On the orthogonality of partial sums of the generalized hypergeometric series

• Datum: 2022
• Hauptverfasser: Zagorodnyuk, S. M., Загороднюк , С. М.
• Format: Artikel
• Sprache: Ukrainisch
• Veröffentlicht: Institute of Mathematics, NAS of Ukraine 2022
• Online Zugang: https://umj.imath.kiev.ua/index.php/umj/article/view/6989
• Назва журналу: Ukrains’kyi Matematychnyi Zhurnal

UDC 517.587 It turns out that the partial sums $g_n(z)=\displaystyle\sum\nolimits_{k=0}^n\dfrac{(a_1)_k\ldots(a_p)_k}{(b_1)_k\ldots(b_q)_k}\,\dfrac{z^k}{k!}$ 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.The corresponding monic polynomials $G_n(z)$ are $R_I$-type polynomials, and therefore, they are related to biorthogonal rational functions.The polynomials $g_n$ satisfy a differential equation (in $z$) and a recurrence relation (in $n$).In this paper, we study the integral representations for $g_n$ and their basic properties.It is shown that partial sums of arbitrary power series with non-zero coefficients are also related to biorthogonal rational functions.For polynomials $g_n(z),$ we obtain a relation to Jacobi-type pencils and their associated polynomials.