Infinite and finite series involving central binomial coefficients and closed forms of generalized hypergeometric functions
Abstract
Let ℤ-={−1,−2,dotsc} and ℕ={1,2,dotsc}. In 2023, Qi and Lim gave two claims for summing the infinite series ∑k=1∞binom2kk1/(α+k)(±1/4)k for α∈ℂ∖ℤ-. In this paper, we confirm these claims by expressing the infinite series ∑k=1∞binom2kk1/(α+k)(z/4)k and the finite series ∑k=1nbinom2kk1/(α+k)(z/4)k in terms of the Gauss hypergeometric function 2F1[1/2,α;1+α;z] and a generalized hypergeometric function 3F2, for |z|≤1 and n∈ℕ. Using the Euler integral representation, we derive closed forms, in terms of finite sums of sines and cosines of arcsin√z, for 2F1[1/2,1/2+n;3/2+n;z], 2F1[1/2,1+n;2+n;z], two related 3F2 functions, and the incomplete beta functions Bz(1/2+n,1/2) and Bz(1+n,1/2). By the Pfaff transformation, we obtain closed forms of several further Gauss hypergeometric functions with parameters shifted by n. We also give a closed form of [(1−z)d/dz(1−z)]n(arcsin√z )/(√(z(1−z)) ) and a combinatorial identity.
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BibTeX
@misc{basnet2026infinite,
title = {{Infinite and finite series involving central binomial coefficients and closed forms of generalized hypergeometric functions}},
author = {Ganesh Bahadur Basnet and Narayan Prasad Pahari and Feng Qi and Arjun Kumar Rathie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.05193},
eprint = {2608.05193},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.05193},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Infinite and finite series involving central binomial coefficients and closed forms of generalized hypergeometric functions
AU - Ganesh Bahadur Basnet
AU - Narayan Prasad Pahari
AU - Feng Qi
AU - Arjun Kumar Rathie
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2608.05193
UR - https://arxiv.org/abs/2608.05193
AB - Let ℤ^-={−1,−2,dotsc} and ℕ={1,2,dotsc}. In 2023, Qi and Lim gave two claims for summing the infinite series ∑_k=1^∞binom2kk1/(α+k)(±1/4)^k for α∈ℂ∖ℤ^-. In this paper, we confirm these claims by expressing the infinite series ∑_k=1^∞binom2kk1/(α+k)(z/4)^k and the finite series ∑_k=1^nbinom2kk1/(α+k)(z/4)^k in terms of the Gauss hypergeometric function _2F_1[1/2,α;1+α;z] and a generalized hypergeometric function _3F_2, for |z|≤1 and n∈ℕ. Using the Euler integral representation, we derive closed forms, in terms of finite sums of sines and cosines of arcsin√z, for _2F_1[1/2,1/2+n;3/2+n;z], _2F_1[1/2,1+n;2+n;z], two related _3F_2 functions, and the incomplete beta functions B_z(1/2+n,1/2) and B_z(1+n,1/2). By the Pfaff transformation, we obtain closed forms of several further Gauss hypergeometric functions with parameters shifted by n. We also give a closed form of [(1−z)d/dz(1−z)]^n(arcsin√z )/(√(z(1−z)) ) and a combinatorial identity.
ER -