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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 MIT Professor Kac over the last few years at MIT.
Front Matter
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’s Binomial Formula and a Noncommutative 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’s 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 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
Back Matter