Quantum Calculus

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