@article{Zagorodnyuk_2022, title={On the orthogonality of partial sums of the generalized hypergeometric series}, volume={74}, url={http://umj.imath.kiev.ua/index.php/umj/article/view/6989}, DOI={10.37863/umzh.v74i1.6989}, abstractNote={<p>UDC 517.587</p> <p>It turns out that the partial sums $g_n(z)=\displaystyle\sum
olimits_{k=0}^n\dfrac{(a_1)_k\ldots(a_p)_k}{(b_1)_k\ldots(b_q)_k}\,\dfrac{z^k}{k!}$ of the generalized hypergeometric series ${}_p F_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z)$ with parameters $a_j, b_l\in\mathbb{C}\backslash\{0,-1,-2,\ldots\}$ are Sobolev orthogonal polynomials.<br>The corresponding monic polynomials $G_n(z)$ are $R_I$-type polynomials, and therefore, they are related to biorthogonal rational functions.<br>The polynomials $g_n$ satisfy a differential equation (in $z$) and a recurrence relation (in $n$).<br>In this paper, we study the integral representations for $g_n$ and their basic properties.<br>It is shown that partial sums of arbitrary power series with non-zero coefficients are also related to biorthogonal rational functions.<br>For polynomials $g_n(z),$ we obtain a relation to Jacobi-type pencils and their associated polynomials.</p>}, number={1}, journal={Ukrainsâ€™kyi Matematychnyi Zhurnal}, author={Zagorodnyuk, S. M.}, year={2022}, month={Jan.}, pages={36 - 44} }