qintegral

qintegral refers to a q-calculus concept known as the Jackson q-integral, a q-analogue of the definite integral. It plays the role of summing a function over a geometric lattice and is foundational in the study of q-series, basic hypergeometric functions, and related areas. The concept was developed in the framework of quantum calculus, with F. H. Jackson contributing to its development in the early 20th century.

For 0 < q < 1 and a > 0, the Jackson q-integral of a function f on the interval [0, a] is defined by

∫_0^a f(x) d_q x = a (1 − q) ∑_{n=0}^∞ q^n f(a q^n).

More generally, the q-integral on [0, b] is ∫_0^b f(x) d_q x = b (1 − q) ∑_{n=0}^∞ q^n f(b q^n), and intervals can be handled by subtracting integrals over subintervals, ∫_a^b f(x) d_q x = ∫_0^b f(x) d_q x − ∫_0^a f(x) d_q x.

As q approaches 1 from below, the Jackson q-integral converges to the ordinary Riemann integral, linking the q-calculus framework to classical analysis. The q-integral is naturally paired with the q-difference operator, forming a q-calculus analogous to the relationship between differentiation and integration in standard calculus.

Applications of the q-integral appear in the theory of q-orthogonal polynomials, basic hypergeometric series, and various areas of mathematical physics where discrete, geometrically spaced structures are studied. It also serves as a tool for deriving q-analogues of classical integral identities and transforms.