Matrices (A) under bar and (B) under bar of arbitrary rank N, given by simple expressions in terms of the N zeros of certain Laguerre or para-Jacobi polynomials of degree N, feature a Diophantine property. In the Laguerre case, this property states that the 2N zeros of the polynomial p(2N) (lambda) = det[lambda(2) + lambda(A) under bar + (B) under bar] are all integers; indeed we conjecture that det [lambda(2) + lambda(A) under bar + (B) under bar] = Pi(N)(k=1) (lambda(2) - k(2)). The results in the para-Jacobi case are somewhat analogous; they refer to the zeros of the general solution of the ODE generally characterizing Jacobi polynomials, in the special case in which this solution is a para- Jacobi polynomial featuring an additional, arbitrary parameter.
Diophantine properties of the zeros of certain Laguerre and para-Jacobi polynomials / Calogero, Francesco; Yi, Ge. - In: JOURNAL OF PHYSICS. A, MATHEMATICAL AND THEORETICAL. - ISSN 1751-8113. - 45:9(2012), p. 095207. [10.1088/1751-8113/45/9/095207]
Diophantine properties of the zeros of certain Laguerre and para-Jacobi polynomials
CALOGERO, Francesco;
2012
Abstract
Matrices (A) under bar and (B) under bar of arbitrary rank N, given by simple expressions in terms of the N zeros of certain Laguerre or para-Jacobi polynomials of degree N, feature a Diophantine property. In the Laguerre case, this property states that the 2N zeros of the polynomial p(2N) (lambda) = det[lambda(2) + lambda(A) under bar + (B) under bar] are all integers; indeed we conjecture that det [lambda(2) + lambda(A) under bar + (B) under bar] = Pi(N)(k=1) (lambda(2) - k(2)). The results in the para-Jacobi case are somewhat analogous; they refer to the zeros of the general solution of the ODE generally characterizing Jacobi polynomials, in the special case in which this solution is a para- Jacobi polynomial featuring an additional, arbitrary parameter.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.