On zero behavior of higher-order Sobolev-type discrete q-Hermite I orthogonal polynomials
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866910318947467264 |
|---|---|
| author | Huertas, Edmundo J. Lastra, Alberto Soria-Lorente, Anier Soto-Larrosa, Víctor |
| author_facet | Huertas, Edmundo J. Lastra, Alberto Soria-Lorente, Anier Soto-Larrosa, Víctor |
| contents | In this work, we investigate the sequence of monic q-Hermite I-Sobolev type orthogonal polynomials of higher-order, denoted as $\{\mathbb{H}_{n}(x;q)\}_{n\geq 0}$, which are orthogonal with respect to the following non-standard inner product involving q-differences: \begin{equation*} \langle p,q\rangle_{λ}=\int_{-1}^{1}f\left( x\right) g\left(x\right) (qx,-qx;q)_{\infty }d_{q}(x)+λ\,(\mathscr{D}_{q}^{j}f)(α)(\mathscr{D}_{q}^{j}g)(α), \end{equation*} where $α\in \mathbb{R}\backslash (-1,1)$, $λ$ belongs to the set of positive real numbers, $\mathscr{D}_{q}^{j}$ denotes the $j$-th $q $-discrete analogue of the derivative operator, and $(qx,-qx;q)_{\infty}d_{q}(x)$ denotes the orthogonality weight with its points of increase in a geometric progression. We proceed to obtain the hypergeometric representation of $\mathbb{H}_{n}(x;q)$ and explicit expressions for the corresponding ladder operators. From the latter, we obtain a novel kind of three-term recurrence formula with rational coefficients associated with these polynomial family. Moreover, for certain real values of $α$, we present some results concerning the location of the zeros of $\mathbb{H}_n(x;q)$ and we perform a comprehensive analysis of their asymptotic behavior as the parameter $λ$ varies from zero to infinity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_03381 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On zero behavior of higher-order Sobolev-type discrete q-Hermite I orthogonal polynomials Huertas, Edmundo J. Lastra, Alberto Soria-Lorente, Anier Soto-Larrosa, Víctor Classical Analysis and ODEs Mathematical Physics In this work, we investigate the sequence of monic q-Hermite I-Sobolev type orthogonal polynomials of higher-order, denoted as $\{\mathbb{H}_{n}(x;q)\}_{n\geq 0}$, which are orthogonal with respect to the following non-standard inner product involving q-differences: \begin{equation*} \langle p,q\rangle_{λ}=\int_{-1}^{1}f\left( x\right) g\left(x\right) (qx,-qx;q)_{\infty }d_{q}(x)+λ\,(\mathscr{D}_{q}^{j}f)(α)(\mathscr{D}_{q}^{j}g)(α), \end{equation*} where $α\in \mathbb{R}\backslash (-1,1)$, $λ$ belongs to the set of positive real numbers, $\mathscr{D}_{q}^{j}$ denotes the $j$-th $q $-discrete analogue of the derivative operator, and $(qx,-qx;q)_{\infty}d_{q}(x)$ denotes the orthogonality weight with its points of increase in a geometric progression. We proceed to obtain the hypergeometric representation of $\mathbb{H}_{n}(x;q)$ and explicit expressions for the corresponding ladder operators. From the latter, we obtain a novel kind of three-term recurrence formula with rational coefficients associated with these polynomial family. Moreover, for certain real values of $α$, we present some results concerning the location of the zeros of $\mathbb{H}_n(x;q)$ and we perform a comprehensive analysis of their asymptotic behavior as the parameter $λ$ varies from zero to infinity. |
| title | On zero behavior of higher-order Sobolev-type discrete q-Hermite I orthogonal polynomials |
| topic | Classical Analysis and ODEs Mathematical Physics |
| url | https://arxiv.org/abs/2402.03381 |