Generalized Bell polynomials
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866912031919046656 |
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| author | Durán, Antonio J. |
| author_facet | Durán, Antonio J. |
| contents | In this paper, generalized Bell polynomials $(\Be_n^ϕ)_n$ associated to a sequence of real numbers $ϕ=(ϕ_i)_{i=1}^\infty$ are introduced. Bell polynomials correspond to $ϕ_i=0$, $i\ge 1$. We prove that when $ϕ_i\ge 0$, $i\ge 1$: (a) the zeros of the generalized Bell polynomial $\Be_n^ϕ$ are simple, real and non positive; (b) the zeros of $\Be_{n+1}^ϕ$ interlace the zeros of $\Be_n^ϕ$; (c) the zeros are decreasing functions of the parameters $ϕ_i$. We find a hypergeometric representation for the generalized Bell polynomials. As a consequence, it is proved that the class of all generalized Bell polynomials is actually the same class as that of all Laguerre multiple polynomials of the first kind. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_11344 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized Bell polynomials Durán, Antonio J. Classical Analysis and ODEs In this paper, generalized Bell polynomials $(\Be_n^ϕ)_n$ associated to a sequence of real numbers $ϕ=(ϕ_i)_{i=1}^\infty$ are introduced. Bell polynomials correspond to $ϕ_i=0$, $i\ge 1$. We prove that when $ϕ_i\ge 0$, $i\ge 1$: (a) the zeros of the generalized Bell polynomial $\Be_n^ϕ$ are simple, real and non positive; (b) the zeros of $\Be_{n+1}^ϕ$ interlace the zeros of $\Be_n^ϕ$; (c) the zeros are decreasing functions of the parameters $ϕ_i$. We find a hypergeometric representation for the generalized Bell polynomials. As a consequence, it is proved that the class of all generalized Bell polynomials is actually the same class as that of all Laguerre multiple polynomials of the first kind. |
| title | Generalized Bell polynomials |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2409.11344 |