We consider the differential equations y = λ0(x)y ′ + s0(x)y, where λ0(x), s0(x) are C −functions. We prove (i) if the differential equation, has a polynomial solution of degree n > 0, then δn = λnsn−1 − λn−1sn = 0, where λn = λ ′ n−1 + sn−1 + λ0λn−1 and sn = s ′ n−1 + s0λk−1, n = 1, 2, . . . . Conversely (ii) if λnλn−1 6= 0 and δn = 0, then the differential equation has a polynomial solution o...

Hough functions are the eigenfunctions of the Laplace tidal equation governing fluid motion on a rotating sphere with a resting basic state. Several numerical methods have been used in the past. In this paper, we compare two of those methods: normalized associated Legendre polynomial expansion and Chebyshev collocation. Both methods are not widely used, but both have some advantages over the co...

We give a remarkable additional othogonality property of the classical Legendre polynomials on the real interval [−1, 1]: polynomials up to degree n from this family are mutually orthogonal under the arcsine measure weighted by the degree-n normalized Christoffel function.

We show how lattice paths and the reflection principle can be used to give easy proofs of unimodality results. In particular, we give a “one-line” combinatorial proof of the unimodality of the binomial coefficients. Other examples include products of binomial coefficients, polynomials related to the Legendre polynomials, and a result connected to a conjecture of Simion.