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Lambert Series
In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form :S(q)=\sum_^\infty a_n \frac . It can be resumed formally by expanding the denominator: :S(q)=\sum_^\infty a_n \sum_^\infty q^ = \sum_^\infty b_m q^m where the coefficients of the new series are given by the Dirichlet convolution of ''a''''n'' with the constant function 1(''n'') = 1: :b_m = (a*1)(m) = \sum_ a_n. \, This series may be inverted by means of the Möbius inversion formula, and is an example of a Möbius transform. Examples Since this last sum is a typical number-theoretic sum, almost any natural multiplicative function will be exactly summable when used in a Lambert series. Thus, for example, one has :\sum_^\infty q^n \sigma_0(n) = \sum_^\infty \frac where \sigma_0(n)=d(n) is the number of positive divisors of the number ''n''. For the higher order sum-of-divisor functions, one has :\sum_^\infty q^n \sigma_\alpha(n) = \sum_^\infty \frac where \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramanujan Theta Function
In mathematics, particularly -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) = \sum_^\infty a^\frac \; b^\frac for . The Jacobi triple product identity then takes the form :f(a,b) = (-a; ab)_\infty \;(-b; ab)_\infty \;(ab;ab)_\infty. Here, the expression (a;q)_n denotes the -Pochhammer symbol. Identities that follow from this include :\varphi(q) = f(q,q) = \sum_^\infty q^ = and :\psi(q) = f\left(q,q^3\right) = \sum_^\infty q^\frac = and :f(-q) = f\left(-q,-q^2\right) = \sum_^\infty (-1)^n q^\frac = (q;q)_\infty This last being the Euler function, which is closely related to the Dedekind eta function. The Jacobi theta functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
