The paper deals with the problem of construction and investigation of branched continued fraction expansions of special functions of several variables. We give some recurrence relations of Horn hypergeometric functions H3. By these relations the branched continued fraction expansions of Horn’s hypergeometric function H3 ratios have been constructed. We have established some convergence criteria for the above-mentioned branched continued fractions with elements in R2 and C2. In addition, it is proved that the branched continued fraction expansions converges to the functions which are an analytic continuation of the above-mentioned ratios in some domain (here domain is an open connected set). Application for some system of partial differential equations is considered.

We consider polynomials on spaces ℓ (C ), 1 ≤ < +∞, of -summing sequences of -dimensional complex vectors, which are symmetric with respect to permutations of elements of the sequences, and describe algebraic bases of algebras of continuous symmetric polynomials on ℓ (C ).

The paper contains a description of a symmetric convolution of the algebra of block-symmetric analytic functions of bounded type on 1 -sum of the space C 2 . We show that the specrum of such algebra does not coincide of point evaluation functionals and we describe characters of the algebra as functions of exponential type with plane zeros.

We consider so called block-symmetric polynomials on sequence spaces $\ell_1\oplus \ell_{\infty}, \ell_1\oplus c, \ell_1\oplus c_0,$ that is, polynomials which are symmetric with respect to permutations of elements of the sequences. It is proved that every continuous block-symmetric polynomials on $\ell_1\oplus \ell_{\infty}$ can be uniquely represented as an algebraic combination of some special block-symmetric polynomials, which form an algebraic basis. It is interesting to note that the algebra of block-symmetric polynomials is infinite-generated while $\ell_{\infty}$ admits no symmetric polynomials. Algebraic bases of the algebras of block-symmetric polynomials on $\ell_1\oplus \ell_{\infty}$ and $\ell_1\oplus c_0$ are described.

The classical Newton formulas gives recurrent relations between algebraic bases of symmetric polynomials. They are true, of course, for symmetric polynomials on infinite-dimensional sequences Banach space. In this paper we consider block-symmetric polynomials (or MacMahon symmetric polynomials) on Banach spaces $\ell_p(\mathbb{C}^s),$ $1\le p\le \infty.$ We prove an analogue of the Newton formula for the block-symmetric polynomials for the case $p=1.$ In the case $1< p$ we have no classical elementary block-symmetric polynomials. However, we extend the obtained Newton type formula for $\ell_1(\mathbb{C}^s)$ to the case of $\ell_p(\mathbb{C}^s),$ $1< p\le \infty$ and by this way found a natural definition of elementary block-symmetric polynomials on $\ell_p(\mathbb{C}^s).$