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COMBINATORIAL IDENTITIES AND HYPERGEOMETRIC FUNCTIONS
Rocky Mountain Journal of Mathematics, 2022zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Alzer, Horst, Richards, Kendall C.
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On two combinatorial identities
Journal of Computational Methods in Sciences and Engineering, 2017Two combinatorial identities are proved. The first one allows easy to establish connections between Bernoulli and Stirling numbers and values of zeta function of integer arguments and to find numerous identities, including the new ones, for these quantities.
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Publicationes Mathematicae Debrecen, 2022
The author proves that the only solution of the difference equation \[ \Phi (2(v+1))+\sum^{v}_{k=1}\binom{2v}{2k-1} \Phi (2k) \Phi (2(v-k+1))=0 \] subject to the initial condition \(\Phi (2)=1/4\) is \(\Phi (2k)=((2^{2k}-1)/2k) B_{2k}\) where the \(B_{2k}\) are the even-indexed Bernoulli numbers. Using this result, he then proves that (1/k)\(| B_{2k}|^{
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The author proves that the only solution of the difference equation \[ \Phi (2(v+1))+\sum^{v}_{k=1}\binom{2v}{2k-1} \Phi (2k) \Phi (2(v-k+1))=0 \] subject to the initial condition \(\Phi (2)=1/4\) is \(\Phi (2k)=((2^{2k}-1)/2k) B_{2k}\) where the \(B_{2k}\) are the even-indexed Bernoulli numbers. Using this result, he then proves that (1/k)\(| B_{2k}|^{
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Zeon Algebra and Combinatorial Identities
SIAM Review, 2014We show that the ordinary derivative of a real analytic function of one variable can be realized as a Grassmann--Berezin-type integration over the Zeon algebra, the Z-integral. As a by-product of this representation, we give new proofs of the Faa di Bruno formula and Spivey's identity [M. Z. Spivey, J.
Antônio Francisco Neto +1 more
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The Combinatorial Identity on the Jacobian Conjecture
Acta Applicandae Mathematicae, 2005zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Egorychev, Georgy P. +1 more
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Vertex Algebras and Combinatorial Identities [PDF]
Acta Applicandae Mathematica, 2002 In 1980's J. Lepowsky and R. Wilson gave a Lie-theoretic interpretation of Rogers-Ramanujan identities in terms of level 3 representations of affine Lie algebra sl(2, C)~. When applied to other representations and affine Lie algebras, Lepowsky-Wilson's approach yielded series of other combinatorial identities of Rogers-Ramanujan type. At about the same
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Some Combinatorial and Analytical Identities
Annals of Combinatorics, 2012zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Ismail, Mourad E.H., Stanton, Dennis
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Identities for Combinatorial Extremal Theory
Bulletin of the London Mathematical Society, 1997This paper proves new Ahlswede-Zhang type identities for finite set systems.
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Three Identities in Combinatory Analysis
Journal of the London Mathematical Society, 1943Es sei \[ x_n = 1 - x^n, \quad \bar x_n = 1 + x^n, \quad x_n!= \prod_{i=1}^n x_i, \quad \bar x_n! = \prod_{i=1}^n \bar x_i, \] \[ x_n^2 = x_{2n}, \quad x_n^2! = x_2x_4\ldots x_{2n}, \quad x! = x_\infty !,\quad \bar x! = \bar x_\infty!, \quad x^2! = x_\infty^2!, \] dann beweist der Verf.
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A Combinatorial Identity and the World Series
SIAM Review, 1993In this note the author gives a simple probabilistic proof of a combinatorial identity by calculating the winning probability in the World Series. The winning probabilities and the expected length of the championship series are given by the applications of the identity and its generalization.
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