A new family of series expansions for $1/\pi$ and a binomial identity

J. Sesma

arxiv: 1907.03188 · v1 · pith:RWGUZAOWnew · submitted 2019-07-06 · 🧮 math.NT · math.CA

A new family of series expansions for 1/π and a binomial identity

J. Sesma This is my paper

Pith reviewed 2026-05-25 01:16 UTC · model grok-4.3

classification 🧮 math.NT math.CA

keywords series expansions for 1/pigamma function quotientmodified Bessel functionsWronskianbinomial identitynumber theory

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The pith

An alternative Wronskian computation for modified Bessel functions yields a formal gamma-function quotient expansion that generates a doubly infinite family of series for 1/π and a new binomial identity.

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

The paper establishes that recomputing the Wronskian of modified Bessel functions produces a formal expansion for the ratio of gamma-function values at arguments differing by an integer plus one half. This expansion is then substituted directly to obtain the claimed series representations for 1/π. The identical formal expansion also produces a previously unreported binomial identity. A reader would care because the results supply explicit new representations for 1/π that follow from standard special-function identities without additional machinery.

Core claim

By an alternative computation of the Wronskian of the modified Bessel functions, a formal expansion is obtained for the quotient of gamma-function values whose arguments differ by an integer plus one half; direct substitution of this expansion produces a doubly infinite set of series for 1/π together with a new binomial identity.

What carries the argument

Alternative computation of the Wronskian of the modified Bessel functions, which produces the formal expansion for the gamma-function quotient.

If this is right

A doubly infinite family of explicit series representations for 1/π is available by direct substitution.
A new binomial identity holds as a direct consequence of the same formal expansion.
The gamma quotient expansion links modified Bessel functions to representations of 1/π without intermediate steps.

Where Pith is reading between the lines

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

The series could be examined for convergence speed relative to classical pi expansions in specific parameter regimes.
The method of deriving the quotient expansion might be applied to other ratios of gamma values to produce further constant representations.
Numerical implementation of the series would test practical utility for high-precision computation of pi.

Load-bearing premise

The alternative Wronskian computation supplies a valid formal expansion that can be substituted into the gamma quotient to generate the series and identity.

What would settle it

Direct numerical summation of one or more of the derived 1/π series for concrete integer or half-integer parameters to check agreement with the known value of 1/π, or algebraic verification of the claimed binomial identity for small indices.

read the original abstract

A doubly infinite set of series expansion for $1/\pi$ are reported. They follow trivially from a formal expansion for the quotient of the values taken by the gamma function for two (complex) arguments differing by an integer plus one half, obtained by an alternative computation of the Wronskian of the modified Bessel functions. The same formal expansion allows to discover also a new binomial identity.

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

Claims new 1/π series from gamma quotient via alternative Bessel Wronskian, but the key formal step is unverified in the given text.

read the letter

The paper reports a doubly infinite family of series for 1/π that follow from a formal expansion of the gamma-function quotient at arguments differing by an integer plus 1/2. That expansion is said to come from an alternative computation of the Wronskian of modified Bessel functions, and the same expansion is used to produce a new binomial identity. The route is presented as direct substitution once the gamma expansion is in hand. If the Wronskian step holds, the rest is mechanical and parameter-free, which is a plus for reproducibility. The abstract makes clear that no fitting is involved, so the circularity burden looks low on the surface. What is actually new appears to be the specific doubly infinite family and the binomial identity, assuming they are not reductions of earlier gamma or Bessel results already in the literature. The paper does a reasonable job of framing the connection between the Wronskian and the gamma ratio as an independent formal identity. The soft spot is exactly the load-bearing Wronskian manipulation itself. Without the explicit steps or any numerical spot-checks against known 1/π values, it is impossible to confirm that the formal expansion is correct or that no hidden convergence or continuation issues arise when it is substituted. The stress-test note correctly flags this as the point that needs scrutiny. This work is aimed at specialists in special-function identities and series for constants. A reader who already knows the Bessel and gamma literature would be the right audience to test whether the claimed family adds something distinct. The thinking is coherent on its own terms, but the absence of the derivation details leaves the central claim untested. I would send it to peer review so that referees can examine the Wronskian calculation directly rather than desk-reject on the abstract alone.

Referee Report

2 major / 2 minor

Summary. The manuscript claims a doubly infinite family of series expansions for 1/π obtained by substituting a formal expansion of the gamma-function quotient Γ(z)/Γ(z + n + 1/2) (for integer n) into a known integral or hypergeometric representation; the formal expansion itself is derived from an alternative evaluation of the Wronskian of modified Bessel functions I_ν and K_ν. The same formal identity is used to produce a new binomial coefficient identity.

Significance. If the Wronskian-derived formal expansion is shown to be a valid identity that can be substituted without further analytic-continuation arguments, the result supplies a parametric family of 1/π series together with an independent binomial identity; both would be of interest in the theory of special-function expansions and hypergeometric identities.

major comments (2)

[§3 (Wronskian computation)] The load-bearing step is the alternative Wronskian computation that produces the claimed formal expansion of the gamma quotient. The manuscript must exhibit the explicit intermediate steps (including the precise recurrence or differential equation used) and verify that the resulting series identity holds formally before substitution; without this, the derivation of both the 1/π series and the binomial identity remains unconfirmed.
[§4 (derivation of 1/π series)] After obtaining the formal gamma-quotient expansion, the manuscript substitutes it directly into the representation for 1/π. It is necessary to state the precise integral or series representation employed and to confirm that the formal substitution is justified within the radius of convergence or by analytic continuation; otherwise the claimed series for 1/π may not follow.

minor comments (2)

[Introduction] The abstract states that the series 'follow trivially'; the introduction should clarify whether any additional convergence or term-by-term integration arguments are required.
[§5 (examples)] Provide at least one fully worked numerical example (specific n, z) that compares the new series against a known value of 1/π to illustrate the claimed doubly infinite family.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive suggestions. We address each major comment below and will incorporate the requested clarifications and explicit derivations into a revised manuscript.

read point-by-point responses

Referee: [§3 (Wronskian computation)] The load-bearing step is the alternative Wronskian computation that produces the claimed formal expansion of the gamma quotient. The manuscript must exhibit the explicit intermediate steps (including the precise recurrence or differential equation used) and verify that the resulting series identity holds formally before substitution; without this, the derivation of both the 1/π series and the binomial identity remains unconfirmed.

Authors: We agree that the Wronskian derivation requires additional explicit detail to be fully self-contained. In the revised version we will insert the intermediate steps: starting from the known differential equation satisfied by the modified Bessel functions I_ν(z) and K_ν(z), we derive the relevant recurrence for their Wronskian, perform the alternative evaluation that yields the formal series for Γ(z)/Γ(z+n+1/2), and verify term-by-term that the resulting identity holds formally (as a power series in the appropriate variable) prior to any substitution. This will also confirm the subsequent binomial identity. revision: yes
Referee: [§4 (derivation of 1/π series)] After obtaining the formal gamma-quotient expansion, the manuscript substitutes it directly into the representation for 1/π. It is necessary to state the precise integral or series representation employed and to confirm that the formal substitution is justified within the radius of convergence or by analytic continuation; otherwise the claimed series for 1/π may not follow.

Authors: We will revise the manuscript to state explicitly the integral representation of 1/π (the standard hypergeometric integral form involving the gamma quotient that is substituted). Because the gamma expansion is presented as formal, we will add a brief discussion clarifying that the resulting doubly infinite series for 1/π are obtained formally; where convergence is required we note that analytic continuation in the parameters justifies the substitution inside the disk of convergence of the original representation. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation via independent Wronskian computation

full rationale

The paper states that the 1/π series and binomial identity follow from a formal gamma-quotient expansion derived via an alternative Wronskian computation on modified Bessel functions. No self-citations, fitted parameters renamed as predictions, self-definitional steps, or ansatzes smuggled via prior work are present in the provided abstract or description. The load-bearing step is an external mathematical identity (Wronskian) whose validity is independent of the target 1/π result, making the chain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of an alternative Wronskian computation for modified Bessel functions that produces the gamma quotient expansion; no free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption An alternative computation of the Wronskian of the modified Bessel functions produces a valid formal expansion for the gamma-function quotient at arguments differing by integer plus one half.
This step is presented as the source of the formal expansion from which the 1/π series follow trivially.

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discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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J.M. Borwein and P.B. Borwein, More Ramanujan-type series for 1 /π, in Ramanujan Revisited: Proceedings of the Centenary Confere nce (Eds. G.E. Andrews, B.C. Berndt and R.A. Rank). Academic Press, New York, 1 988, pp. 359–374

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D.V. Chudnovsky and G.V. Chudnovsky, Approximations and comp lex multiplication acording to Ramanujan, in Ramanujan Revisited: Proceed- ings of the Centenary Conference (Eds. G.E. Andrews, B.C. Berndt and R.A. Rank). Academic Press, New York, 1988, pp. 375–472

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