Using the theory of orthogonal polynomials, their associated recursion relations and differential formulas we develop a new method for evaluating integrals that include orthogonal polynomials. The method is illustrated by obtaining the following integral result that involves the Bessel function and associated Laguerre polynomial: integral(infinity)(0) x(v)e(-x/2) J(v)(mu x)L-n(2v)(x)dx = 2(v)Ga...

Abstract. We use a multidimensional extension of Bailey’s transform to derive two very general q-generating functions, which are q-analogues of a paper by Exton [7]. These expressions are then specialised to give more practical formulae, which are q-analogues of generating relations for Karlssons generalised Kampe de Fériet function. A number of examples are given including q-Laguerre polynomia...

Let Cλ n(x), n = 0, 1, . . . , λ > −1/2, be the ultraspherical (Gegenbauer) polynomials, orthogonal in (−1, 1) with respect to the weight function (1−x2)λ−1/2. Denote by xnk(λ), k = 1, . . . , n, the zeros of Cλ n(x) enumerated in decreasing order. In this short note we prove that, for any n ∈ IN , the product (λ+1)xn1(λ) is a convex function of λ if λ ≥ 0. The result is applied to obtain some ...

Orthogonality of the Jacobi and of Laguerre polynomials, P (α,β) n and L (α) n , is established for α, β ∈ C \ Z−, α + β 6= −2,−3, . . . using the Hadamard finite part of the integral which gives their orthogonality in the classical cases. Riemann-Hilbert problems that these polynomials satisfy are found. The results are formally similar to the ones in the classical case (when Rα,Rβ > −1).

"A simple result deserves a simple proof" I Objects lying in four different boxes are rearranged in such a way that the number of objects where the bottom is a rearrangement of the top. The weight W (T) of the multiset permutation T is defined to be (-I) ~-where U, is the number of columns of the form I: 111122333444 , x # y. For example, if n-= 441223133411 then U, = 7 and w (n-) = (-~) ~ =-1....

Arithmetic Asian or average price options deliver payoffs based on the average underlying price over a prespecified time period. Asian options are an important family of derivative contracts with a wide variety of applications in currency, equity, interest rate, commodity, energy, and insurance markets. We derive two analytical formulas for the value of the continuously sampled arithmetic Asian...

A set of orthogonal polynomials with 8 independent “q’s” is defined which generalizes the Laguerre polynomials. The moments of the measure for these polynomials are the generating functions for permutations according to eight different statistics. Specializing these statistics gives many other well-known sets of combinatorial objects and relevant statistics. The specializations are studied, wit...

In this note, we have obtained a novel extension of a bilateral generating relationsinvolvingbiorthogonal polynomials ; from the existence of quasi-bilinear generating relation by group theoretic method. As particular cases, we obtain the corresponding results on generalised Laguerre polynomials.