exp_q

The function expq(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 ω = exp(2πi / q). Then This lets us find closed-form expressions for expq(x). For example, when q = 4, ω = 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 expq(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…