Semiclassical Multiple Orthogonal Polynomials and the Properties of Jacobi–Bessel Polynomials

Abstract

This paper deals with Hermite Pade polynomials in the case where the multiple orthogonality condition is related to semiclassical functionals. The polynomials, introduced in such a way, are a generalization of classical orthogonal polynomials (Jacobi, Laguerre, Hermite, and Bessel polynomials). They satisfy a Rodrigues type formula and an (s+2)-order differential equation, where s is the class of the semiclassical functional. A special case of polynomials, multiple orthogonal with respect to the semiclassical weight function w(x)=x : 0 (x&a) : 1 e #Âx (a combination of the classical weights of Jacobi and Bessel), is analyzed in order to obtain the strong (Szego type) asymptotics and the zero distribution.1997 Academic Press

Automorphisms of Algebras and Bochner's Property for Vector Orthogonal Polynomials

“…Applications of d-orthogonal polynomials include the simultaneous Padé approximation problem where the multiple orthogonal polynomials appear [2,3,4,5,14]. Multiple orthogonal polynomials play an important role in random matrix theory [4,10].…”

Classical 2‐orthogonal polynomials and differential equations

“…From the above propositions (i.e., C n = 0), the coefficient of P (3) n+3 (x) is independent of n (a fortiori S 3,n (x) is independent of n).…”

“…In conclusion, we have just shown that there are four types of linear third-order differential equations R 4,n (x)P (3) n+3 (x) + R 3,n (x)P n+3 (x) + R 2,n (x)P n+3 (x) + R 1,n (x)P n+3 (x) = 0, n ≥ 0, (4.114) having as solutions classical 2-orthogonal polynomials, namely, (i) equation ( . Furthermore, the coefficients of (4.83) and (4.102) and the coefficients of the four-term recurrence relations associated with the solutions of these equations are derived.…”

“…For the type II multiple orthogonal polynomials, there is a rich bibliography [2][3][4]20]. Here, we quote some classical 2-orthogonal polynomial sequences which were a subject of a deep study and whose generating functions and integral representations of the linear forms £ 0 and £ 1 have been established [6][7][8][9][10][11].…”

Об Асимптотическом Поведении Полуклассических Полиномов Совместной Ортогональности Типа Бесселя

“…Здесь мы будем рассматривать случай s = 1. В работе [7] дана классификация 7 полуклассических весов, которые определяются взаимным расположением особых точек уравнения Пирсона (3):…”