Función theta de Ramanujan

En matemáticas, particularmente na teoría de q-análogos, a función theta de Ramanujan xeneraliza a forma das funcións theta de Jacobi, ao tempo que captura as súas propiedades xerais. En particular, o produto triplo Jacobi adquire unha forma particularmente elegante cando se escribe en termos do theta Ramanujan. A función leva o nome do matemático Srinivasa Ramanujan. Definición
A función theta de Ramanujan defínese como 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}}} para |ab| < 1. A identidade do produto triplo de Jacobi toma logo a forma 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 }. } Aquí, a expresión ( a; q ) n {\displaystyle (a; q)_{n}} denota o q-símbolo de Pochhammer. As identidades que se derivan deste inclúen 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 }}} e 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 }}} e 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 }} Esta última é a función de Euler, que está moi relacionada coa función eta de Dedekind. A función theta de Jacobi pódese escribir en termos da función theta de Ramanujan como: ϑ 00 ( w, q )
f ( q w 2, q w − 2 ) {\displaystyle \vartheta _{00}(w, q)
f\left(qw^{2}, qw^{-2}\right)} Representacións con integrais
Temos a seguinte representación con integral para a forma completa de dous parámetros da función theta de Ramanujan: f ( a, b )
1 + ∫ 0 ∞ 2 a e − 12 t 22 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 − 12 t 22 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 {\begin{aligned}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\end{aligned}}} Os casos especiais das funcións theta de Ramanujan dados por φ(q):
f(q, q) (secuencia A000122 na OEIS) e ψ(q):
f(q, q3) (secuencia A010054 na OEIS) tamén teñen as seguintes representacións con integrais: φ ( q )
1 + ∫ 0 ∞ e − 12 t 22 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 − 12 t 22 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}}} Isto leva a varias integrais de casos especiais para constantes definidas por estas funcións cando q:
e−kπ. En particular, temos que φ ( e − k π )
1 + ∫ 0 ∞ e − 12 t 22 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 14 Γ ( 34 )
1 + ∫ 0 ∞ e − 12 t 22 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 14 Γ ( 34 ) ⋅ 2 + 22
1 + ∫ 0 ∞ e − 12 t 22 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 14 Γ ( 34 ) ⋅ 1 + 3214338
1 + ∫ 0 ∞ e − 12 t 22 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 14 Γ ( 34 ) ⋅ 5 + 25534
1 + ∫ 0 ∞ e − 12 t 22 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}}} e que ψ ( e − k π )
∫ 0 ∞ e − 12 t 22 π [ cos ⁡ ( k π t ) − e k π 2 cos ⁡ ( k π t ) − cosh ⁡ k π 2 ] d t π 14 Γ ( 34 ) ⋅ e π 8258
∫ 0 ∞ e − 12 t 22 π [ cos ⁡ ( π t ) − e π 2 cos ⁡ ( π t ) − cosh ⁡ π 2 ] d t π 14 Γ ( 34 ) ⋅ e π 4254
∫ 0 ∞ e − 12 t 22 π [ cos ⁡ ( 2 π t ) − e π cos ⁡ ( 2 π t ) − cosh ⁡ π ] d t π 14 Γ ( 34 ) ⋅ 1 + 24 e π 162716
∫ 0 ∞ e − 12 t 22 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}}}