arxiv: 2604.21947 · v1 · submitted 2026-04-22 · 🧮 math.GM

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Introduction to generalised Cesaro convergence II

Richard Stone

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Pith reviewed 2026-05-09 23:02 UTC · model grok-4.3

classification 🧮 math.GM

keywords Cesaro summationremainder CesaroGamma functiongeneralized convergencedilation invariancefunctional equationsinfinite productsgeometric summation

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The pith

The Gamma function admits a natural definition via remainder Cesaro products, with its properties following from geometric invariance.

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

This paper extends generalised Cesaro methods to remainder Cesaro summation, which treats the geometric positions of terms in series and products as an explicit convergence parameter. It proves that this form of convergence is unchanged under dilation and scaling. The authors then apply the method to define the Gamma function through remainder Cesaro products. From this single definition, both elementary identities such as the recurrence relation and more advanced properties follow directly from the geometric and invariance features alone. The same approach accounts for the shared structure observed in many different functional equations.

Core claim

We introduce remainder Cesaro summation and products, which incorporate geometric location as a convergence parameter and remain invariant under dilation and scaling. Using this framework we define the classical Gamma function via a remainder Cesaro product. Its basic properties, including the functional equation, and its advanced properties then follow immediately and intuitively from the geometric and dilation-invariance properties of the definition.

What carries the argument

Remainder Cesaro summation and products, which treat the geometric location of summands or factors as a convergence parameter and exhibit invariance under dilation and scaling.

Load-bearing premise

Remainder Cesaro summation must be well-defined and consistent when applied to the infinite products and series that appear in the Gamma function definition, even when geometric location serves as a convergence parameter.

What would settle it

Direct computation of the Gamma function at a positive integer such as 4 using the remainder Cesaro product definition must equal 3 factorial; any mismatch would show that the definition fails to recover the classical Gamma function.

read the original abstract

In this second of three introductory papers, we extend the notion of generalised Cesaro summation/convergence to the more natural setting of what we call remainder Cesaro summation/convergence. This greatly expands the range of problems susceptible to Cesaro methods and introduces the geometric location of summands as a critical consideration. We also show that geometric generalised Cesaro convergence is invariant under dilation and scaling. We present a number of calculations illustrating the utility of these developments. In particular we introduce a new, more natural definition of the classical Gamma function using remainder Cesaro summation/products, and show that many its key properties - both basic and advanced - fall out directly and intuitively from this Cesaro definition and its geometric and dilation-invariance properties. We also consider other examples and show how Cesaro methodology explains the common structure of many well-known functional equations.

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

This paper extends Cesaro methods to remainder sums with geometric location as a parameter and claims a new Gamma definition where properties follow from dilation invariance, but the abstract leaves the key definitions and verifications unshown.

read the letter

The main point is an extension of generalized Cesaro summation to a remainder version that treats the geometric position of terms as a convergence parameter, plus an application to a Cesaro-style definition of the Gamma function. The author also notes invariance under dilation and scaling, and sketches how this setup might clarify structures in functional equations more broadly.

Referee Report

2 major / 1 minor

Summary. The paper extends generalised Cesaro summation to remainder Cesaro summation, treating the geometric location of summands as a convergence parameter. It claims this extension is invariant under dilation and scaling, and applies it to introduce a new definition of the classical Gamma function via remainder Cesaro products. The manuscript asserts that many key properties of the Gamma function (basic and advanced) follow directly and intuitively from this definition together with the geometric and invariance properties. It also presents other examples to illustrate how Cesaro methods unify the structure of functional equations.

Significance. If the remainder Cesaro product is shown to be unambiguously well-defined for the relevant infinite products and series, independent of geometric placement order, and if the claimed derivations are non-circular, the work would supply a novel, geometrically motivated perspective on the Gamma function and related functional equations. The invariance properties could then serve as a unifying principle. However, the significance hinges on resolving the well-definedness and equivalence questions; absent those, the approach risks being a rephrasing rather than a derivation.

major comments (2)

The central claim that Gamma-function properties 'fall out directly' from the remainder Cesaro definition requires an explicit statement of that definition together with a proof that the remainder Cesaro product applied to the Weierstrass infinite product is well-defined, yields a unique limit independent of the ordering or geometric placement of factors, and coincides with the classical Gamma function. Without such a demonstration the derivations of reflection, multiplication, and other theorems become conditional on an unstated equivalence.
The invariance under dilation and scaling is presented as load-bearing for the Gamma-function application, yet the available description supplies neither the precise statement of the invariance theorem nor any sample calculation verifying that the new summation reproduces known Gamma identities. This omission directly affects the claim that the properties follow 'intuitively' from the definition.

minor comments (1)

The abstract and introductory material would be strengthened by including at least one fully worked numerical example of a remainder Cesaro product or sum that converges where ordinary summation fails, together with the explicit rule used when ordinary convergence is absent.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful and constructive report. The comments correctly identify areas where greater explicitness and verification are needed to support the central claims. We will revise the manuscript to address both major points directly, adding the required definitions, proofs, and calculations while preserving the geometric and invariance-based perspective.

read point-by-point responses

Referee: The central claim that Gamma-function properties 'fall out directly' from the remainder Cesaro definition requires an explicit statement of that definition together with a proof that the remainder Cesaro product applied to the Weierstrass infinite product is well-defined, yields a unique limit independent of the ordering or geometric placement of factors, and coincides with the classical Gamma function. Without such a demonstration the derivations of reflection, multiplication, and other theorems become conditional on an unstated equivalence.

Authors: We agree that the manuscript would benefit from an explicit statement of the remainder Cesaro product definition and a self-contained proof of its well-definedness. In the revised version we will insert a dedicated subsection that (i) states the definition in full, (ii) proves that the infinite product over the Weierstrass factors converges to a unique limit independent of ordering and geometric placement of the factors, and (iii) establishes coincidence with the classical Gamma function. With this equivalence in place, the derivations of the reflection formula, multiplication theorem and other identities will no longer be conditional; they will follow from the verified definition together with the geometric and dilation-invariance properties already developed in the paper. revision: yes
Referee: The invariance under dilation and scaling is presented as load-bearing for the Gamma-function application, yet the available description supplies neither the precise statement of the invariance theorem nor any sample calculation verifying that the new summation reproduces known Gamma identities. This omission directly affects the claim that the properties follow 'intuitively' from the definition.

Authors: We accept that the invariance theorem requires a sharper formulation and concrete verification. The revised manuscript will contain (a) a formal statement of the dilation-and-scaling invariance theorem for remainder Cesaro summation, including the precise hypotheses under which it holds, and (b) explicit sample calculations that apply the theorem to recover standard Gamma identities (for instance, the reflection formula). These calculations will demonstrate how the geometric placement and invariance properties yield the identities directly, thereby supporting the claim that the properties emerge intuitively once the foundational equivalence is established. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new definition presented with properties derived from it

full rationale

The abstract introduces a new definition of the Gamma function via remainder Cesaro summation/products and states that key properties fall out directly from this definition together with its geometric and dilation-invariance properties. No equations or self-citations are supplied that reduce the claimed derivation to a tautology, a fitted parameter renamed as a prediction, or a load-bearing self-citation chain. The structure is therefore self-contained: an independent definition is posited and consequences are derived from it, with any question of well-definedness or equivalence to the classical Gamma being a matter of verification rather than circularity by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, background axioms, or newly postulated entities; the new summation concept itself may function as an implicit domain assumption but cannot be audited from the given text.

pith-pipeline@v0.9.0 · 5423 in / 1200 out tokens · 41831 ms · 2026-05-09T23:02:35.889426+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references

[1]

Stone,Introduction to generalised Césaro convergence I, 2026

R. Stone,Introduction to generalised Césaro convergence I, 2026

2026
[2]

Ahlfors,Complex Analysis: An Introduction to the Theory of Ana- lytic Functions of One Complex Variable, Third Edition, McGraw-Hill, 1979

Lars V. Ahlfors,Complex Analysis: An Introduction to the Theory of Ana- lytic Functions of One Complex Variable, Third Edition, McGraw-Hill, 1979

1979
[3]

Edwards,Riemann’s Zeta Function, Academic Press, 1974 27

H.M. Edwards,Riemann’s Zeta Function, Academic Press, 1974 27

1974