منابع مشابه
A Combinatorial Identity and Applications
An identity for the finite sum ∑N 1 z qn−r is given. Related sums (or series) were studied by Scherk, Clausen, Ramanujan, Shanks, Andrews, and others. We use such identity to give new formulas for ∑∞ 1 z qn−r , the Riemann zeta function and the Euler–Mascheroni constant. An irrationality result is also proved.
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Let α = (α1, α2, · · · , αm) ∈ R>0. Let αi,j be the vector obtained from α by deleting the entries αi and αj . Besser and Moree [1] introduced some invariants and near invariants related to the solutions ∈ {±1}m−2 of the linear inequality |αi −αj | < 〈 , αi,j〉 < αi +αj , where 〈, 〉 denotes the usual inner product and αi,j the vector obtained from α by deleting αi and αj . The main result of Bes...
متن کاملA Combinatorial Identity and the World Series
In this note the author gives a simple probabilistic proof of a combinatorial identity by calculatingthe winning probabilityin the World Series. The winning probabilitiesand the expected length of the championship series are given by the applications of the identity and its generalization.
متن کاملA Combinatorial Proof for Cayley's Identity
Abstract In a recent paper, Caracciolo, Sokal and Sportiello presented, inter alia, an algebraic/combinatorial proof for Cayley’s identity. The purpose of the present paper is to give a “purely combinatorial” proof for this identity; i.e., a proof involving only combinatorial arguments. Since these arguments eventually employ a generalization of Laplace’s Theorem, we present a “purely combinato...
متن کاملA Combinatorial Proof of Sun's \Curious" Identity
A binomial coeÆcient identity due to Zhi-Wei Sun is the subject of half a dozen recent papers that prove it by various analytic techniques and establish a generalization. Here we give a simple proof that uses weight-reversing involutions on suitable con gurations involving dominos and colorings. With somewhat more work, the method extends to the generalization also. 0 Introduction The identity
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ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series A
سال: 1977
ISSN: 0097-3165
DOI: 10.1016/0097-3165(77)90012-7