Zeros distribution and interlacing property for certain polynomial sequences
In this article, we first prove that the Hankel determinant of order three of the polynomial sequence {Pn(x)=∑k≥0P(n,k)xk}n≥0{\left\{{P}_{n}\left(x)={\sum }_{k\ge 0}P\left(n,k){x}^{k}\right\}}_{n\ge 0} is weakly (Hurwitz) stable, where P(n,k)P\left(n,k) satisfies the recurrence relation P(n,k)=(a1n+...
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De Gruyter
2024-11-01
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| Online Access: | https://doi.org/10.1515/math-2024-0085 |
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| author | Guo Wan-Ming |
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| author_sort | Guo Wan-Ming |
| collection | DOAJ |
| description | In this article, we first prove that the Hankel determinant of order three of the polynomial sequence {Pn(x)=∑k≥0P(n,k)xk}n≥0{\left\{{P}_{n}\left(x)={\sum }_{k\ge 0}P\left(n,k){x}^{k}\right\}}_{n\ge 0} is weakly (Hurwitz) stable, where P(n,k)P\left(n,k) satisfies the recurrence relation P(n,k)=(a1n+a2)P(n−1,k)+(b1n+b2)P(n−1,k−1),P\left(n,k)=\left({a}_{1}n+{a}_{2})P\left(n-1,k)+\left({b}_{1}n+{b}_{2})P\left(n-1,k-1), with P(n,k)=0P\left(n,k)=0 wherever k∉{0,1,…,n}.k\notin \left\{0,1,\ldots ,n\right\}. The stability of a polynomial is closely associated with the interlacing property, which is based on the Hermite-Biehler theorem. We also show the interlacing property of the polynomial sequence (Un(x))n≥0,{\left({U}_{n}\left(x))}_{n\ge 0}, which satisfies the following recurrence relation: Un(x)=(αnx+βn)Un−1(x)+(unx2+vnx)Un−1′(x){U}_{n}\left(x)=\left({\alpha }_{n}x+{\beta }_{n}){U}_{n-1}\left(x)+\left({u}_{n}{x}^{2}+{v}_{n}x){U}_{n-1}^{^{\prime} }\left(x) based on the Hermite-Biehler theorem. As applications, we obtain the weak (Hurwitz) stability of the Hankel determinant of order three for the row polynomials of the (unsigned) Stirling numbers of the first kind, the Whitney numbers of the first kind, and show the interlacing property of Eulerian polynomials, Bell polynomials, and Dowling polynomials. |
| format | Article |
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| institution | OA Journals |
| issn | 2391-5455 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | De Gruyter |
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| series | Open Mathematics |
| spelling | doaj-art-27ee9e39c3fa428680d7347bcaa315492025-08-20T01:48:37ZengDe GruyterOpen Mathematics2391-54552024-11-0122150053510.1515/math-2024-0085Zeros distribution and interlacing property for certain polynomial sequencesGuo Wan-Ming0School of Mathematical Sciences, Qufu Normal University, Qufu 273165, P. R. ChinaIn this article, we first prove that the Hankel determinant of order three of the polynomial sequence {Pn(x)=∑k≥0P(n,k)xk}n≥0{\left\{{P}_{n}\left(x)={\sum }_{k\ge 0}P\left(n,k){x}^{k}\right\}}_{n\ge 0} is weakly (Hurwitz) stable, where P(n,k)P\left(n,k) satisfies the recurrence relation P(n,k)=(a1n+a2)P(n−1,k)+(b1n+b2)P(n−1,k−1),P\left(n,k)=\left({a}_{1}n+{a}_{2})P\left(n-1,k)+\left({b}_{1}n+{b}_{2})P\left(n-1,k-1), with P(n,k)=0P\left(n,k)=0 wherever k∉{0,1,…,n}.k\notin \left\{0,1,\ldots ,n\right\}. The stability of a polynomial is closely associated with the interlacing property, which is based on the Hermite-Biehler theorem. We also show the interlacing property of the polynomial sequence (Un(x))n≥0,{\left({U}_{n}\left(x))}_{n\ge 0}, which satisfies the following recurrence relation: Un(x)=(αnx+βn)Un−1(x)+(unx2+vnx)Un−1′(x){U}_{n}\left(x)=\left({\alpha }_{n}x+{\beta }_{n}){U}_{n-1}\left(x)+\left({u}_{n}{x}^{2}+{v}_{n}x){U}_{n-1}^{^{\prime} }\left(x) based on the Hermite-Biehler theorem. As applications, we obtain the weak (Hurwitz) stability of the Hankel determinant of order three for the row polynomials of the (unsigned) Stirling numbers of the first kind, the Whitney numbers of the first kind, and show the interlacing property of Eulerian polynomials, Bell polynomials, and Dowling polynomials.https://doi.org/10.1515/math-2024-0085hankel determinantinterlacing propertyhurwitz stabilityhermite-biehler theorem05a1515a1511b8326c1005a20 |
| spellingShingle | Guo Wan-Ming Zeros distribution and interlacing property for certain polynomial sequences Open Mathematics hankel determinant interlacing property hurwitz stability hermite-biehler theorem 05a15 15a15 11b83 26c10 05a20 |
| title | Zeros distribution and interlacing property for certain polynomial sequences |
| title_full | Zeros distribution and interlacing property for certain polynomial sequences |
| title_fullStr | Zeros distribution and interlacing property for certain polynomial sequences |
| title_full_unstemmed | Zeros distribution and interlacing property for certain polynomial sequences |
| title_short | Zeros distribution and interlacing property for certain polynomial sequences |
| title_sort | zeros distribution and interlacing property for certain polynomial sequences |
| topic | hankel determinant interlacing property hurwitz stability hermite-biehler theorem 05a15 15a15 11b83 26c10 05a20 |
| url | https://doi.org/10.1515/math-2024-0085 |
| work_keys_str_mv | AT guowanming zerosdistributionandinterlacingpropertyforcertainpolynomialsequences |