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The aim is to remove the positivity hypothesis related to the dominant Borel singularity of generating functions in polynomial continued fractions.

June 8, 2026Open Access

Removing the Positivity Hypothesis: Edge2013Borel Location and Type for Sign-Varying Polynomial Continued Fractions

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The aim is to remove the positivity hypothesis related to the dominant Borel singularity of generating functions in polynomial continued fractions.
Established the dominant Borel singularity location for arbitrary lower coefficients' signs.
Utilized classical continued-fraction theory to identify the dominant solution of the Wallis recurrence.
Verified results across eleven sign-varying families through numerical corroboration.
The dominant singularity retains its location at s = R = d^k/β_d regardless of lower coefficients' signs.
G is holonomic with Fuchsian type singularity at R with a local exponent -γ, confirming conclusions of EBR-I.
The absence of an entire Borel transform is established through the nonvanishing of the connection coefficient C.

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Randomized trial examines singularity characteristics in polynomial continued fractions, highlighting implications for Borel transforms.

Papanokechi (2026) studied this question.

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