Difference between revisions of "Ramanujan theta function"

From specialfunctionswiki
Jump to: navigation, search
 
(One intermediate revision by the same user not shown)
Line 1: Line 1:
Let $|ab|<1$. The Ramanujan theta function $f$ is defined by
+
The Ramanujan theta function, $f$, is defined for $|ab|<1$ by
$$f(a,b)=\displaystyle\sum_{k=-\infty}^{\infty} a^{\frac{n(n+1)}{2}} b^{\frac{n(n-1)}{2}}.$$
+
$$f(a,b)=\displaystyle\sum_{k=-\infty}^{\infty} a^{\frac{k(k+1)}{2}} b^{\frac{k(k-1)}{2}}.$$
 +
 
 +
=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]]
 
[[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

References