TY - JOUR
AU - S. M. Zagorodnyuk
PY - 2022/01/24
Y2 - 2022/08/09
TI - On the orthogonality of partial sums of the generalized hypergeometric series
JF - Ukrainsâ€™kyi Matematychnyi Zhurnal
JA - Ukr. Mat. Zhurn.
VL - 74
IS - 1
SE - Research articles
DO - 10.37863/umzh.v74i1.6989
UR - http://umj.imath.kiev.ua/index.php/umj/article/view/6989
AB - UDC 517.587It 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.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.
ER -