Combinatorial Identities Involving The Binomial Coefficients And Several Integer Sequences

The research of combinatorial identities is one important part of combinatorial math-ematics. In this paper, we mainly discuss some combinatorial identities which involve thereciprocal of binomial coe?cients. It is well known that computing the values of combi-natorial sums involving the reciprocal of binomial coe?cients is a di?cult subject, but itattracts more attention and the subject has already been fully researched. In this paper,we make furthermore research into combinatorial sums like rn=m (λnrr) and r≥m (tnn+r+ rr),and we obtain some new interesting results.In the first part of the second chapter, the authors make use of the integral identityfor the reciprocal of binomial coe?cientsto extend the sums rn=m (λnrr), which appeared in theorem 2.1 of the paper(B. Sury, T.Wang and F.Z.Zhao, Identities involving reciprocals of bonomial coe?cients, Journal ofInteger Sequence, 7(2004), Article 04.2.8.). In the second part, the author will use thetheory of generating functions to consider some combinatorial sums involving the reciprocalof binomial coe?cients, especially, for the theorem 3.6 of the above paper, we obtain abetter result An(t) for the infinite sums r≥m (tnn+r+ rr), and we also obtain some interestingresults about Harmonic numbers.In the third chapter, we first introduce the definition of Riordan arrays, and then usingit we obtain some combinatorial identities involving several classical integer sequences.At last, making use of the expansions of trigonometric functions, the author obtainssome results about binomial coe?cients, some of which are the known results, but othersare new.