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Register a new userOn certain definite integrals and infinite series
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E.E. Kummer
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A translation of Kummer`s paper On certain definite integrals and infinite series
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An operatorial method, already employed to formulate a generalization of the Ramanujan master theorem, is applied to the evaluation of integrals of various type. This technique provide a very flexible and powerful tool yielding new results encompassing various aspects of the special function theory.
This is the translation of Leonhard Eulers paper De Seriebus divergentibus written in Latin into English. Leonhard Euler defines and discusses divergent series. He is especially interested in the example $1!-2!+3!-text{etc.}$ and uses different methods to sum it. He finds a value of about $0.59...$.
E661 in the Enestrom index. This was originally published as Variae considerationes circa series hypergeometricas (1776). In this paper Euler is looking at the asymptotic behavior of infinite products that are similar to the Gamma function. He looks at the relations between some infinite products and integrals. He takes the logarithm of these infinite products, and expands these using the Euler-Maclaurin summation formula. In section 14, Euler seems to be rederiving some of the results he already proved in the paper. However I do not see how these derivations are different. If any readers think they understand please I would appreciate it if you could email me. I am presently examining Eulers work on analytic number theory. The two main topics I want to understand are the analytic continuation of analytic functions and the connection to divergent series, and the asymptotic behavior of the Gamma function.
Let $X$ be a real prehomogeneous vector space under a reductive group $G$, such that $X$ is an absolutely spherical $G$-variety with affine open orbit. We define local zeta integrals that involve the integration of Schwartz-Bruhat functions on $X$ against generalized matrix coefficients of admissible representations of $G(mathbb{R})$, twisted by complex powers of relative invariants. We establish the convergence of these integrals in some range, the meromorphic continuation as well as a functional equation in terms of abstract $gamma$-factors. This subsumes the Archimedean zeta integrals of Godement-Jacquet, those of Sato-Shintani (in the spherical case), and the previous works of Bopp-Rubenthaler. The proof of functional equations is based on Knops results on Capelli operators.
In real Hilbert spaces, this paper generalizes the orthogonal groups $mathrm{O}(n)$ in two ways. One way is by finite multiplications of a family of operators from reflections which results in a group denoted as $Theta(kappa)$, the other is by considering the automorphism group of the Hilbert space denoted as $O(kappa)$. We also try to research the algebraic relationship between the two generalizations and their relationship to the stable~orthogonal~group~$mathrm{O}=varinjlimmathrm{O}(n)$ in terms of topology. In this paper we mainly show that : (a) $Theta(kappa)$ is a topological and normal subgroup of $O(kappa)$; (b) $O^{(n)}(kappa) to O^{(n+1)}(kappa) stackrel{pi}{to} S^{kappa}$ is a fibre bundle where $O^{(n)}(kappa)$ is a subgroup of $O(kappa)$ and $S^{kappa}$ is a generalized sphere.
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