Bispectrality of Multivariable Racah–Wilson Polynomials

Bispectrality of Multivariable Racah-wilson Polynomials

We construct a commutative algebra Ax of difference operators in R, depending on p + 3 parameters which is diagonalized by the multivariable Racah polynomials Rp(n; x) considered by Tratnik [27]. It is shown that for specific values of the variables x = (x1, x2, . . . , xp) there is a hidden duality between n and x. Analytic continuation allows us to construct another commutative algebra An in ...

Squarefree Values of Multivariable Polynomials

Given f ∈ Z[x1, . . . , xn], we compute the density of x ∈ Z such that f(x) is squarefree, assuming the abc conjecture. Given f, g ∈ Z[x1, . . . , xn], we compute unconditionally the density of x ∈ Z such that gcd(f(x), g(x)) = 1. Function field analogues of both results are proved unconditionally. Finally, assuming the abc conjecture, given f ∈ Z[x], we estimate the size of the image of f({1, ...

Multivariable Construction of Extended Jacobi Matrix Polynomials

The main aim of this paper is to construct a multivariable extension with the help of the extended Jacobi matrix polynomials (EJMPs). Generating matrix functions and recurrence relations satisfied by these multivariable matrix polynomials are derived. Furthermore, general families of multilinear and multilateral generating matrix functions are obtained and their applications are presented.

Miscellaneous properties of some multivariable polynomials

Bispectrality of Kp Solitons

It is by now well known that the wave functions of rational solutions to the KP hierarchy which can be achieved as limits of the pure n-soliton solutions satisfy an eigenvalue equation for ordinary differential operators in the spectral parameter. This property is known as “bispectrality” and has proved to be both interesting and useful. In this note, it is shown that certain (non-rational) sol...