Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-18T17:26:32.572Z Has data issue: false hasContentIssue false

Geometric and probabilistic aspects of some combinatorial identities

Published online by Cambridge University Press:  09 April 2009

T. P. Speed
Affiliation:
Department of Mathematics, University of Western Australia, Nedlands, W.A. 6009.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For positive integers a, b and n define the combinational expression We give geometric and probabilistic interpretations of these expressions (and their multidimensional extensions) and find new, simple proofs of the convolution identities known to hold for such expressions.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1976

References

Blackwell, David and Dubins, Lester (1966), ‘An elementary proof of an identity of Gould's’, Bol. Soc. Mat. Mexicana Ser. 2 11, 108110.Google Scholar
Gould, H. W. (1956), ‘Some generalisations of Vandermonde's convolution’, Amer. Math. Monthly 63, 8491.CrossRefGoogle Scholar
Gould, H. W. (1957), ‘Final analysis of Vandermonde's convolution’, Amer. Math. Monthly 64, 409415.CrossRefGoogle Scholar
Gould, H. W. (1974), ‘Coefficient identities for powers of Taylor and Dirichlet series’, Amer. Math. Monthly 81, 313.CrossRefGoogle Scholar
Gould, H. W. and Kaucky, J. (1966), ‘Evaluation of a class of binomial coefficient summations’, J. Combinatorial Theory 1, 233247.CrossRefGoogle Scholar
Harris, Theodore E. (1963), The theory of branching processes. Springer-Verlag Berlin, Göttingen, Heidelberg.CrossRefGoogle Scholar
Mohanty, S. G. (1966a), ‘Some convolutions with multinomial coefficients and related probability distributions’, SIAM Review 8, 501509.CrossRefGoogle Scholar
Mohanty, S. G. (1966b), ‘An urn problem related to the ballot problem’, Amer. Math. Monthly 73, 526528.CrossRefGoogle Scholar
Mohanty, S. G. (1972), ‘On queues involving batches’, J. Applied Probability 9, 430435.CrossRefGoogle Scholar
Mohanty, S. G. and Narayana, T. V. (1961), ‘Some properties of composition and their application to probability and statistics I’, Biom. Zeit. 3, 252258.CrossRefGoogle Scholar
Raney, George N. (1964), ‘A formal solution of ’, Canad. J. Math. 16, 755762.CrossRefGoogle Scholar
Riordan, John (1958), An introduction to combinatorial analysis. John Wiley and Sons, New York, London, Sydney.Google Scholar
Takács, Lajos (1967) Combinatorial methods in the theory of stochastic processes. John Wiley and Sons, Inc., New York, London, Sydney.Google Scholar