exp_q

The function exp_q(x) is defined by taking the power series for exp(x) and keeping only the terms whose index is a multiple of q. For example, exp₂(x) keeps only the even-numbered terms in the exponential power series and so equals cosh(x).

In general,

The first sum uses Iverson's bracket notation: a Boolean expression in brackets denotes the function that returns 1 when the expression is true and zero when it is false. Here the bracket equals 1 when q divides n and is zero otherwise.

Closed forms

Let = exp(2i / q). Then

This lets us find closed-form expressions for exp_q(x). For example, when q = 4, = i and

Here's a proof of the identity above:

In the proof we used the identity

which is important in deriving the properties of the discrete Fourier transform.

Differential equations

The first time I saw the function exp_q(x) was in differential equations, though I didn't know at the time the function had a name.

When a course in differential equations gets to power series solutions, a common example or homework problem is to solve

for k = 3 or 4, i.e. to find a function that equals its third or fourth derivative.

If the initial conditions are

and

the unique solution to

is y(x) = exp_k(x).

Mathematica and Mittag-Leffler

Mathematica does not have a built-in function implementing exp_q(x), but it does have an implementation of the Mittag-Leffler function, and so thanks to a relation between this function and exp_q(x) you can implement the latter as

expq[x_, q_] := MittagLefflerE[q, x^q]

Combinatorics

The first time I saw the notation exp_q(x) was in combinatorics. I had intended to include an application from that book here, but I make that the topic for the next post.

The post exp_q first appeared on John D. Cook.