By G. P. Egorychev
ISBN-10: 0821845128
ISBN-13: 9780821845127
This monograph will be of curiosity to a wide spectrum of readers: experts in discrete and non-stop arithmetic, physicists, engineers, and others attracted to computing sums and using advanced research in discrete arithmetic. It comprises investigations at the challenge of discovering quintessential representations for and computing finite and endless sums (generating functions); those come up in perform in combinatorial research, the idea of algorithms and programming on a working laptop or computer, chance concept, workforce concept, and serve as thought, in addition to in physics and different components of data. A basic procedure is gifted for computing sums and different expressions in closed shape by way of lowering them to one-dimensional and a number of integrals, commonly to contour integrals.
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Sample text
An e−n/x = O x 1+ . n=1 Thus, for any individual term in the sum, we have an e−n/x = O x 1+ . Choosing x = n, we deduce that an = O(n1+ ). If we let π be an automorphic representation on GL(2) with local parameters, αp , βp , then one can give an intuitive description of the symmetric power Lfunctions attached to π in the following way. Consider, with rm denoting the m-th symmetric power of π , m m−j 1− Lm (s) := L(s, π, rm ) = p j =1 αp j βp −1 ps where we are ignoring the finitely many Euler factors that need to be modified corresponding to the ramified factors.
The subspace S is invariant under both of these actions. Diagonalizing the Hecke action, we get a decomposition S = ⊕f S (f ) into submodules, where each S (f ) is a rank two module over Kf [Gal(Q/Q)], and Kf is a certain number field (the field generated by the Fourier coefficients of f ). 5 Geometric Realization of Modular Forms of Higher Weight 37 These are the Galois representations attached to modular forms of higher weight whose existence had been conjectured by Serre and constructed by Deligne [38].
1007/978-81-322-0770-2_4, © Springer India 2013 39 40 4 The Ramanujan Conjecture from GL(2) to GL(n) wait until 1974 when Deligne proved it as a consequence of his proof of the Weil conjectures. We outline the proofs of these conjectures. As mentioned in the introduction to Chap. 3, the essential property of (z) is that it is a modular form of weight 12 for the full modular group SL2 (Z), which is the group of 2 × 2 matrices with integer entries and determinant 1. This means that for z in the upper half-plane h, we have az + b cz + d = (cz + d)12 (z) a c ∀ b ∈ SL2 (Z).
Integral representations and computation of combinatorial sums by G. P. Egorychev
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