Generalized $f$-Eulerian polynomials: zeros and hypergeometric representations with applications
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916954670891008 |
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| author | Karp, Dmitrii Vishnyakova, Anna |
| author_facet | Karp, Dmitrii Vishnyakova, Anna |
| contents | In this paper, we explore (slightly generalized) $f$-Eulerian polynomials introduced by Stanley and frequently appearing in combinatorics. Notable special cases include the classical Eulerian polynomials, the generating polynomials of order polynomials for certain labeled posets, and the $d$-Narayana polynomials. We establish simple sufficient conditions for the reality (and sign) of their zeros and present implications for total positivity of sequences generated by values of polynomials at integers. We further relate these polynomials to generalized Euler's transformations for the generalized hypergeometric functions with integral parameter differences. Exploiting this and other hypergeometric connections, we provide purely hypergeometric proofs for various known and some new properties of $d$-Narayana polynomials. Another family encompassed by our definition of the generalized $f$-Eulerian polynomials is that of Jacobi-Piñeiro type II multiple orthogonal polynomials. Their zero location can thus be analyzed, for both canonical and non-canonical parameter values, without invoking orthogonality. Finally, we present several connection formulas relating $d$-Narayana polynomials to particular Jacobi-Piñeiro polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_13897 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized $f$-Eulerian polynomials: zeros and hypergeometric representations with applications Karp, Dmitrii Vishnyakova, Anna Combinatorics Classical Analysis and ODEs 33C20, 05A15, 05A19, 26C10, 30C15, 12D10, 33C47, 11B83, 15B48 In this paper, we explore (slightly generalized) $f$-Eulerian polynomials introduced by Stanley and frequently appearing in combinatorics. Notable special cases include the classical Eulerian polynomials, the generating polynomials of order polynomials for certain labeled posets, and the $d$-Narayana polynomials. We establish simple sufficient conditions for the reality (and sign) of their zeros and present implications for total positivity of sequences generated by values of polynomials at integers. We further relate these polynomials to generalized Euler's transformations for the generalized hypergeometric functions with integral parameter differences. Exploiting this and other hypergeometric connections, we provide purely hypergeometric proofs for various known and some new properties of $d$-Narayana polynomials. Another family encompassed by our definition of the generalized $f$-Eulerian polynomials is that of Jacobi-Piñeiro type II multiple orthogonal polynomials. Their zero location can thus be analyzed, for both canonical and non-canonical parameter values, without invoking orthogonality. Finally, we present several connection formulas relating $d$-Narayana polynomials to particular Jacobi-Piñeiro polynomials. |
| title | Generalized $f$-Eulerian polynomials: zeros and hypergeometric representations with applications |
| topic | Combinatorics Classical Analysis and ODEs 33C20, 05A15, 05A19, 26C10, 30C15, 12D10, 33C47, 11B83, 15B48 |
| url | https://arxiv.org/abs/2509.13897 |