Тета-функция Рамануджана

Тета-функция Рамануджана обобщает тета-функции Якоби, не разрушая основные их свойства. В частности, тройное произведение Якоби принимает особенно элегантный вид, когда записывается в терминах тета-функции Рамануджана. Функция носит имя Сриниваса Рамануджана Айенгора.

Определение

Тета-функция Рамануджана определяется как

f(a,b)=\sum _{n=-\infty }^{\infty }a^{n(n+1)/2}\;b^{n(n-1)/2}

для |ab| < 1. Тождество тройного произведения Якоби тогда принимает вид

f(a,b)=(-a;ab)_{\infty }\;(-b;ab)_{\infty }\;(ab;ab)_{\infty }.

Здесь выражение $(a;q)_{n}$ означает q-символ Похгаммера. Тождества, вытекающие из этого

f(q,q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}={(-q;q^{2})_{\infty }^{2}(q^{2};q^{2})_{\infty }}

f(q,q^{3})=\sum _{n=0}^{\infty }q^{n(n+1)/2}={(q^{2};q^{2})_{\infty }}{(-q;q)_{\infty }}

f(-q,-q^{2})=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{n(3n-1)/2}=(q;q)_{\infty }

Последнее тождество является функцией Эйлера^[англ.], которая тесно связана с эта-функцией Дедекинда^[англ.]. Тета-функция Якоби может быть записана в терминах тета-функции Рамануджана:

\vartheta (w,q)=f(qw^{2},qw^{-2})

Приложение в теории струн

Тета-функция Рамануджана используется для определения критических размерностей в теории бозонных струн, теории суперструн и М-теории.

Примечания

Литература

Bailey W. N. Generalized Hypergeometric Series. — Cambridge: Cambridge University Press, 1935. — Т. 32. — (Cambridge Tracts in Mathematics and Mathematical Physics).
Basic Hypergeometric Series. — 2nd. — Cambridge: Cambridge University Press, 2004. — Т. 96. — (Encyclopedia of Mathematics and Its Applications). — ISBN 0-521-83357-4.
Hazewinkel, Michiel, ed. (2001), "Ramanujan function", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Michio Kaku. Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension. — Oxford: Oxford University Press, 1994. — ISBN 0-19-286189-1.
Weisstein, Eric W. Ramanujan Theta Functions (англ.) на сайте Wolfram MathWorld.

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