Rising factorials and hypergeometric functions

Carlos M. da Fonseca and Anthony G. Shannon
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 3, Pages 526–535
DOI: 10.7546/nntdm.2026.32.3.526-535
Full paper (PDF, 207 Kb)

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Authors and affiliations

Carlos M. da Fonseca
Kuwait College of Science and Technology
Doha District, Safat 13133, Kuwait
Faculty of Applied Mathematics and Informatics, Technical University of Sofia
Kliment Ohridski Blvd. 8, 1000 Sofia, Bulgaria

Anthony G. Shannon
Warrane College, The University of New South Wales
Kensington, NSW 2033, Australia

Abstract

This note considers factorial binomial coefficients, focusing on both rising and falling factorial powers and their generalisations. We recall a factorial-binomial formulation of classical hypergeometric identities, including Saalschütz’s summation, and highlight the combinatorial structure underlying the coefficients. For Type 2 rising factorial binomial coefficients we establish a Pascal-type recurrence relation, identities for negative indices and some generating functions. We also discuss potential extensions to rising factorial Jordan coefficients and congruence properties, motivated by recent results of Kim and Kim. This unified approach offers a combinatorial and algebraic framework for analysing hypergeometric series and related number-theoretic properties.

Keywords

• Rising factorials
• Hypergeometric functions

2020 Mathematics Subject Classification

• 05A10
• 33C20
• 11B68
• 11B73
• 11B83

References

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Manuscript history

• Received: 26 June 2026
• Revised: 14 October 2026
• Accepted: 19 October 2026
• Online First: 20 October 2026

Copyright information

Ⓒ 2026 by the Authors.
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).