In one sentence, quantum calculus is ordinary calculus without taking limits. This undergraduate text develops two types of quantum calculi, the q-calculus and the h-calculus. It studies quantum calculus along the traditional lines of ordinary calculus and discovers many important notions and results in combinatorics, number theory, and other fields of mathematics. The book assumes only some knowledge of first year calculus and linear algebra and is addressed mainly to undergraduate students.

Inhalt

Introduction * q-derivative and h-derivative * Generalized Taylor's formula for polynomials * q-analogue of (x-a)^n, n an integer, and q-derivatives of binomials * q-Taylor's formula for polynomials * Gauss' binomial formula and a non-commutative binomial formula * Properties of q-binomial coefficients * q-binomial coefficients and linear algebra over finite fields * q-Taylor's formula for formal power series and Heine's binomial formula * Two Euler identities and two q-exponential functions * q-trigonometric functions * Jacobi's triple product identity * Classical partition function and Euler's product formula * q-hypergeometric functions and Heine's formula * More on Heine's formula and the general binomial * Ramanujan's product formula * Explicit formulas for sums of two and of four squares * Explicit formulas for sums of two and of four triangular numbers * q-antiderivative * Jackson integral * Fundamental theorem of q-calculus and integration by parts * q-gamma and q-beta functions * h-derivative and h-integral * Bernoulli polynomials and Bernoulli numbers * Sums of powers * Euler-Maclaurin formula * Symmetric quantum calculus * Appendix: a list of q-antiderivatives * Literature