F. Calogero

Properties of the zeros of the classical polynomials and of the Bbessel functions

SummaryMany algebraic equations satisfied by the zeros of the classical polynomials and of the Bessel functions are reported. Some of them are collected from recent papers; several of them are new; most of them display remarkable diophantine features. Certain matrices constructed with arbitrary numbers rather than the zeros of special functions, but displaying analogous diophantine properties, are also exhibited.RiassuntoSi riportano molte equazioni algebriche soddisfatte dagli zeri dei polinomi classici e delle funzioni di Bessel. Alcune sono riprese da lavori recenti; numerose sono nuove; la maggioranza esibisce notevoli proprietà di carattere diofantino. Si esibiscono inoltre alcune matrici, costruite con numeri arbitrari anziché con gli zeri di funzioni speciali, che anche loro posseggono però delle notevoli proprietà diofantine.РезюмеПриводится большое число алгебраических уравнений, которым удовлетворяют нули классических полиномов и бесселевых функций. Некоторые из уравнений взяты из недавних работ, некоторые из них новые. Большинство из них обнаруживают интересные особенности. Указываются некоторые матрицы, сконструированные с помощью произвольного числа нулей специальных функций.

Lagrangian interpolation and differentiation

SummaryThis paper outlines the connection between some recent results on matrices, differential operators and polynomials, and classical results on the Lagrange interpolation of a function and its derivatives. The focus is on the connotation of the two matrices of ordern,Xij =δjkxj andnn

Interpolation, differentiation and solution of eigenvalue problems for periodic functions

Z_{jk} = delta _{jk} sumlimits_{m = 1}^n {prime left( {x_j - x_m } right)^{ - 1} + } left( {1 - delta _{jk} } right)left( {x_j - x_k } right)^{ - 1}

Interpolation and differentiation for periodic functions

nn, defined in terms ofn arbitrary (different) numbersxj, as representations of the two operatorsx and d/dx.

Finite transformations of certain isospectral matrices

SummaryA technique is presented for computing the eigenvaluesλm of the regular Sturm-Liouville problemd2(x)ψm′’(x) +dI(x)ψmt’(x) +d0(x)ψm(x) = λmψm(x),ψm(b) =ψm(c) = 0. They are approximated by the eigenvaluesglm =λm-δλm of a (n ×n)-matrixA, given by a simple explicit formula in terms of the three functionsdp(x) evaluated atn (arbitrary) pointsxj, and of the distances of these points from the boundariesb andc. The rate of convergence asn → ∞ is estimated to be quite fast, as suggested by the formula, applicable for equally spaced xj’s in the interval (b,c), |δλm/λm|~(πm/2n)n-2.

Disproof of a conjecture

SummaryIt is shown that the (n×n)-matricesX andnn

Coupled nonlinear evolution equations solvable via the inverse spectral transform, and solitons that come back: the boomeron

tilde Z