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| Format: | Preprint |
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2024
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| Accès en ligne: | https://arxiv.org/abs/2401.17687 |
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| _version_ | 1866909313976500224 |
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| author | Brugidou, Vincent |
| author_facet | Brugidou, Vincent |
| contents | Let $λ=\left( λ_{1},λ_{2},...,λ_{r}\right) $ be an integer partition, and $\left[p_{λ}\right] $ the $q$-analog of the symmetric power function $%p_{λ}$. This $q$-analogue has been defined as a special case, in the author's previous article: "A $q$-analog of certain symmetric functions and one of its specializations". Here, we prove that a large part of the classical relations between $p_{λ}$, on one hand, and the elementary and complete symmetric functions $e_{n}$ and $h_{n}$, on the other hand, have $q$-analogues with $\left[ p_{λ}\right] $. In particular, the generating functions $E\left( t\right) =\sum\nolimits_{n\geq 0}e_{n}t^{n}$ and $H\left( t\right) =\sum\nolimits_{n\geq 0}h_{n}t^{n}$ are expressed in terms of $\left[ p_{n}\right] $, using Gessel's $q$-exponential formula and a variant of it. A factorization of these generating functions into infinite $q$-products, which has no classical counterpart, is established. By specializing these results, we show that the $q$-binomial theorem is a special case of these infinite $q$-products. We also obtain new formulas for the tree inversions enumerators and for certain $q$-orthogonal polynomials, detailing the case of dicrete $q$-Hermite polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_17687 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | q-power symmetric functions and q-exponential formula Brugidou, Vincent Combinatorics Commutative Algebra Let $λ=\left( λ_{1},λ_{2},...,λ_{r}\right) $ be an integer partition, and $\left[p_{λ}\right] $ the $q$-analog of the symmetric power function $%p_{λ}$. This $q$-analogue has been defined as a special case, in the author's previous article: "A $q$-analog of certain symmetric functions and one of its specializations". Here, we prove that a large part of the classical relations between $p_{λ}$, on one hand, and the elementary and complete symmetric functions $e_{n}$ and $h_{n}$, on the other hand, have $q$-analogues with $\left[ p_{λ}\right] $. In particular, the generating functions $E\left( t\right) =\sum\nolimits_{n\geq 0}e_{n}t^{n}$ and $H\left( t\right) =\sum\nolimits_{n\geq 0}h_{n}t^{n}$ are expressed in terms of $\left[ p_{n}\right] $, using Gessel's $q$-exponential formula and a variant of it. A factorization of these generating functions into infinite $q$-products, which has no classical counterpart, is established. By specializing these results, we show that the $q$-binomial theorem is a special case of these infinite $q$-products. We also obtain new formulas for the tree inversions enumerators and for certain $q$-orthogonal polynomials, detailing the case of dicrete $q$-Hermite polynomials. |
| title | q-power symmetric functions and q-exponential formula |
| topic | Combinatorics Commutative Algebra |
| url | https://arxiv.org/abs/2401.17687 |