Combinatorial identities derived from explicit formulas of Gauss hypergeometric functions
Abstract
In the present paper, with the help of the Faà di Bruno formula and several identities of partial Bell polynomials, the author establishes explicit formulas of the Gauss hypergeometric functions 2F1biggl((1−n)/2,(2−n)/2;3/2−m;z2biggr), 2F1biggl(−n/2,(1−n)/2;1/2−m;z2biggr), 2F1biggl(a,a+1/2;3/2−m;z2biggr), 2F1biggl(a,a+1/2;1/2−m;z2biggr) for m,n∈ℕ and a∈ℂ, and then derives two combinatorial identities ∑k=0m2k/k! binom2m−2km−k ∑ℓ=0k ((−1)ℓ)/2ℓ ((2k−2ℓ−1)!!)/((n−ℓ)!) binom2k−ℓ−1ℓ−1 =1/n!binom2m−nm and ∑k=1m1/(k!)2binom2m−2km−k ∑ℓ=1k binomkℓℓ(2k−ℓ−1)! (2a)ℓ =binom2m+2am−binom2mm, where m∈ℕ0, n∈ℤ, and a∈ℂ. These newly-established identities extend and generalize the beautiful combinatorial identity ∑k=0n 2k/k!binom2n−2kn−k ∑j=0k((−1)j)/2j ((2k−2j−1)!!)/((n−j)!) binom2k−j−1j−1 =1/n!, n∈ℕ0, which was obtained in Theorem 4 of the paper "F. Qi, C.-Y. He, and D. Lim, Explicit formulas of two Gauss hypergeometric functions and several combinatorial identities, Discrete Appl. Math. Vol. 393 (2026), 215--229. DOI: https://doi.org/10.1016/j.dam.2026.06.023."
Related
- Extensions of several famous combinatorial identities via hypergeometric functions
- Combinatorial formulas for the multivariate Markov-Krein transform
- Combinatorial decompositions of strict partitions and multi-sum Rogers-Ramanujan type identities
- A Combinatorial Proof of Partition Identities for the Number of Distinct Part Sizes
- Combinatorial Goussarov-Polyak-Viro Formulas for the Linking Number and Low Degree Coefficients of the Conway Polynomial
BibTeX
@misc{qi2026combinatorial,
title = {{Combinatorial identities derived from explicit formulas of Gauss hypergeometric functions}},
author = {Feng Qi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.10643},
eprint = {2607.10643},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.10643},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Combinatorial identities derived from explicit formulas of Gauss hypergeometric functions
AU - Feng Qi
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2607.10643
UR - https://arxiv.org/abs/2607.10643
AB - In the present paper, with the help of the Faà di Bruno formula and several identities of partial Bell polynomials, the author establishes explicit formulas of the Gauss hypergeometric functions _2F_1biggl((1−n)/2,(2−n)/2;3/2−m;z^2biggr), _2F_1biggl(−n/2,(1−n)/2;1/2−m;z^2biggr), _2F_1biggl(a,a+1/2;3/2−m;z^2biggr), _2F_1biggl(a,a+1/2;1/2−m;z^2biggr) for m,n∈ℕ and a∈ℂ, and then derives two combinatorial identities ∑_k=0^m2^k/k! binom2m−2km−k ∑_ℓ=0^k ((−1)^ℓ)/2^ℓ ((2k−2ℓ−1)!!)/((n−ℓ)!) binom2k−ℓ−1ℓ−1 =1/n!binom2m−nm and ∑_k=1^m1/(k!)^2binom2m−2km−k ∑_ℓ=1^k binomkℓℓ(2k−ℓ−1)! (2a)_ℓ =binom2m+2am−binom2mm, where m∈ℕ_0, n∈ℤ, and a∈ℂ. These newly-established identities extend and generalize the beautiful combinatorial identity ∑_k=0^n 2^k/k!binom2n−2kn−k ∑_j=0^k((−1)^j)/2^j ((2k−2j−1)!!)/((n−j)!) binom2k−j−1j−1 =1/n!, n∈ℕ_0, which was obtained in Theorem 4 of the paper "F. Qi, C.-Y. He, and D. Lim, Explicit formulas of two Gauss hypergeometric functions and several combinatorial identities, Discrete Appl. Math. Vol. 393 (2026), 215--229. DOI: https://doi.org/10.1016/j.dam.2026.06.023."
ER -