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Terminating Zero-Balanced Hypergeometric Series Using Divided Differences

  • Fatma Zürnacı Yetiş
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Abstract

There is a close relationship between divided differences and hypergeometric series, and recent studies have shown that divided differences can be used effectively to derive terminating hypergeometric identities. In this paper, we apply this approach to terminating zero-balanced hypergeometric series. Using explicit product evaluations arising from divided differences, we give alternative proofs of zero-balanced 3F2(1) and 4F3(1) summation formulas and then extend the argument to the general terminating r+1Fr(1) case. We also combine the Lagrange representation with the Leibniz rule for divided differences to derive a finite convolution transformation for a terminating r+2Fr+1(1) series. Thus, the divided-difference approach not only reproduces the known zero-balanced summation formulas but also yields a finite convolution transformation for terminating hypergeometric series, together with its zero-balanced specialization.

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Cite

BibTeXRIS
BibTeX
@misc{yetis2026terminating,
  title = {{Terminating Zero-Balanced Hypergeometric Series Using Divided Differences}},
  author = {Fatma Zürnacı Yetiş},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06168},
  eprint = {2609.06168},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06168},
}
RIS
TY  - UNPB
TI  - Terminating Zero-Balanced Hypergeometric Series Using Divided Differences
AU  - Fatma Zürnacı Yetiş
PY  - 2026
DA  - 2026/09/05
PB  - arXiv
DO  - 10.48550/arxiv.2609.06168
UR  - https://arxiv.org/abs/2609.06168
AB  - There is a close relationship between divided differences and hypergeometric series, and recent studies have shown that divided differences can be used effectively to derive terminating hypergeometric identities. In this paper, we apply this approach to terminating zero-balanced hypergeometric series. Using explicit product evaluations arising from divided differences, we give alternative proofs of zero-balanced _3F_2(1) and _4F_3(1) summation formulas and then extend the argument to the general terminating _r+1F_r(1) case. We also combine the Lagrange representation with the Leibniz rule for divided differences to derive a finite convolution transformation for a terminating _r+2F_r+1(1) series. Thus, the divided-difference approach not only reproduces the known zero-balanced summation formulas but also yields a finite convolution transformation for terminating hypergeometric series, together with its zero-balanced specialization.
ER  -