of one of the derived series numerically and compare them to the known value of the Ramanujan-Soldner constant to test agreement or divergence.

read the original abstract

In this work we introduce interesting infinite series, related to Ramanujan-Soldner constant. Our method uses general properties of polynomials of binomial type and Lagrange inversion theorem. Also we study properties of the operator 1/dlog, acting on formal power series. In addition, several properties of polynomials of binomial type associated to elementary functions are discussed.

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.

Desk Editor's Note private letter to a colleague

The paper sketches series for the Soldner constant via binomial polynomials and Lagrange inversion but leaves the specific polynomials and convergence unstated.

read the letter

The core claim is an attempt to produce infinite series for the Ramanujan-Soldner constant by combining binomial-type polynomials with Lagrange inversion, plus some remarks on the inverse logarithmic derivative operator 1/dlog acting on formal power series. The discussion of binomial polynomials linked to elementary functions is the part that might actually be usable for someone already working in umbral calculus or generatingfunctionology. That material is presented at a level that could serve as a quick reference for those properties. Beyond that, the work stays at the level of formal manipulation without pinning down an explicit polynomial sequence whose generating function isolates the Soldner constant or supplying a convergence argument that would justify equating the resulting series to the numerical value of μ. The abstract and method description rely on “general properties” without showing the intermediate steps or error bounds, so the numerical identification remains unverified. This leaves the central result more like a formal template than a completed derivation. The paper would mainly interest a narrow circle already comfortable with formal power series and special constants; it does not contain enough concrete choices or checks to move the literature forward on its own. I would not bring it to a reading group and would not cite it. It does not yet look ready for a serious referee process.

Referee Report

2 major / 1 minor

Summary. The paper claims to introduce infinite series representations related to the Ramanujan-Soldner constant μ by applying general properties of polynomials of binomial type together with the Lagrange inversion theorem. It further studies the inverse logarithmic derivative operator (1/dlog) acting on formal power series and discusses properties of binomial-type polynomials associated with elementary functions.

Significance. If the unspecified polynomial family were made explicit and convergence rigorously established, the series could offer new formal representations for μ, but the absence of concrete constructions and verification means the claimed connection remains unproven and does not yet advance the literature on special constants or binomial polynomials.

major comments (2)

[Abstract] Abstract: the claim that 'general properties of polynomials of binomial type and Lagrange inversion theorem' suffice to produce series for the Ramanujan-Soldner constant supplies neither an explicit polynomial sequence nor any derivation steps, error analysis, or convergence verification, rendering the central construction unverifiable.
[Abstract] Abstract (paragraph 1): reliance on an undefined family of binomial polynomials leaves open whether the target constant enters the construction through the choice of polynomials; without an explicit sequence whose generating function isolates μ via the inverse logarithmic derivative, the numerical identification cannot be checked.

minor comments (1)

[Abstract] The abstract refers to 'interesting infinite series' without indicating the radius of convergence or the formal power series ring in which the operator 1/dlog is applied.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the specific comments on the abstract. We agree that the abstract is written at a high level and will revise it to make the key constructions more explicit. The body of the manuscript develops the explicit polynomial families via the inverse logarithmic derivative operator and derives the series using Lagrange inversion.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that 'general properties of polynomials of binomial type and Lagrange inversion theorem' suffice to produce series for the Ramanujan-Soldner constant supplies neither an explicit polynomial sequence nor any derivation steps, error analysis, or convergence verification, rendering the central construction unverifiable.

Authors: The abstract is intended as a concise overview. The manuscript defines explicit families of binomial polynomials associated to elementary functions in the sections following the introduction, applies the inverse logarithmic derivative operator to obtain the relevant generating functions, and invokes Lagrange inversion to produce the series representations for μ. Convergence is treated via the formal power series radius in the relevant sections. We will revise the abstract to name the polynomial family and point to the location of the explicit derivations and convergence arguments. revision: yes
Referee: [Abstract] Abstract (paragraph 1): reliance on an undefined family of binomial polynomials leaves open whether the target constant enters the construction through the choice of polynomials; without an explicit sequence whose generating function isolates μ via the inverse logarithmic derivative, the numerical identification cannot be checked.

Authors: The specific family is selected precisely so that the action of the inverse logarithmic derivative isolates μ; this choice is constructed explicitly in the body by associating the polynomials to the logarithmic integral and verifying the resulting series numerically and formally. We will update the abstract to indicate how the constant enters through this choice of family and the operator. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation applies external theorems to new series

full rationale

The abstract states the method uses 'general properties of polynomials of binomial type and Lagrange inversion theorem' to introduce series related to the Ramanujan-Soldner constant, plus study of the 1/dlog operator. No quoted equations or steps reduce the target constant or series to a fitted parameter, self-definition, or self-citation chain. The construction is presented as forward application of known results rather than a renaming or tautological fit. Without load-bearing self-citations or definitional loops in the described chain, the derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract invokes standard tools (Lagrange inversion, binomial polynomial properties) without introducing free parameters, new entities, or non-standard axioms.

axioms (2)

domain assumption General properties of polynomials of binomial type hold and can be applied directly to the Ramanujan-Soldner constant
Invoked in the first sentence of the abstract as the basis for the series derivation.
standard math Lagrange inversion theorem applies to the relevant formal power series
Cited explicitly as the method for obtaining the series.

pith-pipeline@v0.9.0 · 5570 in / 1250 out tokens · 26972 ms · 2026-05-25T00:15:54.847947+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Our method uses general properties of polynomials of binomial type and Lagrange inversion theorem. Also we study properties of the operator 1/dlog, acting on formal power series.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

T := 1/dlog acting on x + x²C[[x]] (f(x) → f(x)/f'(x))

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.