Quantum calculus

Simply put, quantum calculus is ordinary calculus without taking limits. This undergraduate text develops two types of quantum calculi, the q-calculus and the h-calculus. As this book develops quantum calculus along the lines of traditional calculus, the reader discovers, with a remarkable inevitability, many important notions and results of classical mathematics. This book is written at the level of a first course in calculus and linear algebra and is aimed at undergraduate and beginning graduate students in mathematics, computer science, and physics. It is based on lectures and seminars given by Professor Kac over the last few years at MIT.

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

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