Basic hypergeometric series (original) (raw)
数学において、q超幾何級数（qちょうきかきゅうすう、英: q-hypergeometric series, basic hypergeometric series）は、超幾何級数のq類似である。q超幾何級数は の形式で表される級数である。中でも が多く研究されている。但し、 であり、ここで はqポッホハマー記号である。なお、厳密にいうと、右辺の級数がq超幾何級数であり、左辺の記号は級数の和によって定義されるq超幾何関数を表すものである。
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| dbo:abstract | In mathematics, basic hypergeometric series, or q-hypergeometric series, are q-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series xn is called hypergeometric if the ratio of successive terms xn+1/xn is a rational function of n. If the ratio of successive terms is a rational function of qn, then the series is called a basic hypergeometric series. The number q is called the base. The basic hypergeometric series was first considered by Eduard Heine. It becomes the hypergeometric series in the limit when base . (en) En mathématiques, les séries hypergéométriques basiques de Heine, ou q-séries hypergéométriques, sont des généralisations q-analogues des , à leur tour étendues par les .Une série xn est appelée hypergéométrique si le rapport de deux termes successifs xn+1/xn est une fraction rationnelle de n. Si le rapport de deux termes successifs de est une fraction rationnelle en qn, alors la série est dite hypergéométrique basique, et le nombre q est appelé base. La série hypergéométriques basique 2ϕ1(qα,qβ;qγ;q,x) a d'abord été introduite par . On retrouve la série hypergéométrique F(α,β;γ;x) à la limite si la base q vaut 1. (fr) In matematica, le q-serie ipergeometriche, chiamate anche serie ipergeometriche basiche, sono generalizzazioni delle serie ipergeometriche ordinarie. Si definiscono comunemente due tipi di q-serie, le q-serie ipergeometriche unilaterali e le q-serie ipergeometriche bilaterali. La terminologia viene stabilita in analogia con quella delle serie ipergeometriche ordinarie. Una serie ordinaria viene detta serie ipergeometrica (ordinaria) se il rapporto fra termini successivi è una funzione razionale di n. Se invece il rapporto fra termini successivi è una funzione razionale di , la serie corrispondente viene detta q-serie ipergeometrica. Le q-serie ipergeometriche sono state analizzate per la prima volta da Eduard Heine nel XIX secolo, al fine di individuare caratteristiche comuni alle di Jacobi e alle funzioni ellittiche. (it) 数学において、q超幾何級数（qちょうきかきゅうすう、英: q-hypergeometric series, basic hypergeometric series）は、超幾何級数のq類似である。q超幾何級数は の形式で表される級数である。中でも が多く研究されている。但し、 であり、ここで はqポッホハマー記号である。なお、厳密にいうと、右辺の級数がq超幾何級数であり、左辺の記号は級数の和によって定義されるq超幾何関数を表すものである。 (ja) 基本超几何函数是广义超几何函数的q模拟。 (zh) |
| dbo:wikiPageExternalLink | http://fa.its.tudelft.nl/~koekoek/askey/ https://web.archive.org/web/20050410204356/http:/www.labri.fr/Perso/~lovejoy/1psi1.pdf https://web.archive.org/web/20050530142121/http:/cfc.nankai.edu.cn/publications/04-accepted/Chen-Fu-04A/semi.pdf http://mathworld.wolfram.com/q-HypergeometricFunction.html https://www.ams.org/bookstore%3Ffn=20&arg1=survseries&ikey=SURV-27 http://resolver.sub.uni-goettingen.de/purl%3FGDZPPN002145391 |
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| dbp:authorlink | Eduard Heine (en) |
| dbp:first | Eduard (en) G. E. (en) |
| dbp:id | 17 (xsd:integer) |
| dbp:last | Andrews (en) Heine (en) |
| dbp:title | q-Hypergeometric and Related Functions (en) |
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| rdfs:comment | 数学において、q超幾何級数（qちょうきかきゅうすう、英: q-hypergeometric series, basic hypergeometric series）は、超幾何級数のq類似である。q超幾何級数は の形式で表される級数である。中でも が多く研究されている。但し、 であり、ここで はqポッホハマー記号である。なお、厳密にいうと、右辺の級数がq超幾何級数であり、左辺の記号は級数の和によって定義されるq超幾何関数を表すものである。 (ja) 基本超几何函数是广义超几何函数的q模拟。 (zh) In mathematics, basic hypergeometric series, or q-hypergeometric series, are q-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series xn is called hypergeometric if the ratio of successive terms xn+1/xn is a rational function of n. If the ratio of successive terms is a rational function of qn, then the series is called a basic hypergeometric series. The number q is called the base. (en) In matematica, le q-serie ipergeometriche, chiamate anche serie ipergeometriche basiche, sono generalizzazioni delle serie ipergeometriche ordinarie. Si definiscono comunemente due tipi di q-serie, le q-serie ipergeometriche unilaterali e le q-serie ipergeometriche bilaterali. Le q-serie ipergeometriche sono state analizzate per la prima volta da Eduard Heine nel XIX secolo, al fine di individuare caratteristiche comuni alle di Jacobi e alle funzioni ellittiche. (it) En mathématiques, les séries hypergéométriques basiques de Heine, ou q-séries hypergéométriques, sont des généralisations q-analogues des , à leur tour étendues par les .Une série xn est appelée hypergéométrique si le rapport de deux termes successifs xn+1/xn est une fraction rationnelle de n. Si le rapport de deux termes successifs de est une fraction rationnelle en qn, alors la série est dite hypergéométrique basique, et le nombre q est appelé base. (fr) |
| rdfs:label | Basic hypergeometric series (en) Série hypergéométrique basique (fr) Q-serie ipergeometrica (it) Q超幾何級数 (ja) 基本超几何函数 (zh) |
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