In mathematics, a q-series is defined as

usually considered first as a formal power series; it is also an analytic function of q, in the unit disc.

The Euler function is given by

The coefficient of q^k in the Maclaurin series for 1 / φ(q) gives the number of all partitions of k. That is,

where p(k) is the partition function of k.

The Euler identity is

Note that (3n² - n) / 2 is a pentagon number.

The Euler function is related to the Dedekind eta function through a Ramanujan identity as

φ(q) = q ^{- 1 / 24}η(τ)

where q = e^2πiτ is the square of the nome.

Note that both functions have the symmetry of the modular group. The Euler function also plays a role in describing the interior of the Mandelbrot set.

Q-analogs

There is a substantial theory constructing q-analogs of results, in particular in combinatorics and the theory of special functions. A q-analog, roughly speaking, is a theorem or identity for a q-series that gives back a known result in the limit, as q → 1, inside the unit circle. For convenience this is written as the limit q → 1⁻, which suggests the limit through real values tending up to 1, which is in fact more restricted, though the difference is not usually significant.

Noticing that

we define the q-analog of n, also known as the q-bracket of n to be

From this one can define the q-analog of the factorial, the q-factorial, as

Again, one recovers the usual factorial by taking the limit as .

From the q-factorials, one can move on to define the q-binomial coefficients, also known as Gaussian coefficients, Gaussian polynomials, or Gaussian binomials:

See also

• Gaussian binomial
• Stirling number
• Young tableau
• Jacobi theta function
• Modular forms

References

• Eugene Mukhin Symmetric Polynomials and Partitions (undated, 2004 or earlier)
• Linas Vepstas Spectral Analysis of Mandelbrot Interior (2000) describes the appearance of q-series in the Mandelbrot Set.

External links

• q-analog from MathWorld
• q-bracket from MathWorld
• q-factorial from MathWorld
• q-binomial coefficient from MathWorld