The problem of evaluating integrals is well known to every student who has had a year of calculus. It was an especially important subject in 19th century analysis and it has now been revived with the appearance of symbolic languages. In this book, the authors use the problem of exact evaluation of definite integrals as a starting point for exploring many areas of mathematics. The questions discussed in this book, first published in 2004, are as old as calculus itself. In presenting the combination of methods required for the evaluation of most integrals, the authors take the most interesting, rather than the shortest, path to the results. Along the way, they illuminate connections with many subjects, including analysis, number theory, algebra and combinatorics. This will be a guided tour of exciting discovery for undergraduates and their teachers in mathematics, computer science, physics, and engineering.
S. Ramanujan introduced a technique, known as Ramanujan's Master Theorem, which provides an analytic expression for the Mellin transform of a function. The main identity of this theorem involves the extrapolation of the sequence of coefficients of the integrand, defined originally as a function on N to C. The history and proof of this result are reviewed. Applications to the evaluation of a variety of definite integrals is presented.
The focus of this paper is the study of generalized Fibonacci polynomials and Fibonomial coefficients. The former are polynomials {n} in variables s, t given by {0} = 0, {1} = 1, and {n} = s{n−1}+t{n−2} for n ≥ 2. The latter are defined byThese quotients are also polynomials in s, t and specializations give the ordinary binomial coefficients, the Fibonomial coefficients, and the q-binomial coefficients. We present some of their fundamental properties, including a more general recursion for {n}, an analogue of the binomial theorem, a new proof of the Euler-Cassini identity in this setting with applications to estimation of tails of series, and valuations when s and t take on integral values. We also study a corresponding analogue of the Catalan numbers. Conjectures and open problems are scattered throughout the paper.
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