On series of orthogonal polynomials and systems of classical type polynomials
UDC 517.587 If $\displaystyle\sum\nolimits_{k=0}^\infty c_k g_k(x),$ is a formal series of orthonormal polynomials $g_k(x)$ on the real line that has positive coefficients $c_k,$ then its partial sums $u_n(x)$ are associated with Jacobi type pencils.Therefore, they possess a recurrence relation and...
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| Date: | 2021 |
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| Main Authors: | , , |
| Format: | Article |
| Language: | Ukrainian |
| Published: |
Institute of Mathematics, NAS of Ukraine
2021
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/6527 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | UDC 517.587
If $\displaystyle\sum\nolimits_{k=0}^\infty c_k g_k(x),$ is a formal series of orthonormal polynomials $g_k(x)$ on the real line that has positive coefficients $c_k,$ then its partial sums $u_n(x)$ are associated with Jacobi type pencils.Therefore, they possess a recurrence relation and special orthonormality conditions.The cases where $g_k(x)$ are Jacobi or Laguerre polynomials will be of a special interest.For a suitable choice of parameters~$c_k,$ the partial sums $u_n(x)$ are Sobolev orthogonal polynomials with a $(3\times 3)$ matrix measure.A~further selection of parameters gives differential equations for $u_n.$In this case, polynomials $u_n(x)$ are solutions to generalized eigenvalue problems both in $x$ and in $n.$ |
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| DOI: | 10.37863/umzh.v73i6.6527 |