Euler function

In mathematics, the Euler function is given by Named after Leonhard Euler, it is a model example of a q-series and provides the prototypical example of a relation between combinatorics and complex analysis. The coefficient in the formal power series expansion for gives the number of partitions of k. That is, where is the partition function. The Euler identity, also known as the Pentagonal number theorem, is is a pentagonal number. The Euler function is related to the Dedekind eta function as The Euler function may be expressed as a q-Pochhammer symbol: The logarithm of the Euler function is the sum of the logarithms in the product expression, each of which may be expanded about q = 0, yielding which is a Lambert series with coefficients -1/n. The logarithm of the Euler function may therefore be expressed as where -[1/1, 3/2, 4/3, 7/4, 6/5, 12/6, 8/7, 15/8, 13/9, 18/10, ...

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Pentagonal Number Theorem and Jacobi Identity MATH-511: Number theory II.a - Modular forms

Covers the pentagonal number theorem, Jacobi identity, and modularity of eta and theta functions.

In mathematical area of combinatorics, the q-Pochhammer symbol, also called the q-shifted factorial, is the product with It is a q-analog of the Pochhammer symbol , in the sense that The q-Pochhammer symbol is a major building block in the construction of q-analogs; for instance, in the theory of basic hypergeometric series, it plays the role that the ordinary Pochhammer symbol plays in the theory of generalized hypergeometric series.

In mathematics, the pentagonal number theorem, originally due to Euler, relates the product and series representations of the Euler function. It states that In other words, The exponents 1, 2, 5, 7, 12, ... on the right hand side are given by the formula gk = k(3k − 1)/2 for k = 1, −1, 2, −2, 3, ... and are called (generalized) pentagonal numbers . (The constant term 1 corresponds to .) This holds as an identity of convergent power series for , and also as an identity of formal power series.

In number theory, the partition function p(n) represents the number of possible partitions of a non-negative integer n. For instance, p(4) = 5 because the integer 4 has the five partitions 1 + 1 + 1 + 1, 1 + 1 + 2, 1 + 3, 2 + 2, and 4. No closed-form expression for the partition function is known, but it has both asymptotic expansions that accurately approximate it and recurrence relations by which it can be calculated exactly. It grows as an exponential function of the square root of its argument.