The derivative connecting problems for some classical polynomials
Given two polynomial sets $\{ P_n(x) \}_{n\geq 0},$ and $\{ Q_n(x) \}_{n\geq 0}$ such that $$\deg ( P_n(x) ) =n, \deg ( Q_n(x) )=n.$$ The so-called the connecting problem between them asks to find the coefficients $\alpha_{n,k}$ in the expression $\displaystyle Q_n(x) =\sum_{k=0}^{n} \alpha_{n,k} P_...
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Vasyl Stefanyk Precarpathian National University
2019-12-01
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doaj-6830865e235b490d8e2e9429d6046d5a2020-11-25T03:15:13ZengVasyl Stefanyk Precarpathian National UniversityKarpatsʹkì Matematičnì Publìkacìï2075-98272313-02102019-12-0111243144110.15330/cmp.11.2.431-4412121The derivative connecting problems for some classical polynomialsA. Ramskyi0N. Samaruk1O. Poplavska2Khmelnytskyi National University, 11 Instytytska str., 29016, Khmelnytskyi, UkraineKhmelnytskyi National University, 11 Instytytska str., 29016, Khmelnytskyi, UkraineKhmelnytskyi National University, 11 Instytytska str., 29016, Khmelnytskyi, UkraineGiven two polynomial sets $\{ P_n(x) \}_{n\geq 0},$ and $\{ Q_n(x) \}_{n\geq 0}$ such that $$\deg ( P_n(x) ) =n, \deg ( Q_n(x) )=n.$$ The so-called the connecting problem between them asks to find the coefficients $\alpha_{n,k}$ in the expression $\displaystyle Q_n(x) =\sum_{k=0}^{n} \alpha_{n,k} P_k(x).$ Let $\{ S_n(x) \}_{n\geq 0}$ be another polynomial set with $\deg ( S_n(x) )=n.$ The general connection problem between them consists in finding the coefficients $\alpha^{(n)}_{i,j}$ in the expansion $$Q_n(x) =\sum_{i,j=0}^{n} \alpha^{(n)}_{i,j} P_i(x) S_{j}(x).$$ The connection problem for different types of polynomials has a long history, and it is still of interest. The connection coefficients play an important role in many problems in pure and applied mathematics, especially in combinatorics, mathematical physics and quantum chemical applications. For the particular case $Q_n(x)=P'_{n+1}(x)$ the connection problem is called the derivative connecting problem and the general derivative connecting problem associated to $\{ P_n(x) \}_{n\geq 0}.$ In this paper, we give a closed-form expression of the derivative connecting problems for well-known systems of polynomials.https://journals.pnu.edu.ua/index.php/cmp/article/view/2121connection probleminversion problemderivative connecting problemconnecting coefficientsorthogonal polynomials |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
A. Ramskyi N. Samaruk O. Poplavska |
spellingShingle |
A. Ramskyi N. Samaruk O. Poplavska The derivative connecting problems for some classical polynomials Karpatsʹkì Matematičnì Publìkacìï connection problem inversion problem derivative connecting problem connecting coefficients orthogonal polynomials |
author_facet |
A. Ramskyi N. Samaruk O. Poplavska |
author_sort |
A. Ramskyi |
title |
The derivative connecting problems for some classical polynomials |
title_short |
The derivative connecting problems for some classical polynomials |
title_full |
The derivative connecting problems for some classical polynomials |
title_fullStr |
The derivative connecting problems for some classical polynomials |
title_full_unstemmed |
The derivative connecting problems for some classical polynomials |
title_sort |
derivative connecting problems for some classical polynomials |
publisher |
Vasyl Stefanyk Precarpathian National University |
series |
Karpatsʹkì Matematičnì Publìkacìï |
issn |
2075-9827 2313-0210 |
publishDate |
2019-12-01 |
description |
Given two polynomial sets $\{ P_n(x) \}_{n\geq 0},$ and $\{ Q_n(x) \}_{n\geq 0}$ such that $$\deg ( P_n(x) ) =n, \deg ( Q_n(x) )=n.$$ The so-called the connecting problem between them asks to find the coefficients $\alpha_{n,k}$ in the expression $\displaystyle Q_n(x) =\sum_{k=0}^{n} \alpha_{n,k} P_k(x).$ Let $\{ S_n(x) \}_{n\geq 0}$ be another polynomial set with $\deg ( S_n(x) )=n.$ The general connection problem between them consists in finding the coefficients $\alpha^{(n)}_{i,j}$ in the expansion $$Q_n(x) =\sum_{i,j=0}^{n} \alpha^{(n)}_{i,j} P_i(x) S_{j}(x).$$ The connection problem for different types of polynomials has a long history, and it is still of interest. The connection coefficients play an important role in many problems in pure and applied mathematics, especially in combinatorics, mathematical physics and quantum chemical applications. For the particular case $Q_n(x)=P'_{n+1}(x)$ the connection problem is called the derivative connecting problem and the general derivative connecting problem associated to $\{ P_n(x) \}_{n\geq 0}.$
In this paper, we give a closed-form expression of the derivative connecting problems for well-known systems of polynomials. |
topic |
connection problem inversion problem derivative connecting problem connecting coefficients orthogonal polynomials |
url |
https://journals.pnu.edu.ua/index.php/cmp/article/view/2121 |
work_keys_str_mv |
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