Ramanujans thetafunktion definieras som
f ( a , b ) = ∑ n = − ∞ ∞ a n ( n + 1 ) / 2 b n ( n − 1 ) / 2 {\displaystyle f(a,b)=\sum _{n=-\infty }^{\infty }a^{n(n+1)/2}\;b^{n(n-1)/2}} då |ab | < 1. Jacobis trippelprodukt tar då formen
f ( a , b ) = f ( b , a ) = ( − a ; a b ) ∞ ( − b ; a b ) ∞ ( a b ; a b ) ∞ {\displaystyle f(a,b)=f(b,a)=(-a;ab)_{\infty }\;(-b;ab)_{\infty }\;(ab;ab)_{\infty }} där ( a ; q ) n {\displaystyle (a;q)_{n}} är q-Pochhammersymbolen . Tre viktiga specialfall av Ramanujans thetafunktion är
f ( q , q ) = ∑ n = − ∞ ∞ q n 2 = ( − q ; q 2 ) ∞ 2 ( q 2 ; q 2 ) ∞ {\displaystyle f(q,q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}={(-q;q^{2})_{\infty }^{2}(q^{2};q^{2})_{\infty }}} och
f ( q , q 3 ) = ∑ n = 0 ∞ q n ( n + 1 ) / 2 = ( q 2 ; q 2 ) ∞ ( − q ; q ) ∞ {\displaystyle f(q,q^{3})=\sum _{n=0}^{\infty }q^{n(n+1)/2}={(q^{2};q^{2})_{\infty }}{(-q;q)_{\infty }}} och
f ( − q ) := f ( − q , − q 2 ) = ∑ n = − ∞ ∞ ( − 1 ) n q n ( 3 n − 1 ) / 2 = ( q ; q ) ∞ {\displaystyle f(-q):=f(-q,-q^{2})=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{n(3n-1)/2}=(q;q)_{\infty }} Speciella värden
redigera
φ ( e − π ) = π 4 Γ ( 3 4 ) {\displaystyle \varphi \left(e^{-\pi }\right)={\frac {\sqrt[{4}]{\pi }}{\Gamma ({\frac {3}{4}})}}} φ ( e − 2 π ) = 6 π + 4 2 π 4 2 Γ ( 3 4 ) {\displaystyle \varphi \left(e^{-2\pi }\right)={\frac {\sqrt[{4}]{6\pi +4{\sqrt {2}}\pi }}{2\Gamma ({\frac {3}{4}})}}} φ ( e − 3 π ) = 27 π + 18 3 π 4 3 Γ ( 3 4 ) {\displaystyle \varphi \left(e^{-3\pi }\right)={\frac {\sqrt[{4}]{27\pi +18{\sqrt {3}}\pi }}{3\Gamma ({\frac {3}{4}})}}} φ ( e − 4 π ) = 8 π 4 + 2 π 4 4 Γ ( 3 4 ) {\displaystyle \varphi \left(e^{-4\pi }\right)={\frac {{\sqrt[{4}]{8\pi }}+2{\sqrt[{4}]{\pi }}}{4\Gamma ({\frac {3}{4}})}}} φ ( e − 5 π ) = 225 π + 100 5 π 4 5 Γ ( 3 4 ) {\displaystyle \varphi \left(e^{-5\pi }\right)={\frac {\sqrt[{4}]{225\pi +100{\sqrt {5}}\pi }}{5\Gamma ({\frac {3}{4}})}}} φ ( e − 6 π ) = 3 2 + 3 3 4 + 2 3 − 27 4 + 1728 4 − 4 3 ⋅ 243 π 2 8 6 1 + 6 − 2 − 3 6 Γ ( 3 4 ) {\displaystyle \varphi \left(e^{-6\pi }\right)={\frac {{\sqrt[{3}]{3{\sqrt {2}}+3{\sqrt[{4}]{3}}+2{\sqrt {3}}-{\sqrt[{4}]{27}}+{\sqrt[{4}]{1728}}-4}}\cdot {\sqrt[{8}]{243{\pi }^{2}}}}{6{\sqrt[{6}]{1+{\sqrt {6}}-{\sqrt {2}}-{\sqrt {3}}}}{\Gamma ({\frac {3}{4}})}}}}
W.N. Bailey, Generalized Hypergeometric Series , (1935) Cambridge Tracts in Mathematics and Mathematical Physics, No.32, Cambridge University Press, Cambridge.
George Gasper and Mizan Rahman, Basic Hypergeometric Series, 2nd Edition , (2004), Encyclopedia of Mathematics and Its Applications, 96 , Cambridge University Press, Cambridge. ISBN 0-521-83357-4 .
Hazewinkel, Michiel, red. (2001), ”Ramanujan function” , Encyclopedia of Mathematics , Springer, ISBN 978-1556080104
Weisstein, Eric W. , "Ramanujan Theta Functions ", MathWorld . (engelska)