Polynomial Gosper multipliers for hypergeometric terms

Kwang-Wu Chen; Kwang-Wu Chen

doi:10.3934/math.20261144

AIMS Mathematics

2026, Volume 11, Issue 9: 28740-28761. doi: 10.3934/math.20261144

Previous Article Next Article

Research article

Polynomial Gosper multipliers for hypergeometric terms

Kwang-Wu Chen ^,

Department of Data Science and Mathematics, University of Taipei, Taipei 100234, Taiwan

Received: 08 May 2026 Revised: 23 August 2026 Accepted: 02 September 2026 Published: 08 September 2026
MSC : 05A19, 33C20, 68W30

We study polynomial multipliers that transform hypergeometric terms into Gosper-summable terms. For a broad class of hypergeometric terms whose consecutive-term ratio has a constant numerator and a polynomial denominator, we introduce a linear operator whose image coincides with the full space of polynomial Gosper multipliers, thereby obtaining a complete characterization of all such multipliers. We construct a normalized basis for this space and derive explicit recurrence relations for the basis coefficients and the associated inverse change-of-basis matrices. These results yield closed-form evaluations of the corresponding finite sums and are illustrated by several classical factorial-type hypergeometric terms.
- hypergeometric terms,
- Gosper summation,
- telescoping sums,
- polynomial multipliers,
- closed-form evaluations
Citation: Kwang-Wu Chen. Polynomial Gosper multipliers for hypergeometric terms[J]. AIMS Mathematics, 2026, 11(9): 28740-28761. doi: 10.3934/math.20261144

Related Papers:

Abstract

We study polynomial multipliers that transform hypergeometric terms into Gosper-summable terms. For a broad class of hypergeometric terms whose consecutive-term ratio has a constant numerator and a polynomial denominator, we introduce a linear operator whose image coincides with the full space of polynomial Gosper multipliers, thereby obtaining a complete characterization of all such multipliers. We construct a normalized basis for this space and derive explicit recurrence relations for the basis coefficients and the associated inverse change-of-basis matrices. These results yield closed-form evaluations of the corresponding finite sums and are illustrated by several classical factorial-type hypergeometric terms.

References

[1]	R. W. Gosper, Decision procedure for indefinite hypergeometric summation, Proc. Natl. Acad. Sci. USA, 75 (1978), 40–42. https://doi.org/10.1073/pnas.75.1.40 doi: 10.1073/pnas.75.1.40
[2]	M. Petkovšek, H. S. Wilf, D. Zeilberger, $ A = B $, A K Peters/CRC Press, 1997.
[3]	Q. H. Hou, Y. Wei, Telescoping method, summation formulas, and inversion pairs, Electron. Res. Arch., 29 (2021), 2657–2671. https://doi.org/10.3934/era.2021007
[4]	K. W. Chen, Some double $ q $-series by telescoping, Mathematics, 11 (2023), 2949. https://doi.org/10.3390/math11132949 doi: 10.3390/math11132949
[5]	C. Koutschan, E. Wong, Creative telescoping on multiple sums, Math. Comput. Sci., 15 (2021), 483–498. https://doi.org/10.1007/s11786-021-00514-3 doi: 10.1007/s11786-021-00514-3
[6]	P. Paule, C. Schneider, Creative telescoping for hypergeometric double sums, J. Symbolic Comput., 128 (2025), 102394. https://doi.org/10.1016/j.jsc.2024.102394 doi: 10.1016/j.jsc.2024.102394
[7]	Q. H. Hou, G. J. Li, Gosper summability of rational multiples of hypergeometric terms, J. Differ. Equ. Appl., 27 (2021), 1723–1733. https://doi.org/10.1080/10236198.2021.2007903 doi: 10.1080/10236198.2021.2007903
[8]	R. Dougherty-Bliss, Gosper's algorithm and Bell numbers, arXiv, 2022. https://doi.org/10.48550/arXiv.2210.13520
[9]	OEIS Foundation Inc., The on-line encyclopedia of integer sequences. Available from: https://oeis.org.
[10]	L. Comtet, Advanced combinatorics: the art of finite and infinite expansions, Springer Dordrecht, 1974. https://doi.org/10.1007/978-94-010-2196-8

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)