On the orthogonality of partial sums of the generalized hypergeometric series - Zagorodnyuk, S. M. (Ukrains’kyi Matematychnyi Zhurnal, 2022)
UDC 517.587 It turns out that the partial sums g_n(z)=olimits_k=0^n(a_1)_k…(a_p)_k(b_1)_k…(b_q)_k z^kk! of the generalized hypergeometric series _p F_q(a_1,…,a_p;b_1,…,b_q;z) with parameters a_j, b_lC{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.