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arXiv · 2409.11344

Generalized Bell polynomials

Antonio J. Durán
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Abstract

In this paper, generalized Bell polynomials $(\Be_n^\phi)_n$ associated to a sequence of real numbers $\phi=(\phi_i)_{i=1}^\infty$ are introduced. Bell polynomials correspond to $\phi_i=0$, $i\ge 1$. We prove that when $\phi_i\ge 0$, $i\ge 1$: (a) the zeros of the generalized Bell polynomial $\Be_n^\phi$ are simple, real and non positive; (b) the zeros of $\Be_{n+1}^\phi$ interlace the zeros of $\Be_n^\phi$; (c) the zeros are decreasing functions of the parameters $\phi_i$. 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.

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Antonio J. Durán. 2024-09-17. Generalized Bell polynomials. https://arxiv.org/abs/2409.11344

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