**INTRODUCTION**

Suppose X_{1}, X_{2}, ..., X_{n} are n i.i.d. variates,
each with c.d.f. F(x) and p.d.f. f(x). Rearranging X_{i}’s in increasing
order of magnitude, we obtain corresponding order statistics X_{1:n}≤X_{2:n}≤...≤X_{n:n}.
Then the c.d.f. of the rth order statistic X_{r:n}, 1≤r≤n, is
given by Arnold *et al*. (1992),

We denote the k-th moment of X_{r:n} by
i.e.,

Several recurrence relations satisfied by the single and product moments of
order statistics are available in the literature, which are highly useful for
the computation of the moments of order statistics in a simple recursive manner.
Likewise, several identities satisfied by these moments are also available which
are quite useful in checking the computation of the moments of order statistics.
For example, Joshi (1973) has given two simple identities
among single moments of order statistics and applied them in proving some combinatorial
identities. Joshi and Balakrishnan (1981) 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 (1968).
Joshi and Shubha (1991) 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 (1996)
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 (1998) have
derived some new identities among c.d.f.’s and single moments of order
statistics by using Legendre polynomials in the interval [-1,1] and applied
these identities to obtain some combinatorial identities. For similar other
work, one may refer to Joshi and Balakrishnan (1982)
and Balakrishnan and Sultan (1998).

In this study, we propose to establish some general identities among c.d.f.’s
and single moments of order statistics by using Legendre polynomials in the
interval [a,b]. These identities are then applied to obtain some new combinatorial
identities. These results generalize some of the results of
Joshi and Balakrishnan (1981) and Saran and Pushkarna
(1998).

**APPLICATIONS OF LEGENDRE POLYNOMIALS**

The Legendre polynomials L_{n}(t) in the finite interval [a,b] are
defined (Sansone, 1959) as:

It follows that:

the first expression coming from the application of Leibnitz rule and the binomial
theorem in Eq. 3, the second expression coming from Eq.
3 by expanding (t-a)^{n} binomially in powers of t and a and writing
(t-b)^{n} as [(t-1)+(1-b)]^{n} and expanding it binomially in
powers of (t-1) and (1-b) and the third expression coming from Eq.
3 by writing (t-a)^{n} as [(t-1)+(1-a)]^{n} and (t-b)^{n}
as [(t-1)+(1-b)]^{n} and expanding each of them binomially.

Integrating Eq. 4 from 0 to F(x) and using Eq.
1, we get an identity among c.d.f.’s of order statistics given in the
following theorem.

**Theorem 1 **For an arbitrary c.d.f. F(x):

The corresponding identities in terms of moments of order statistics are given
below:

where, k =1, 2,… .

**Deductions**

Setting and
t = 1 in Eq. 4, we get, respectively, the following combinatorial
identities:

and

Similarly, letting x→∞ in Eq. 5, or, equivalently,
putting k = 0 in Eq. 6, we get:

Further, some other combinatorial identities can also be derived by applying
Eq. 6 to some specific distributions for which the moments
of order statistics are known to have an explicit expression. For example, consider
the exponential distribution with density function f(x) = e¯^{x},
x≥0, for which (David and Nagaraja, 2003)

where, T_{0} = 0 and for t≥1,

Relation in Eq. 6, for k = 1, then applied to these moments
gives the identity:

Similarly, for power function distribution with density function f(x) = ux^{u-1},
0≤x≤1, u>0 (for u = 1, it is the uniform distribution in [0,1]), Malik
(1967) has shown that, for k≥0:

(Also, David and Nagaraja, 2003).

Relation in Eq. 6 then applied to these moments gives for
k = 0, 1, 2,… the combinatorial identity:

**Remark 1**

It may be noted that for a = -1 and b = 1, the results of this section will
reduce to the corresponding results of Saran and Pushkarna
(1998).

**LINEAR COMBINATION OF LEGENDRE POLYNOMIALS**

Legendre polynomials L_{n}(t) relative to the interval [a,b] defined
in Eq. 3, satisfies the following relation:

(Sansone, 1959).

Differentiating the above equation with respect to t, k times, we get:

where c = 1-b and d = 1-a.

Integrating Eq. 16 from 0 to F(x) and using
Eq. 1, we get an identity among c.d.f.’s of order statistics given
in the following theorem.

**Theorem 2**

For an arbitrary c.d.f. F(x):

The corresponding identity in terms of moments of order statistics is given
below:

where c and d are as defined in Eq. 16.

**Remark 2**

It may be noted that some combinatorial identities can be derived by applying
the relation given in Eq. 18 to the moments of order statistics
from some specific distributions such as exponential and power function distributions,
as discussed above.

**FOURIER COEFFICIENTS**

Expanding (t-a)^{n} and (t-b)^{n} binomially in powers of t,
a and b and then differentiating term by term, Eq. 3 implies:

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. 19 by pt^{p-1} and integrating
with respect to t, from a to b and then equating it with Eq.
20, we get the following combinatorial identity:

**Remark 3**

It may be noted that by setting a = 0 and b = 1, the results of this section
will reduce to the corresponding results of Joshi and Balakrishnan
(1981).

**CONCLUSION**

In this study, some general identities among c.d.f.’s and single moments
of order statistics have been established by using Legendre polynomials in the
interval [a,b]. These identities are then applied to obtain some new combinatorial
identities. Further, these results generalize some of the results of Joshi
and Balakrishnan (1981) and Saran and Pushkarna (1998).

**ACKNOWLEDGMENT**

The authors are grateful to the referees for giving valuable comments that led to an improvement in the presentation of the study.