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After recalling the most important properties of the Bell polynomials, we show how to approximate a positive compact operator by a suitable matrix. Then, we derive a representation formula for functions of the obtained matrix, which can be considered as an approximate value for the functions of the corresponding operator.

In this paper both upper bounds and lower bounds are given for the first eigenvalues of a rhombical membrane with free boundary. The upper bounds are obtained through an application of the Rayleigh-Ritz method. The lower bounds are calculated by means of a procedure relying on the Orthogonal Invariants method, which is inspired, in view of the lack of a known Green function, by that one developed by G. Fichera in a work on the Neumann eigenvalue problem.

By using representation formulas based on the Bell polynomials we derive some relations verified by the orthogonal invariants of the eigenvalues of hermitian compact operators, including some preceding results of Didier Robert.

After recalling the most important properties of the Bell polynomials, we show how to approximate a positive compact operator by a suitable matrix. Then, we derive a representation formula for functions of the obtained matrix, which can be considered as an approximate value for the functions of the corresponding operator. [ABSTRACT FROM AUTHOR]