Ramanujan theta function

Mathematical function

In mathematics, particularly q-analog theory, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties. In particular, the Jacobi triple product takes on a particularly elegant form when written in terms of the Ramanujan theta. The function is named after mathematician Srinivasa Ramanujan.

Definition

The Ramanujan theta function is defined as

f ( a , b ) = ∑ n = − ∞ ∞ a n ( n + 1 ) 2 b n ( n − 1 ) 2 {\displaystyle f(a,b)=\sum _{n=-\infty }^{\infty }a^{\frac {n(n+1)}{2}}\;b^{\frac {n(n-1)}{2}}}

for |ab| < 1. The Jacobi triple product identity then takes the form

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

Here, the expression ( a ; q ) n {\displaystyle (a;q)_{n}} denotes the q-Pochhammer symbol. Identities that follow from this include

φ ( q ) = f ( q , q ) = ∑ n = − ∞ ∞ q n 2 = ( − q ; q 2 ) ∞ 2 ( q 2 ; q 2 ) ∞ {\displaystyle \varphi (q)=f(q,q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}={\left(-q;q^{2}\right)_{\infty }^{2}\left(q^{2};q^{2}\right)_{\infty }}}

and

ψ ( q ) = f ( q , q 3 ) = ∑ n = 0 ∞ q n ( n + 1 ) 2 = ( q 2 ; q 2 ) ∞ ( − q ; q ) ∞ {\displaystyle \psi (q)=f\left(q,q^{3}\right)=\sum _{n=0}^{\infty }q^{\frac {n(n+1)}{2}}={\left(q^{2};q^{2}\right)_{\infty }}{(-q;q)_{\infty }}}

and

f ( − q ) = f ( − q , − q 2 ) = ∑ n = − ∞ ∞ ( − 1 ) n q n ( 3 n − 1 ) 2 = ( q ; q ) ∞ {\displaystyle f(-q)=f\left(-q,-q^{2}\right)=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{\frac {n(3n-1)}{2}}=(q;q)_{\infty }}

This last being the Euler function, which is closely related to the Dedekind eta function. The Jacobi theta function may be written in terms of the Ramanujan theta function as:

ϑ ( w , q ) = f ( q w 2 , q w − 2 ) {\displaystyle \vartheta (w,q)=f\left(qw^{2},qw^{-2}\right)}

Integral representations

We have the following integral representation for the full two-parameter form of Ramanujan's theta function:[1]

f ( a , b ) = 1 + ∫ 0 ∞ 2 a e − 1 2 t 2 2 π [ 1 − a a b cosh ⁡ ( log ⁡ a b t ) a 3 b − 2 a a b cosh ⁡ ( log ⁡ a b t ) + 1 ] d t + ∫ 0 ∞ 2 b e − 1 2 t 2 2 π [ 1 − b a b cosh ⁡ ( log ⁡ a b t ) a b 3 − 2 b a b cosh ⁡ ( log ⁡ a b t ) + 1 ] d t {\displaystyle f(a,b)=1+\int _{0}^{\infty }{\frac {2ae^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {1-a{\sqrt {ab}}\cosh \left({\sqrt {\log ab}}\,t\right)}{a^{3}b-2a{\sqrt {ab}}\cosh \left({\sqrt {\log ab}}\,t\right)+1}}\right]dt+\int _{0}^{\infty }{\frac {2be^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {1-b{\sqrt {ab}}\cosh \left({\sqrt {\log ab}}\,t\right)}{ab^{3}-2b{\sqrt {ab}}\cosh \left({\sqrt {\log ab}}\,t\right)+1}}\right]dt}

The special cases of Ramanujan's theta functions given by φ(q) := f(q, q) OEIS: A000122 and ψ(q) := f(q, q3) OEIS: A010054 [2] also have the following integral representations:[1]

φ ( q ) = 1 + ∫ 0 ∞ e − 1 2 t 2 2 π [ 4 q ( 1 − q 2 cosh ⁡ ( 2 log ⁡ q t ) ) q 4 − 2 q 2 cosh ⁡ ( 2 log ⁡ q t ) + 1 ] d t ψ ( q ) = ∫ 0 ∞ 2 e − 1 2 t 2 2 π [ 1 − q cosh ⁡ ( log ⁡ q t ) q − 2 q cosh ⁡ ( log ⁡ q t ) + 1 ] d t {\displaystyle {\begin{aligned}\varphi (q)&=1+\int _{0}^{\infty }{\frac {e^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {4q\left(1-q^{2}\cosh \left({\sqrt {2\log q}}\,t\right)\right)}{q^{4}-2q^{2}\cosh \left({\sqrt {2\log q}}\,t\right)+1}}\right]dt\\[6pt]\psi (q)&=\int _{0}^{\infty }{\frac {2e^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {1-{\sqrt {q}}\cosh \left({\sqrt {\log q}}\,t\right)}{q-2{\sqrt {q}}\cosh \left({\sqrt {\log q}}\,t\right)+1}}\right]dt\end{aligned}}}

This leads to several special case integrals for constants defined by these functions when q := e−kπ (cf. theta function explicit values). In particular, we have that [1]

φ ( e − k π ) = 1 + ∫ 0 ∞ e − 1 2 t 2 2 π [ 4 e k π ( e 2 k π − cos ⁡ ( 2 π k t ) ) e 4 k π − 2 e 2 k π cos ⁡ ( 2 π k t ) + 1 ] d t π 1 4 Γ ( 3 4 ) = 1 + ∫ 0 ∞ e − 1 2 t 2 2 π [ 4 e π ( e 2 π − cos ⁡ ( 2 π t ) ) e 4 π − 2 e 2 π cos ⁡ ( 2 π t ) + 1 ] d t π 1 4 Γ ( 3 4 ) ⋅ 2 + 2 2 = 1 + ∫ 0 ∞ e − 1 2 t 2 2 π [ 4 e 2 π ( e 4 π − cos ⁡ ( 2 π t ) ) e 8 π − 2 e 4 π cos ⁡ ( 2 π t ) + 1 ] d t π 1 4 Γ ( 3 4 ) ⋅ 1 + 3 2 1 4 3 3 8 = 1 + ∫ 0 ∞ e − 1 2 t 2 2 π [ 4 e 3 π ( e 6 π − cos ⁡ ( 6 π t ) ) e 12 π − 2 e 6 π cos ⁡ ( 6 π t ) + 1 ] d t π 1 4 Γ ( 3 4 ) ⋅ 5 + 2 5 5 3 4 = 1 + ∫ 0 ∞ e − 1 2 t 2 2 π [ 4 e 5 π ( e 10 π − cos ⁡ ( 10 π t ) ) e 20 π − 2 e 10 π cos ⁡ ( 10 π t ) + 1 ] d t {\displaystyle {\begin{aligned}\varphi \left(e^{-k\pi }\right)&=1+\int _{0}^{\infty }{\frac {e^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {4e^{k\pi }\left(e^{2k\pi }-\cos \left({\sqrt {2\pi k}}\,t\right)\right)}{e^{4k\pi }-2e^{2k\pi }\cos \left({\sqrt {2\pi k}}\,t\right)+1}}\right]dt\\[6pt]{\frac {\pi ^{\frac {1}{4}}}{\Gamma \left({\frac {3}{4}}\right)}}&=1+\int _{0}^{\infty }{\frac {e^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {4e^{\pi }\left(e^{2\pi }-\cos \left({\sqrt {2\pi }}\,t\right)\right)}{e^{4\pi }-2e^{2\pi }\cos \left({\sqrt {2\pi }}\,t\right)+1}}\right]dt\\[6pt]{\frac {\pi ^{\frac {1}{4}}}{\Gamma \left({\frac {3}{4}}\right)}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}&=1+\int _{0}^{\infty }{\frac {e^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {4e^{2\pi }\left(e^{4\pi }-\cos \left(2{\sqrt {\pi }}\,t\right)\right)}{e^{8\pi }-2e^{4\pi }\cos \left(2{\sqrt {\pi }}\,t\right)+1}}\right]dt\\[6pt]{\frac {\pi ^{\frac {1}{4}}}{\Gamma \left({\frac {3}{4}}\right)}}\cdot {\frac {\sqrt {1+{\sqrt {3}}}}{2^{\frac {1}{4}}3^{\frac {3}{8}}}}&=1+\int _{0}^{\infty }{\frac {e^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {4e^{3\pi }\left(e^{6\pi }-\cos \left({\sqrt {6\pi }}\,t\right)\right)}{e^{12\pi }-2e^{6\pi }\cos \left({\sqrt {6\pi }}\,t\right)+1}}\right]dt\\[6pt]{\frac {\pi ^{\frac {1}{4}}}{\Gamma \left({\frac {3}{4}}\right)}}\cdot {\frac {\sqrt {5+2{\sqrt {5}}}}{5^{\frac {3}{4}}}}&=1+\int _{0}^{\infty }{\frac {e^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {4e^{5\pi }\left(e^{10\pi }-\cos \left({\sqrt {10\pi }}\,t\right)\right)}{e^{20\pi }-2e^{10\pi }\cos \left({\sqrt {10\pi }}\,t\right)+1}}\right]dt\end{aligned}}}

and that

ψ ( e − k π ) = ∫ 0 ∞ e − 1 2 t 2 2 π [ cos ⁡ ( k π t ) − e k π 2 cos ⁡ ( k π t ) − cosh ⁡ k π 2 ] d t π 1 4 Γ ( 3 4 ) ⋅ e π 8 2 5 8 = ∫ 0 ∞ e − 1 2 t 2 2 π [ cos ⁡ ( π t ) − e π 2 cos ⁡ ( π t ) − cosh ⁡ π 2 ] d t π 1 4 Γ ( 3 4 ) ⋅ e π 4 2 5 4 = ∫ 0 ∞ e − 1 2 t 2 2 π [ cos ⁡ ( 2 π t ) − e π cos ⁡ ( 2 π t ) − cosh ⁡ π ] d t π 1 4 Γ ( 3 4 ) ⋅ 1 + 2 4 e π 16 2 7 16 = ∫ 0 ∞ e − 1 2 t 2 2 π [ cos ⁡ ( π 2 t ) − e π 4 cos ⁡ ( π 2 t ) − cosh ⁡ π 4 ] d t {\displaystyle {\begin{aligned}\psi \left(e^{-k\pi }\right)&=\int _{0}^{\infty }{\frac {e^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {\cos \left({\sqrt {k\pi }}\,t\right)-e^{\frac {k\pi }{2}}}{\cos \left({\sqrt {k\pi }}\,t\right)-\cosh {\frac {k\pi }{2}}}}\right]dt\\[6pt]{\frac {\pi ^{\frac {1}{4}}}{\Gamma \left({\frac {3}{4}}\right)}}\cdot {\frac {e^{\frac {\pi }{8}}}{2^{\frac {5}{8}}}}&=\int _{0}^{\infty }{\frac {e^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {\cos \left({\sqrt {\pi }}\,t\right)-e^{\frac {\pi }{2}}}{\cos \left({\sqrt {\pi }}\,t\right)-\cosh {\frac {\pi }{2}}}}\right]dt\\[6pt]{\frac {\pi ^{\frac {1}{4}}}{\Gamma \left({\frac {3}{4}}\right)}}\cdot {\frac {e^{\frac {\pi }{4}}}{2^{\frac {5}{4}}}}&=\int _{0}^{\infty }{\frac {e^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {\cos \left({\sqrt {2\pi }}\,t\right)-e^{\pi }}{\cos \left({\sqrt {2\pi }}\,t\right)-\cosh \pi }}\right]dt\\[6pt]{\frac {\pi ^{\frac {1}{4}}}{\Gamma \left({\frac {3}{4}}\right)}}\cdot {\frac {{\sqrt[{4}]{1+{\sqrt {2}}}}\,e^{\frac {\pi }{16}}}{2^{\frac {7}{16}}}}&=\int _{0}^{\infty }{\frac {e^{-{\frac {1}{2}}t^{2}}}{\sqrt {2\pi }}}\left[{\frac {\cos \left({\sqrt {\frac {\pi }{2}}}\,t\right)-e^{\frac {\pi }{4}}}{\cos \left({\sqrt {\frac {\pi }{2}}}\,t\right)-\cosh {\frac {\pi }{4}}}}\right]dt\end{aligned}}}

Application in string theory

The Ramanujan theta function is used to determine the critical dimensions in bosonic string theory, superstring theory and M-theory.

References

  1. ^ a b c Schmidt, M. D. (2017). "Square series generating function transformations" (PDF). Journal of Inequalities and Special Functions. 8 (2). arXiv:1609.02803.
  2. ^ Weisstein, Eric W. "Ramanujan Theta Functions". MathWorld. Retrieved 29 April 2018.
  • Bailey, W. N. (1935). Generalized Hypergeometric Series. Cambridge Tracts in Mathematics and Mathematical Physics. Vol. 32. Cambridge: Cambridge University Press.
  • Gasper, George; Rahman, Mizan (2004). Basic Hypergeometric Series. Encyclopedia of Mathematics and Its Applications. Vol. 96 (2nd ed.). Cambridge: Cambridge University Press. ISBN 0-521-83357-4.
  • "Ramanujan function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
  • Kaku, Michio (1994). Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension. Oxford: Oxford University Press. ISBN 0-19-286189-1.
  • Weisstein, Eric W. "Ramanujan Theta Functions". MathWorld.