On the representation of analytic functions by series of derived bases of polynomials in hyperelliptical regions

One of the important themes in complex analysis is the expansion of analytic functions by infinite series in a given sequence of bases of polynomials. In the present paper, we investigated the representation of analytic functions in different domains of derived bases of polynomials. The behaviour of the associated representation of whole functions is directly related to determining the convergence properties (effectiveness) of such bases. The representation domains are closed hyperellipses, open hyperellipses, and closed regions surrounding a closed hyperellipse. Also, some results concerning the order of derived bases in hyperellipse are obtained. The results obtained are natural generalisations of the results obtained in hyperspherical regions.