Generalizing Pascal’s row sums to complex orders: A Poisson summation approach

Fabrizio Mancini
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 2, Pages 300–312
DOI: 10.7546/nntdm.2026.32.2.300-312
Full paper (PDF, 808 Kb)

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Authors and affiliations

Fabrizio Mancini
I.I.S. da Vinci
Civitanova Marche, Italy

Abstract

We generalize Pascal’s triangle row sums by defining shifted fractional binomial coefficients with a real scaling parameter and a complex shift. Using the Poisson summation formula, we derive exact summation identities extended via analytic continuation to conditionally convergent regimes. We prove that when the magnitude of the scaling parameter is not greater than 2, the sum collapses to a simple exponential form independent of the shift; otherwise, it generalizes classical series multisections to non-integer steps and complex shifts. These closed forms allow for rapid and exact computation even where symbolic algebra systems fail.

Keywords

• Pascal’s triangle
• Fractional binomial coefficients
• Bernoulli numbers with fractional indices
• Gamma function
• Fourier transform
• Poisson summation formula
• Analytic continuation
• Series multisection

2020 Mathematics Subject Classification

• 11B65
• 33B15
• 42A38

References

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Manuscript history

• Received: 11 February 2026
• Revised: 23 March 2026
• Accepted: 20 April 2026
• Online First: 24 April 2026

Copyright information

Ⓒ 2026 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).