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31.
Zhi-Guo Liu 《The Ramanujan Journal》2002,6(4):429-447
In this paper the author proves a q-expansion formula which utilizes the Leibniz formula for the q-differential operator. This expansion leads to new proofs of the Rogers–Fine identity, the nonterminating 65 summation formula, and Watson's q-analog of Whipple's theorem. Andrews' identities for sums of three squares and sums of three triangular numbers are also derived. Other identities of Andrews and new identities for Hecke type series are also discussed. 相似文献
32.
Moharram A. Khan 《Czechoslovak Mathematical Journal》2002,52(2):401-413
Let p, q and r be fixed non-negative integers. In this note, it is shown that if R is left (right) s-unital ring satisfying
, respectively) where
, then R is commutative. Moreover, commutativity of R is also obtained under different sets of constraints on integral exponents. Also, we provide some counterexamples which show that the hypotheses are not altogether superfluous. Thus, many well-known commutativity theorems become corollaries of our results. 相似文献
33.
34.
XinRongMA 《数学学报(英文版)》2004,20(1):157-162
In two centuries ago,Ming Autu discovered the famous Catalan numbers while he tried to expand the function sin(2px) as power series of sin(x) for the case p=1,2,3.Very recently,P.J.Larcombe shows that for any p,sin(2px) can always be expressed as an infinite power series of sin(x) involving precise combinations of Catalan numbers as part of all but the initial p terms and gave all expansions for the case p=4,5.The present paper presents the desired expansion for arbitrary integer p. 相似文献
35.
We study Einstein warped product spaces. As a result, we prove the following: if is an Einstein warped product space with nonpositive scalar curvature and compact base, then is simply a Riemannian product space.
36.
YunYanYANG 《数学学报(英文版)》2003,19(1):63-68
In this paper,we derive an elementary identity for smooth solutions of the following equation:△u(x) K(x)e^2u(x)=0 in R^2 and use it to get some global properties of the solutions. 相似文献
37.
Pavel Rehak 《Czechoslovak Mathematical Journal》2001,51(2):303-321
We study oscillatory properties of the second order half-linear difference equation
. It will be shown that the basic facts of oscillation theory for this equation are essentially the same as those for the linear equation
. We present here the Picone type identity, Reid Roundabout Theorem and Sturmian theory for equation (HL). Some oscillation criteria are also given. 相似文献
38.
Let G be a -compact, locally compact group and I be a closed2-sided ideal with finite codimension in L1(G). It is shownthat there are a closed left ideal L having a right boundedapproximate identity and a closed right ideal R having a leftbounded approximate identity such that I = L + R. The proofuses ideas from the theory of boundaries of random walks ongroups. 2000 Mathematics Subject Classification: primary 43A20;secondary 42A85, 43A07, 46H10, 46H40, 60B11. 相似文献
39.
Cecí lia Ferreira Armando Machado 《Proceedings of the American Mathematical Society》2000,128(7):2181-2186
Let denote the set of all -roots of the identity in a Lie group . We show that is always an embedded submanifold of , having the conjugacy classes of its elements as open submanifolds. These conjugacy classes are examples of -symmetric spaces and we show, more generally, that every -symmetric space of a Lie group is a covering manifold of an embedded submanifold of . We compute also the Hessian of the inclusions of and into , relative to the natural connection on the domain and to the symmetric connection on .
40.
Ramanujan's partition congruences can be proved by first showing that the coefficients in the expansions of (q; q)
r
satisfy certain divisibility properties when r = 4, 6 and 10. We show that much more is true. For these and other values of r, the coefficients in the expansions of (q; q)
r
satisfy arithmetic relations, and these arithmetic relations imply the divisibility properties referred to above. We also obtain arithmetic relations for the coefficients in the expansions of (q; q)
r
(q
t; q
t)
s
, for t = 2, 3, 4 and various values of r and s. Our proofs are explicit and elementary, and make use of the Macdonald identities of ranks 1 and 2 (which include the Jacobi triple product, quintuple product and Winquist's identities). The paper concludes with a list of conjectures. 相似文献