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Correction: Hearing impairment and dementia: cause, catalyst or consequence? [PDF]
J NeurolLevett BA +15 more
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Drinking and pleasure: Interdisciplinarity points the way forward. [PDF]
AddictionNicholls J, Hunt G.
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Stirling Numbers and Eulerian Numbers
2016This chapter is dedicated to counting partitions of sets and partitions of sets into cycles, and also introduces Stirling numbers and Bell numbers. As an application of the concepts discussed here we state Faa di Bruno chain rule for the n-th derivative of a composite of n-times differentiable functions on \(\mathbb R\).
Carlo Mariconda, Alberto Tonolo
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Stirling Numbers for Complex Arguments
SIAM Journal on Discrete Mathematics, 1997The authors define the Stirling numbers for the case of a complex argument by invoking the Cauchy integral formula. Some of the usual identities extend, but others do not. The properties of unimodality and monotonicity do extend.
B. RICHMOND, MERLINI, DONATELLA
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Canadian Mathematical Bulletin, 1961
In his book [1] Combinatorial Analysis, J. Riordan (p. 32) refers to the continual rediscovery of the Stirling numbers. The author of this note has been surprised on many occasions by the number of different environments in which these numbers make a natural appearance and, in fact, this article is concerned with just such an occurrence. The connection
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In his book [1] Combinatorial Analysis, J. Riordan (p. 32) refers to the continual rediscovery of the Stirling numbers. The author of this note has been surprised on many occasions by the number of different environments in which these numbers make a natural appearance and, in fact, this article is concerned with just such an occurrence. The connection
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Aequationes mathematicae
\textit{L. C. Hsu} and \textit{P. J. S. Shiue} [Adv. Appl. Math. 20, No. 3, 366--384 (1998; Zbl 0913.05006)] introduced a far-reaching generalization of Stirling numbers, \(S(n,k;\alpha,\beta,r)\), and they gave eleven known combinatorial sequences as specializations. \textit{B. Bényi} et al. [Integers 22, Paper A79, 28 p.
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\textit{L. C. Hsu} and \textit{P. J. S. Shiue} [Adv. Appl. Math. 20, No. 3, 366--384 (1998; Zbl 0913.05006)] introduced a far-reaching generalization of Stirling numbers, \(S(n,k;\alpha,\beta,r)\), and they gave eleven known combinatorial sequences as specializations. \textit{B. Bényi} et al. [Integers 22, Paper A79, 28 p.
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Log-concavity of stirling numbers and unimodality of stirling distributions
Annals of the Institute of Statistical Mathematics, 1988A series of inequalities involving Stirling numbers of the first and second kind with adjacent indices are obtained, some of which show log- concavity of Stirling numbers in three directions. Some of them are new, others extend or improve earlier results.
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2012
In this chapter we focus on functions of q x , or equivalently functions of the q-binomial coefficients. We systematically find q-analogues of the formulas for Stirling numbers from Jordan and the elementary textbooks by J. Cigler and Schwatt. To this end, various q-difference operators are used. In each of Sections 5.2–5.4, we focus on a certain such △
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In this chapter we focus on functions of q x , or equivalently functions of the q-binomial coefficients. We systematically find q-analogues of the formulas for Stirling numbers from Jordan and the elementary textbooks by J. Cigler and Schwatt. To this end, various q-difference operators are used. In each of Sections 5.2–5.4, we focus on a certain such △
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