expq

Expq, often written as exp_q(x), is a family of q-analogues of the exponential function used in q-calculus, combinatorics, and mathematical physics. The precise form and notation vary by author, but the core idea is to generalize the ordinary exponential with a deformation parameter q. In most treatments, exp_q reduces to the standard exponential e^x when q tends to 1.

A common definition uses the q-integers [n]_q and the q-factorial [n]_q!: [n]_q = (1 - q^n)/(1 - q) and [n]_q! = [1]_q [2]_q ... [n]_q. The q-exponential exp_q(x) is then given by the power series sum_{n=0}^\infty x^n / [n]_q!. This series converges for a range of x depending on q and, in the limit q -> 1, recovers e^x. Other equivalent formulations exist, including versions expressed with q-Pochhammer symbols, leading to related functions often denoted e_q or E_q. Because terminology varies, exp_q is sometimes used interchangeably with these related q-exponentials, though not universally.

Properties of exp_q include analyticity near x = 0 and the satisfaction of q-difference equations rather than the standard addition formula. In general, exp_q(x + y) ≠ exp_q(x) exp_q(y), except in certain special cases or under limiting conditions.

Applications of exp_q appear across q-calculus, quantum groups, and combinatorics. It appears in generating functions, deformations of algebraic structures, and models with q-deformed symmetries. The concept originates in early 20th-century q-series work by F. H. Jackson and has been developed further in various mathematical and physical contexts.