On the orthogonality of partial sums of the generalized hypergeometric series - Zagorodnyuk, S. M. (Ukrains’kyi Matematychnyi Zhurnal, 2022)
UDC 517.587It turns out that the partial sums $g_n(z)=\displaystyle\sum olimits_{k=0}^n\dfrac{(a_1)_k…(a_p)_k}{(b_1)_k…(b_q)_k}\,\dfrac{z^k}{k!}$ of the generalized hypergeometric series ${}_p F_q(a_1,…,a_p;b_1,…,b_q;z)$ with parameters $a_j, b_l\in\mathbb{C}\backslash\{0,-1,-2,…\}$ 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.