Difference between revisions of "Ramanujan theta function"
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(Created page with "Let $|ab|<1$. The Ramanujan theta function $f$ is defined by $$f(a,b)=\displaystyle\sum_{k=-\infty}^{\infty} a^{\frac{n(n+1)}{2}} b^{\frac{n(n-1)}{2}}.$$") |
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| − | + | The Ramanujan theta function, $f$, is defined for $|ab|<1$ by | |
| − | $$f(a,b)=\displaystyle\sum_{k=-\infty}^{\infty} a^{\frac{ | + | $$f(a,b)=\displaystyle\sum_{k=-\infty}^{\infty} a^{\frac{k(k+1)}{2}} b^{\frac{k(k-1)}{2}}.$$ |
| + | |||
| + | =Properties= | ||
| + | [[RamanujanTheta(a,b)=(-a;ab)_inf (-b;ab)_inf (ab;ab)_inf]]<br /> | ||
| + | [[RamanujanTheta(q,q)=sum q^(k^2)]]<br /> | ||
| + | [[RamanujanTheta(q,q)=(-q;q^2)_inf^2 (q^2;q^2)_inf]]<br /> | ||
| + | |||
| + | =References= | ||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Latest revision as of 16:02, 10 July 2017
The Ramanujan theta function, $f$, is defined for $|ab|<1$ by $$f(a,b)=\displaystyle\sum_{k=-\infty}^{\infty} a^{\frac{k(k+1)}{2}} b^{\frac{k(k-1)}{2}}.$$
Properties
RamanujanTheta(a,b)=(-a;ab)_inf (-b;ab)_inf (ab;ab)_inf
RamanujanTheta(q,q)=sum q^(k^2)
RamanujanTheta(q,q)=(-q;q^2)_inf^2 (q^2;q^2)_inf
