We discuss the symmetric homogeneous polynomial solutions of the generalized Laplace’s equation which arises in the context of the Calogero-Sutherland model on a line. The solutions are expressed as linear combinations of Jack polynomials and the constraints on the coefficients of expansion are derived. These constraints involve generalized binomial coefficients defined through Jack polynomials...

A systematic procedure for generating certain identities involving elementary symmetric functions is proposed. These identities, as particular cases, lead to new identities for binomial and q-binomial coefficients.

If p is a prime and n a positive integer, let νp(n) denote the exponent of p in n, and up(n) = n/p νp(n) the unit part of n. If α is a positive integer not divisible by p, we show that the p-adic limit of (−1) up((αp)!) as e → ∞ is a well-defined p-adic integer, which we call zα,p. In terms of these, we then give a formula for the p-adic limit of ( ap+c bpe+d ) as e → ∞, which we call ( ap∞+c b...

The ^-binomial coefficient is a polynomial in q . Given an integer t and a residue class r modulo ;, a sieved ^-binomial coefficient is the sum of those terms whose exponents are congruent to r modulo /. In this paper explicit polynomial identities in q are given for sieved ij-binomial coefficients. As a limiting case, generating functions for the sieved partition function are found as multidim...

Abstract The Levinson–Durbin recursion is used to construct the coefficients which define the minimum mean square error predictor of a new observation for a discrete time, second-order stationary stochastic process. As the sample size varies, the coefficients determine what is called a Levinson–Durbin sequence. A generalized Levinson– Durbin sequence is also defined, and we note that binomial c...

This paper introduces a method for finding closed forms for certain sums involving squares of binomial coefficients. We use this method to present an alternative approach to a problem of evaluating a different type of sums containing squares of the numbers from Catalan's triangle.

Let p > 3 be a prime. We show that T p−1 ≡ p 3 3 p−1 (mod p 2), where the central trinomial coefficient T n is the constant term in the expansion of (1+x+x −1) n. We also prove three congruences conjectured by Sun one of which is as follows: p−1 k=0 p − 1 k 2k k ((−1) k − (−3) −k) ≡ p 3 (3 p−1 − 1) (mod p 3).

Let p be a prime and let a be a positive integer. In this paper we determine ∑pa−1 k=0 ( 2k k+d ) /mk and ∑p−1 k=1 ( 2k k+d ) /(kmk−1) modulo p for all d = 0, . . . , pa, where m is any integer not divisible by p. For example, we show that if p 6= 2, 5 then p−1 ∑

In this paper we obtain some congruences involving central binomial coefficients and Lucas sequences. For example, we show that if p > 5 is a prime then p−1

We prove that, for every integer a, real numbers k and ℓ, and nonnegative integers n, i and j, i+j=n a i + k − ℓ i a j + ℓ j = i+j=n a i + k i a j j , by presenting explicit expressions for its value. We use the identity to generalize a recent result of Chang and Xu, and end the paper by presenting, in explicit form, the ordinary generating function of the sequence 2n+k n ∞ n=0 , where k ∈ R.