Combinatorics sum and its applications in probability statistic

In this thesis, generating functions and combinatorial sums and their applications, which are very important tools in Probability and in Statistics, have been studied. Combinatorics sums including binomial coefficients are given. Relations between these sums and special numbers have be investigated. The results of this thesis will be contribute not only Mathematics, Probability and Statistics, and but also other science. In thesis some formulas and relations which are related to the some special numbers and combinatoricssums are given. For instance, relationship between the Stirling numbers of first kind and second kind, the Fibonacci numbers, the Catalan numbers, Bernoulli numbers, Euler numbers and, Bell numbers etc. are given. In this thesis, we give some combinatorial sums including binomial coefficients, the Catalan numbers and the Bernstein basis functions. Moreover by applying the Lagrange inversion formula to the generating functions for the Stirling numbers, we derive some identities including the Stirling numbers and Bernoulli numbers of negative order.