În matematică , prin serie hipergeometrică fundamentală , câteodată numită și q-serie hipergeometrică , se înțelege generalizarea q-seriilor analoage a seriei hipergeometrice ordinare. În mod uzual sunt definite două serii fundamentale: seria hipergeometrică fundamentală unilaterală și seria hipergeometrică fundamentală bilaterală .
Numele i-a fost dat prin analogie cu seria hipergeometrică ordinară. O serie ordinară
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Serie hipergeometrică fundamentală a fost luată în considerație pentru prima dată de Eduard Heine în secolul XIX, ca un mod de a capta caracteristicile comune ale funcției theta a lui Jacobi si ale funcției eliptice .
Seria hipergeometrică fundamentală unilaterală este definită ca:
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{\displaystyle \;_{j}\phi _{k}\left[{\begin{matrix}a_{1}&a_{2}&\ldots &a_{j}\\b_{1}&b_{2}&\ldots &b_{k}\end{matrix}};q,z\right]=\sum _{n=0}^{\infty }{\frac {(a_{1},a_{2},\ldots ,a_{j};q)_{n}}{(b_{1},b_{2},\ldots ,b_{k},q;q)_{n}}}\left((-1)^{n}q^{n \choose 2}\right)^{1+k-j}z^{n}}
unde
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{\displaystyle (a_{1},a_{2},\ldots ,a_{m};q)_{n}=(a_{1};q)_{n}(a_{2};q)_{n}\ldots (a_{m};q)_{n}}
este permutarea q-factorială . Cazul cel mai important se obține atunci când j = k+1
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{\displaystyle \;_{k+1}\phi _{k}\left[{\begin{matrix}a_{1}&a_{2}&\ldots &a_{k+1}\\b_{1}&b_{2}&\ldots &b_{k}\end{matrix}};q,z\right]=\sum _{n=0}^{\infty }{\frac {(a_{1},a_{2},\ldots ,a_{k+1};q)_{n}}{(b_{1},b_{2},\ldots ,b_{k},q;q)_{n}}}z^{n}.}
Seria hipergeometrică fundamentală bilaterală corespounde seriei hipergeometrice bilaterale și este definită ca:
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{\displaystyle \;_{j}\psi _{k}\left[{\begin{matrix}a_{1}&a_{2}&\ldots &a_{j}\\b_{1}&b_{2}&\ldots &b_{k}\end{matrix}};q,z\right]=\sum _{n=-\infty }^{\infty }{\frac {(a_{1},a_{2},\ldots ,a_{j};q)_{n}}{(b_{1},b_{2},\ldots ,b_{k};q)_{n}}}\left((-1)^{n}q^{n \choose 2}\right)^{k-j}z^{n}.}
Cazul cel mai important se obține atunci când j = k
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{\displaystyle \;_{k}\psi _{k}\left[{\begin{matrix}a_{1}&a_{2}&\ldots &a_{k}\\b_{1}&b_{2}&\ldots &b_{k}\end{matrix}};q,z\right]=\sum _{n=-\infty }^{\infty }{\frac {(a_{1},a_{2},\ldots ,a_{k};q)_{n}}{(b_{1},b_{2},\ldots ,b_{k};q)_{n}}}z^{n}.}
Seria unilaterală poate fi obținută ca un caz special al celei bilaterale facând variabila b q a q n < 0
Expresiile câtorva serii simple includ:
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{\displaystyle {\frac {z}{1-q}}\;_{2}\phi _{1}\left[{\begin{matrix}q\;q\\q^{2}\end{matrix}}\;;q,z\right]={\frac {z}{1-q}}+{\frac {z^{2}}{1-q^{2}}}+{\frac {z^{3}}{1-q^{3}}}+\ldots }
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{\displaystyle {\frac {z}{1-q^{1/2}}}\;_{2}\phi _{1}\left[{\begin{matrix}q\;q^{1/2}\\q^{3/2}\end{matrix}}\;;q,z\right]={\frac {z}{1-q^{1/2}}}+{\frac {z^{2}}{1-q^{3/2}}}+{\frac {z^{3}}{1-q^{5/2}}}+\ldots }
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{\displaystyle \;_{2}\phi _{1}\left[{\begin{matrix}q\;-1\\-q\end{matrix}}\;;q,z\right]=1+{\frac {2z}{1+q}}+{\frac {2z^{2}}{1+q^{2}}}+{\frac {2z^{3}}{1+q^{3}}}+\ldots .}
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{\displaystyle \;_{1}\phi _{0}(a;q,z)=\prod _{n=0}^{\infty }{\frac {1-aq^{n}z}{1-q^{n}z}}}
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{\displaystyle \;_{1}\phi _{0}(a;q,z)={\frac {1-az}{1-z}}\;_{1}\phi _{0}(a;q,qz).}
Cazul special
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Ramanujan a dat următoarea identitate:
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{\displaystyle \;_{1}\psi _{1}\left[{\begin{matrix}a\\b\end{matrix}};q,z\right]=\sum _{n=-\infty }^{\infty }{\frac {(a;q)_{n}}{(b;q)_{n}}}={\frac {(b/a;q)_{\infty }\;(q;q)_{\infty }\;(q/az;q)_{\infty }\;(az;q)_{\infty }}{(b;q)_{\infty }\;(b/az;q)_{\infty }\;(q/a;q)_{\infty }\;(z;q)_{\infty }}}}
valabilă pentru
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{\displaystyle \sum _{n=-\infty }^{\infty }q^{n(n+1)/2}z^{n}=(q;q)_{\infty }\;(-1/z;q)_{\infty }\;(-zq;q)_{\infty }.}
Ken Ono a dat următoarea serie de puteri formală :
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{\displaystyle A(z;q){\stackrel {\rm {def}}{=}}{\frac {1}{1+z}}\sum _{n=0}^{\infty }{\frac {(z;q)_{n}}{(-zq;q)_{n}}}z^{n}=\sum _{n=0}^{\infty }(-1)^{n}z^{2n}q^{n^{2}}.}
Eduard Heine , Theorie der Kugelfunctionen , (1878) 1 , pp 97-125.Eduard Heine, Handbuch die Kugelfunctionen. Theorie und Anwendung (1898) Springer, Berlin.
W.N. Bailey, Generalized Hypergeometric Series , (1935) Cambridge Tracts in Mathematics and Mathematical Physics, No.32, Cambridge University Press, Cambridge.
Gasper, George; Rahman, Mizan (2004 ), Basic hypergeometric series , Encyclopedia of Mathematics and its Applications, 96 (ed. 2nd), Cambridge University Press , ISBN 978-0-521-83357-8 , MR 2128719 William Y. C. Chen and Amy Fu, Semi-Finite Forms of Bilateral Basic Hypergeometric Series Arhivat în 30 mai 2005 , la Wayback Machine .
Sylvie Corteel and Jeremy Lovejoy, Frobenius Partitions and the Combinatorics of Ramanujan's
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Gwynneth H. Coogan and Ken Ono , A q-series identity and the Arithmetic of Hurwitz Zeta Functions Arhivat în 29 august 2006 , la Wayback Machine .American Mathematical Society 131 , pp. 719-724 
![{\displaystyle \;_{j}\phi _{k}\left[{\begin{matrix}a_{1}&a_{2}&\ldots &a_{j}\\b_{1}&b_{2}&\ldots &b_{k}\end{matrix}};q,z\right]=\sum _{n=0}^{\infty }{\frac {(a_{1},a_{2},\ldots ,a_{j};q)_{n}}{(b_{1},b_{2},\ldots ,b_{k},q;q)_{n}}}\left((-1)^{n}q^{n \choose 2}\right)^{1+k-j}z^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/274d2ef79218e289c26f62120cb6dcfcc6636248)
![{\displaystyle \;_{k+1}\phi _{k}\left[{\begin{matrix}a_{1}&a_{2}&\ldots &a_{k+1}\\b_{1}&b_{2}&\ldots &b_{k}\end{matrix}};q,z\right]=\sum _{n=0}^{\infty }{\frac {(a_{1},a_{2},\ldots ,a_{k+1};q)_{n}}{(b_{1},b_{2},\ldots ,b_{k},q;q)_{n}}}z^{n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1afeea0f107b1f1d1975b1a31fd6683ba6d67ba0)
![{\displaystyle \;_{j}\psi _{k}\left[{\begin{matrix}a_{1}&a_{2}&\ldots &a_{j}\\b_{1}&b_{2}&\ldots &b_{k}\end{matrix}};q,z\right]=\sum _{n=-\infty }^{\infty }{\frac {(a_{1},a_{2},\ldots ,a_{j};q)_{n}}{(b_{1},b_{2},\ldots ,b_{k};q)_{n}}}\left((-1)^{n}q^{n \choose 2}\right)^{k-j}z^{n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/501e00fbffa51da9bab6469656fbf853df7d794c)
![{\displaystyle \;_{k}\psi _{k}\left[{\begin{matrix}a_{1}&a_{2}&\ldots &a_{k}\\b_{1}&b_{2}&\ldots &b_{k}\end{matrix}};q,z\right]=\sum _{n=-\infty }^{\infty }{\frac {(a_{1},a_{2},\ldots ,a_{k};q)_{n}}{(b_{1},b_{2},\ldots ,b_{k};q)_{n}}}z^{n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba22c851d35068e0fdf285feb4ba9a3d68f066fd)
![{\displaystyle {\frac {z}{1-q}}\;_{2}\phi _{1}\left[{\begin{matrix}q\;q\\q^{2}\end{matrix}}\;;q,z\right]={\frac {z}{1-q}}+{\frac {z^{2}}{1-q^{2}}}+{\frac {z^{3}}{1-q^{3}}}+\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ade77e25842ebcf32016af8a9b42c32aeaf9fefb)
![{\displaystyle {\frac {z}{1-q^{1/2}}}\;_{2}\phi _{1}\left[{\begin{matrix}q\;q^{1/2}\\q^{3/2}\end{matrix}}\;;q,z\right]={\frac {z}{1-q^{1/2}}}+{\frac {z^{2}}{1-q^{3/2}}}+{\frac {z^{3}}{1-q^{5/2}}}+\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/015a6d86d3f197d4b4bd71f2d416b4af29b20b24)
![{\displaystyle \;_{2}\phi _{1}\left[{\begin{matrix}q\;-1\\-q\end{matrix}}\;;q,z\right]=1+{\frac {2z}{1+q}}+{\frac {2z^{2}}{1+q^{2}}}+{\frac {2z^{3}}{1+q^{3}}}+\ldots .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f2d19a913425e0c8f743c86adc581dc53295c7b)
![{\displaystyle \;_{1}\psi _{1}\left[{\begin{matrix}a\\b\end{matrix}};q,z\right]=\sum _{n=-\infty }^{\infty }{\frac {(a;q)_{n}}{(b;q)_{n}}}={\frac {(b/a;q)_{\infty }\;(q;q)_{\infty }\;(q/az;q)_{\infty }\;(az;q)_{\infty }}{(b;q)_{\infty }\;(b/az;q)_{\infty }\;(q/a;q)_{\infty }\;(z;q)_{\infty }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22e002c5f142c5f5ccd5e4aca799c6fadfaac7e0)
