Using a recent Mergelyan type theorem for products of planar compact sets, we establish generic existence universal Taylor series on simply connected domains \({\varOmega }_i\), \(i=1,\ldots ,d\). The approximation is realized by partial sums the development function sets \(K_i\), ,d\) such that \({\mathbb {C}}-K_i\) and at least one \(i_0\) set \(K_{i_0}\) disjoint from }_{i_0}\).

In this paper, we deal with the convolution series that are a far reaching generalization of conventional power and fractional exponents including Mittag-Leffler type functions. Special attention is given to most interesting case generated by Sonine kernels. first formulate prove second fundamental theorem for general integrals $n$-fold sequential derivatives both Riemann-Liouville Caputo types...

Abstract We describe an expansion of Legendre polynomials, analogous to the Taylor expansion, to approximate arbitrary functions. We show that the polynomial coefficients in Legendre expansion, thus, the whole series, converge to zero much more rapidly compared to the Taylor expansion of the same order. Furthermore, using numerical analysis with sixth-order polynomial expansion, we demonstrate ...