On differential equations for Sobolev-type Laguerre polynomials

Abstract

Abstract. The Sobolev-type Laguerre polynomials {L α,M,N n (x)} ∞ n=0 are orthogonal with respect to the inner product,In 1990 the first and second author showed that in the case M > 0 and N = 0 the polynomials are eigenfunctions of a unique differential operator of the formwhereare independent of n. This differential operator is of order 2α + 4 if α is a nonnegative integer, and of infinite order otherwise.In this paper we construct all differential equations of the formwhere the coefficientsare independent of n and the coefficients a 0 (x), b 0 (x) and c 0 (x) are independent of x, satisfied by the Sobolev-type Laguerre polynomials {L α,M,N n (x)} ∞ n=0 . Further, we show that in the case M = 0 and N > 0 the polynomials are eigenfunctions of a linear differential operator, which is of order 2α + 8 if α is a nonnegative integer and of infinite order otherwise.Finally, we show that in the case M > 0 and N > 0 the polynomials are eigenfunctions of a linear differential operator, which is of order 4α + 10 if α is a nonnegative integer and of infinite order otherwise.

Differential equations for generalized Jacobi polynomials

“…Hence we have (12). Note that in the preceding section we have already determined the 'eigenvalue' coefficients a 0 (n, α, β), b 0 (n, α, β) and c 0 (n, α, β).…”

“…but this is not really necessary in view of (12). In order to prove (8) we apply the definition (21) to…”

“…See also [11] for more details. In [12] we used this inversion method to find all differential equations of the form…”

Bispectral Laguerre type polynomials

“…Other examples of bispectral polynomials are the Krall-Sobolev polynomials (see [24,1,9,10]), the exceptional polynomials (see [14,4,5,6,7,13], and references therein) or the Grünbaum and Haine extension of Krall polynomials ( [16]; see also [19,3]). In these cases, the associated operators (in the discrete and continuous variable) have order greater than 2.…”

Untitled

“…досліджувалися в багатьох роботах, серед яких [1][2][3][4]. У цих та інших відомих нам публікаціях розглядалися узагальнення всіх класичних ортогональних поліномів, окрім узагальнених поліномів Ерміта.…”