Worpitzky Convergence Disk

• Worpitzky Convergence Disk is an analytic region that guarantees absolute convergence and geometric error control for limit-periodic continued fractions.
• It applies the Worpitzky criterion to bound parameters so that all convergents lie within a closed disk, ensuring uniform geometric error decay.
• Its analysis establishes the minimal integer coefficient realization in continued fractions, underpinning analytic identities such as the series for π/4.

The Worpitzky convergence disk is a rigorous analytic region in the complex plane that prescribes absolute convergence and geometric control of all approximants of a limit-periodic continued fraction. It arises from the Worpitzky criterion, which provides explicit quantitative bounds for the convergence and location of convergents of continued fractions with suitably bounded parameters. In the context of polynomial continued fractions representing values such as $π/4$, as recently formalized in connection with the Ramanujan Machine conjectures, the Worpitzky disk plays a central role in establishing not only convergence but also the symbolic minimality and uniqueness of the coefficient structures underlying these analytic identities (Wang, 13 Jan 2026).

1. The Worpitzky Convergence Criterion

Let $F_∞ = b_0 + K_{n=1}^∞(a_n/b_n)$ denote a continued fraction with nonzero denominators $b_n$. The Worpitzky parameters are defined as $ρ_n := a_n/(b_n b_{n-1})$, with $b_0$ normalized to $1$. The Worpitzky criterion asserts that if for all $n \geq 1$,

$|ρ_n| \leq R < 1/4,$

then:

• The continued fraction converges absolutely;
• For every $N$, the tail $T_N := K_{n=N}^∞(a_n/b_n)$ and every finite convergent lie in the closed disk centered at the origin of radius

$F_∞ = b_0 + K_{n=1}^∞(a_n/b_n)$0

• The Worpitzky disk $F_∞ = b_0 + K_{n=1}^∞(a_n/b_n)$1 is strictly contained in the open unit disk, providing explicit geometric error control.

These properties ensure uniform geometric decay of truncation errors and absolute convergence of the continued fraction whenever the Worpitzky bound is respected (Wang, 13 Jan 2026).

2. Analytic Realization in Ramanujan Machine Conjectures

A recently formalized example concerns the continued fraction:

$F_∞ = b_0 + K_{n=1}^∞(a_n/b_n)$2

with $F_∞ = b_0 + K_{n=1}^∞(a_n/b_n)$3, $F_∞ = b_0 + K_{n=1}^∞(a_n/b_n)$4; $F_∞ = b_0 + K_{n=1}^∞(a_n/b_n)$5, $F_∞ = b_0 + K_{n=1}^∞(a_n/b_n)$6 for $F_∞ = b_0 + K_{n=1}^∞(a_n/b_n)$7. Analytically, this structure is equated with the ratio of contiguous Gaussian hypergeometric functions,

$F_∞ = b_0 + K_{n=1}^∞(a_n/b_n)$8

where

$F_∞ = b_0 + K_{n=1}^∞(a_n/b_n)$9

and for $b_n$0, $b_n$1, the hypergeometric function reduces to the classical alternating series for $b_n$2: $b_n$3 (Wang, 13 Jan 2026).

3. Disk Radius and Stability Analysis

For this class, the key parameter is

$b_n$4

Taylor expansion as $b_n$5 yields $b_n$6, so $b_n$7 for all $b_n$8. The supremum $b_n$9 then gives, via the Worpitzky formula,

$ρ_n := a_n/(b_n b_{n-1})$0

Consequently, every tail $ρ_n := a_n/(b_n b_{n-1})$1 and all finite convergents $ρ_n := a_n/(b_n b_{n-1})$2 satisfy $ρ_n := a_n/(b_n b_{n-1})$3. As $ρ_n := a_n/(b_n b_{n-1})$4, absolute convergence is geometric with error decay rate

$ρ_n := a_n/(b_n b_{n-1})$5

The absolute bound $ρ_n := a_n/(b_n b_{n-1})$6 strictly contains all convergents within the disk $ρ_n := a_n/(b_n b_{n-1})$7, the Worpitzky convergence disk for this system.

4. Minimality of Integer Coefficient Realizations

The coefficients $ρ_n := a_n/(b_n b_{n-1})$8 and $ρ_n := a_n/(b_n b_{n-1})$9 for $b_0$0 arise via the unique equivalence transformation $b_0$1 applied to the base Gaussian continued fraction coefficients $b_0$2, $b_0$3. The integer realization is symbolically minimal: no further common integer factor can be removed from all $b_0$4 while preserving the limiting ratio $b_0$5, and any reduction in polynomial degree for $b_0$6 would force $b_0$7, thereby destroying convergence. Specifically, the boundary data $b_0$8 are the minimal integer initial values selecting the negative branch $b_0$9 (Wang, 13 Jan 2026).

5. Structural Implications and Absolute Convergence

The containment of all approximants and partial sums within the closed Worpitzky disk $1$0 provides uniform geometric error control and establishes that the associated limit-periodic continued fraction converges absolutely to $1$1. This result demonstrates the nontrivial analytic structure of continued fractions discovered by algorithmic induction, situating them as explicit consequences of classical hypergeometric transformation theory rather than as isolated numerical phenomena.

6. Table: Parameters and Constraints in Worpitzky Disk Analysis

Parameter	Value/Formulation	Condition
$1$2	$1$3	$1$4 for all $1$5
Worpitzky disk radius $1$6	$1$7	$1$8
Tail and convergent bound	$1$9	$n \geq 1$0
Error decay rate $n \geq 1$1	$n \geq 1$2	$n \geq 1$3 for $n \geq 1$4

All such findings are direct corollaries of the Worpitzky criterion as applied to the specific continued fraction system, confirming that algorithmically discovered analytic identities for constants such as $n \geq 1$5 are deeply subsumed under the classical analytic theory of continued fraction convergence (Wang, 13 Jan 2026).