$q$-Analogs of $\pi$-related formulas from Jackson's $_8\phi_7$-series via the inversion approach
UDC 517.588, 517.52 By using the multiplicative form of the extended Carlitz inverse-series relations, we establish two general ``dual'' theorems on Jackson's summation formula for the well-poised $_8\phi_7$-series. Their duplicate forms under the partition pattern $n = \B...
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| Дата: | 2026 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
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Institute of Mathematics, NAS of Ukraine
2026
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/9405 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1866663722775216128 |
|---|---|
| author | Chen, Xiaojing Chu, Wenchang Chen, Xiaojing Chu, Wenchang |
| author_facet | Chen, Xiaojing Chu, Wenchang Chen, Xiaojing Chu, Wenchang |
| author_institution_txt_mv | [
{
"author": "Xiaojing Chen",
"institution": "School of Statistics and Data Science, Qufu Normal University, Qufu, China"
},
{
"author": "Wenchang Chu",
"institution": "Independent Researcher: Via Dalmazio Birago 9\/E, Lecce, Puglia, Italy"
}
] |
| author_sort | Chen, Xiaojing |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2026-05-30T12:43:59Z |
| description | UDC 517.588, 517.52
By using the multiplicative form of the extended Carlitz inverse-series relations, we establish two general ``dual'' theorems on Jackson's summation formula for the well-poised $_8\phi_7$-series. Their duplicate forms under the partition pattern $n = \Big\lfloor\dfrac{n}2 \Big\rfloor + \Big\lfloor\dfrac{n + 1}2 \Big\rfloor $ are explored and yield numerous $q$-series identities whose limiting cases as $q\to1$ result in the classical $\pi$-related Ramanujan-like series with convergence rate ``"$dfrac1{16}$", including the series for $1/\pi^2$ discovered by Guillera (2003). The triplicate dual formulas under the partition pattern $n = \Big\lfloor{\dfrac{n}3}\Big\rfloor + \Big\lfloor{\dfrac{n + 1}3} \Big\rfloor + \Big\lfloor{\dfrac{n + 2}3} \Big\rfloor $ are examined via the "reverse bisection method", which leads us to twenty new $q$-series identities together with their classical counterparts with the convergence rate "$\dfrac{-1}{27}$" as $q\to1.$ |
| doi_str_mv | 10.3842/umzh.v78i5-6.9405 |
| first_indexed | 2026-05-30T01:00:57Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-9405 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-05-31T01:00:49Z |
| publishDate | 2026 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | |
| spelling | umjimathkievua-article-94052026-05-30T12:43:59Z $q$-Analogs of $\pi$-related formulas from Jackson's $_8\phi_7$-series via the inversion approach $q$-Analogs of $\pi$-related formulas from Jackson's $_8\phi_7$-series via the inversion approach Chen, Xiaojing Chu, Wenchang Chen, Xiaojing Chu, Wenchang Basic hypergeometric series; Bisection series; Jackson’s formula for well–poised 8phi7-series Duplicate inversions Triplicate inversion Ramanujan–like pi-related series Guillera’s infinite series for 1/pi^2 33D15 33C20 65B10 UDC 517.588, 517.52 By using the multiplicative form of the extended Carlitz inverse-series relations, we establish two general ``dual'' theorems on Jackson's summation formula for the well-poised $_8\phi_7$-series. Their duplicate forms under the partition pattern $n = \Big\lfloor\dfrac{n}2 \Big\rfloor + \Big\lfloor\dfrac{n + 1}2 \Big\rfloor $ are explored and yield numerous $q$-series identities whose limiting cases as $q\to1$ result in the classical $\pi$-related Ramanujan-like series with convergence rate ``"$dfrac1{16}$", including the series for $1/\pi^2$ discovered by Guillera (2003). The triplicate dual formulas under the partition pattern $n = \Big\lfloor{\dfrac{n}3}\Big\rfloor + \Big\lfloor{\dfrac{n + 1}3} \Big\rfloor + \Big\lfloor{\dfrac{n + 2}3} \Big\rfloor $ are examined via the "reverse bisection method", which leads us to twenty new $q$-series identities together with their classical counterparts with the convergence rate "$\dfrac{-1}{27}$" as $q\to1.$ УДК 517.588, 517.52 $q$-Аналоги формул, що пов'язані з $\pi$ для $_8\phi_7$-рядів Джексона за допомогою методу обернення З використанням мультиплікативної форми розширених обернених рядів Карліца встановлено дві загальні ``подвійні'' теореми щодо формули сумування Джексона для добре врівноважених $_8\phi_7$-рядів. Досліджено їхні подвоєні форми за схемою розбиття $n = \Big\lfloor\dfrac{n}{2}\Big\rfloor + \Big\lfloor\dfrac{n+1}{2}\Big\rfloor,$ що призводить до численних тотожностей для $q$-рядів, граничні випадки яких при $q\to1$ відтворюють класичні ряди Рамануджана, пов'язані з $\pi,$ зі швидкістю збіжності "$\dfrac1{16}$", включаючи ряд для $1/\pi^2,$ відкритий Гільєрою (2003). Триплікатні подвійні формули за схемою розбиття $n = \Big\lfloor\dfrac{n}{3}\Big\rfloor + \Big\lfloor\dfrac{n+1}{3}\Big\rfloor + \Big\lfloor\dfrac{n+2}{3}\Big\rfloor$ досліджено за допомогою "методу зворотної бісекції", що дозволяє отримати двадцять нових тотожностей для $q$-рядів разом із їхніми класичними аналогами зі швидкістю збіжності "$\dfrac{-1}{27}$" при $q\to1.$ Institute of Mathematics, NAS of Ukraine 2026-05-29 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/9405 10.3842/umzh.v78i5-6.9405 Ukrains’kyi Matematychnyi Zhurnal; Vol. 78 No. 5-6 (2026); 364–365 Український математичний журнал; Том 78 № 5-6 (2026); 364–365 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/9405/10655 Copyright (c) 2026 Xiaojing Chen, Wenchang Chu |
| spellingShingle | Chen, Xiaojing Chu, Wenchang Chen, Xiaojing Chu, Wenchang $q$-Analogs of $\pi$-related formulas from Jackson's $_8\phi_7$-series via the inversion approach |
| title | $q$-Analogs of $\pi$-related formulas from Jackson's $_8\phi_7$-series via the inversion approach |
| title_alt | $q$-Analogs of $\pi$-related formulas from Jackson's $_8\phi_7$-series via the inversion approach |
| title_full | $q$-Analogs of $\pi$-related formulas from Jackson's $_8\phi_7$-series via the inversion approach |
| title_fullStr | $q$-Analogs of $\pi$-related formulas from Jackson's $_8\phi_7$-series via the inversion approach |
| title_full_unstemmed | $q$-Analogs of $\pi$-related formulas from Jackson's $_8\phi_7$-series via the inversion approach |
| title_short | $q$-Analogs of $\pi$-related formulas from Jackson's $_8\phi_7$-series via the inversion approach |
| title_sort | $q$-analogs of $\pi$-related formulas from jackson's $_8\phi_7$-series via the inversion approach |
| topic_facet | Basic hypergeometric series; Bisection series Jackson’s formula for well–poised 8phi7-series Duplicate inversions Triplicate inversion Ramanujan–like pi-related series Guillera’s infinite series for 1/pi^2 33D15 33C20 65B10 |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/9405 |
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