Handbook of statistics 16 Order statistics: theory and methods
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Chen N. Article First Online: 28 September This is a preview of subscription content, log in to check access. Arnold, N. Balakrishnan, and H. Google Scholar. Balakrishnan and A. Basu Eds.
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Balakrishnan and C. Rao Eds. Ahsanullah and M. David and H. Nagaraja, Order Statistics , 3rd ed. Wiley, Hoboken, New Jersey, Joshi and Balakrishnan used some well-known recurrence relations among moments of order statistics and Legendre polynomials in order to obtain some interesting combinatorial identities, some of which agree with the known identities in Riordan Joshi and Shubha gave some new identities which are more general in nature and are applicable when moments of some extreme order statistics do not exist.
Saran and Pushkarna have derived some identities among moments of order statistics, when moments of some lower order statistics do not exist by using generalized and extended forms of Legendre polynomials. These identities are also applicable even when all the moments exist. Saran and Pushkarna have derived some new identities among c. For similar other work, one may refer to Joshi and Balakrishnan and Balakrishnan and Sultan In this study, we propose to establish some general identities among c.
These results generalize some of the results of Joshi and Balakrishnan and Saran and Pushkarna The Legendre polynomials L n t in the finite interval [a,b] are defined Sansone, as:. Integrating Eq. Further, some other combinatorial identities can also be derived by applying Eq. Relation in Eq. Also, David and Nagaraja, Legendre polynomials L n t relative to the interval [a,b] defined in Eq.
Sansone, Remark 2 It may be noted that some combinatorial identities can be derived by applying the relation given in Eq. Expanding t-a n and t-b n binomially in powers of t, a and b and then differentiating term by term, Eq. It can easily be shown by repeated integration by parts that the Fourier coefficient of pt p-1 with respect to L n t is given by:. Now multiplying Eq. In this study, some general identities among c. Further, these results generalize some of the results of Joshi and Balakrishnan and Saran and Pushkarna The authors are grateful to the referees for giving valuable comments that led to an improvement in the presentation of the study.
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