A further q-analogue of Gosper's strange series
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908914158665728 |
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| author | Campbell, John M. Yamaguchi, Yuka |
| author_facet | Campbell, John M. Yamaguchi, Yuka |
| contents | Recently, the second author [Ramanujan J. 2026] introduced and proved a $q$-series identity that appears to provide the first
known $q$-analogue of an evaluation for a ${}_{2}F_{1}$-series known as \emph{Gosper's strange series}. Yamaguchi's derivation of this
$q$-analogue relies on three-term relations for ${}_{2}ϕ_{1}$-series along with Heine's transformation of ${}_{2}ϕ_{1}$-series. In this
note, we introduce and prove, using a $q$-analogue of a
series evaluation technique relying on an Abel-type summation lemma, a further $q$-analogue of Gosper's ${}_{2}F_{1}$-identity that is
inequivalent to Yamaguchi's $q$-analogue, and we also apply this technique to construct an alternative and simplified proof of
Yamaguchi's $q$-analogue, together with a ${}_{3}ϕ_{2}$-series variant of Heine's $q$-analogue
of Gauss's hypergeometric formula, a ${}_{6}ϕ_{5}$-series variant and two ${}_{4}ϕ_{3}$-series variants
of the $q$-analogue of Kummer's identity due to Bailey and Daum,
along with
a $q$-analogue of a result obtained by Cantarini
[Ramanujan J. 2022] via Fourier-Legendre theory and related to Ramanujan's first series for $\frac{1}π$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_24845 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A further q-analogue of Gosper's strange series Campbell, John M. Yamaguchi, Yuka Classical Analysis and ODEs 33D15 Recently, the second author [Ramanujan J. 2026] introduced and proved a $q$-series identity that appears to provide the first known $q$-analogue of an evaluation for a ${}_{2}F_{1}$-series known as \emph{Gosper's strange series}. Yamaguchi's derivation of this $q$-analogue relies on three-term relations for ${}_{2}ϕ_{1}$-series along with Heine's transformation of ${}_{2}ϕ_{1}$-series. In this note, we introduce and prove, using a $q$-analogue of a series evaluation technique relying on an Abel-type summation lemma, a further $q$-analogue of Gosper's ${}_{2}F_{1}$-identity that is inequivalent to Yamaguchi's $q$-analogue, and we also apply this technique to construct an alternative and simplified proof of Yamaguchi's $q$-analogue, together with a ${}_{3}ϕ_{2}$-series variant of Heine's $q$-analogue of Gauss's hypergeometric formula, a ${}_{6}ϕ_{5}$-series variant and two ${}_{4}ϕ_{3}$-series variants of the $q$-analogue of Kummer's identity due to Bailey and Daum, along with a $q$-analogue of a result obtained by Cantarini [Ramanujan J. 2022] via Fourier-Legendre theory and related to Ramanujan's first series for $\frac{1}π$. |
| title | A further q-analogue of Gosper's strange series |
| topic | Classical Analysis and ODEs 33D15 |
| url | https://arxiv.org/abs/2603.24845 |