On the orthogonality of partial sums of the generalized hypergeometric series
Abstract
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.
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