On a $_2F_1\big(\frac{1}{4}\big)$-identity due to Gosper
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918430092820480 |
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| author | Hakimoglu-Brown, Cetin |
| author_facet | Hakimoglu-Brown, Cetin |
| contents | It is only in exceptional cases that a $_2F_1(z)$-series with rational parameters and a rational argument, apart from the cases for $z \in \{ \pm 1, \frac{1}{2} \}$ associated with classical hypergeometric identities, admits an evaluation given by a combination of $Γ$-values with rational arguments. In this paper, we present a new and integration-based approach toward the construction of special values for $_2F_1$-series of the desired form. We apply this approach using a $_2F_1\big(\frac{1}{4}\big)$-identity originally due to Gosper and later considered by Vidunas, Ebisu, and Zudilin, to evaluate a ${}_{2}F_{1}$-series of convergence rate $\big(\frac{172872}{185039}\big)^2$. With regard to extant research on so-called ``strange'' ${}_{2}F_{1}$-evaluations, as in the work of Ebisu and Zeilberger, our new series seems to have the largest numerator/denominator in its argument. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_04799 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On a $_2F_1\big(\frac{1}{4}\big)$-identity due to Gosper Hakimoglu-Brown, Cetin Classical Analysis and ODEs 33C05 It is only in exceptional cases that a $_2F_1(z)$-series with rational parameters and a rational argument, apart from the cases for $z \in \{ \pm 1, \frac{1}{2} \}$ associated with classical hypergeometric identities, admits an evaluation given by a combination of $Γ$-values with rational arguments. In this paper, we present a new and integration-based approach toward the construction of special values for $_2F_1$-series of the desired form. We apply this approach using a $_2F_1\big(\frac{1}{4}\big)$-identity originally due to Gosper and later considered by Vidunas, Ebisu, and Zudilin, to evaluate a ${}_{2}F_{1}$-series of convergence rate $\big(\frac{172872}{185039}\big)^2$. With regard to extant research on so-called ``strange'' ${}_{2}F_{1}$-evaluations, as in the work of Ebisu and Zeilberger, our new series seems to have the largest numerator/denominator in its argument. |
| title | On a $_2F_1\big(\frac{1}{4}\big)$-identity due to Gosper |
| topic | Classical Analysis and ODEs 33C05 |
| url | https://arxiv.org/abs/2604.04799 |