Binomial Coefficients and Lucas Sequences
β Scribed by Achim Flammenkamp; Florian Luca
- Publisher
- Elsevier Science
- Year
- 2002
- Tongue
- English
- Weight
- 226 KB
- Volume
- 93
- Category
- Article
- ISSN
- 0022-314X
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## Abstract We consider the expression and investigate for which __k__ and __n__ is __P__(__k__, __n__) an integer. We find integer solutions __k__ and __m__ of the equation __P__(__k__, __n__) = __m__ for a given integer __m__ and approximate real solutions __x__ = __x__(__k__) of the equation __
In this paper, we present a method for obtaining a wide class of combinatorial identities. We give several examples; some of them have already been considered previously, and others are new.  2002 Elsevier Science (USA)
is transcendental over β«ήβ¬ X when t is an integer G 2. This is due to Stanley for t even, and independently to Flajolet and to Woodcock and Sharif for the general case. While Stanley and Flajolet used analytic methods and studied the asymptotics of the coefficients of this series, Woodcock and Shari
Using binomial coefficients the Clebsch-Gordan and Gaunt coefficients were calculated for extremely large quantum numbers. The main advantage of this approach is directly calculating these coefficients, instead of using recursion relations. Accuracy of the results is quite high for quantum numbers \
A counting argument is developed and divisibility properties of the binomial coefficients are combined to prove, among other results, that where K n , resp. K k n , is the complete, resp. complete k-uniform, hypergaph and R(K n , Z p ), R(K k n , Z 2 ) are the corresponding zero-sum Ramsey numbers.