On the orthogonality of partial sums of the generalized hypergeometric series
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...
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| Date: | 2022 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian |
| Published: |
Institute of Mathematics, NAS of Ukraine
2022
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/6989 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | 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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| DOI: | 10.37863/umzh.v74i1.6989 |