Quantum Calculus

by Kac, Victor G.; Cheung, Pokman

• ISBN13: 9780387953410
• ISBN10: 0387953418
• Format: Paperback
• Copyright: 2001-12-01
• Publisher: Springer Nature

Summary

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.

Table of Contents

Introduction	vii
q-Derivative and h-Derivative	1	(4)
Generalized Taylor's Formula for Polynomials	5	(2)
q-Analogue of (x-a)n, n an Integer, and q-Derivatives of Binomials	7	(5)
q-Taylor's Formula for Polynomials	12	(2)
Gauss's Binomial Formula and a Noncommutative Binomial Formula	14	(3)
Properties of q-Binomial Coefficients	17	(4)
q-Binomial Coefficients and Linear Algebra over Finite Fields	21	(6)
q-Taylor's Formula for Formal Power Series and Heine's Binomial Formula	27	(2)
Two Euler's Identities and Two q-Exponential Functions	29	(4)
q-Trigonometric Functions	33	(2)
Jacobi's Triple Product Identity	35	(2)
Classical Partition Function and Euler's Product Formula	37	(6)
q-Hypergeometric Functions and Heine's Formula	43	(4)
More on Heine's Formula and the General Binomial	47	(3)
Ramanujan Product Formula	50	(6)
Explicit Formulas for Sums of Two and of Four Squares	56	(4)
Explicit Formulas for Sums of Two and of Four Triangular Numbers	60	(4)
q-Antiderivative	64	(3)
Jackson Integral	67	(6)
Fundamental Theorem of q-Calculus and Integration by Parts	73	(3)
q-Gamma and q-Beta Functions	76	(4)
h-Derivative and h-Integral	80	(5)
Bernoulli Polynomials and Bernoulli Numbers	85	(5)
Sums of Powers	90	(2)
Euler--Maclaurin Formula	92	(7)
Symmetric Quantum Calculus	99	(7)
Appendix: A List of q-Antiderivatives	106	(3)
Literature	109	(2)
Index	111