تقرير
An elliptic extension of the multinomial theorem
| العنوان: | An elliptic extension of the multinomial theorem |
|---|---|
| المؤلفون: | Schlosser, Michael J. |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Quantum Algebra, Mathematics - Combinatorics, Primary 05A10, Secondary 11B65, 33D67, 33D80, 33E90 |
| الوصف: | We present a multinomial theorem for elliptic commuting variables. This result extends the author's previously obtained elliptic binomial theorem to higher rank. Two essential ingredients are a simple elliptic star-triangle relation, ensuring the uniqueness of the normal form coefficients, and, for the recursion of the closed form elliptic multinomial coefficients, the Weierstra{\ss} type $\mathsf A$ elliptic partial fraction decomposition. From our elliptic multinomial theorem we obtain, by convolution, an identity that is equivalent to Rosengren's type $\mathsf A$ extension of the Frenkel--Turaev ${}_{10}V_9$ summation, which in the trigonometric or basic limiting case reduces to Milne's type $\mathsf A$ extension of the Jackson ${}_8\phi_7$ summation. Interpreted in terms of a weighted counting of lattice paths in the integer lattice $\mathbb Z^r$, our derivation of the $\mathsf A_r$ Frenkel--Turaev summation constitutes the first combinatorial proof of that fundamental identity, and, at the same time, of important special cases including the $\mathsf A_r$ Jackson summation. Comment: 14 pp |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2307.12921 |
| رقم الأكسشن: | edsarx.2307.12921 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |