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A combinatorial approach to a general two-term recurrence

机译：一般两期复发的组合方法

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摘要

We provide combinatorial proofs of explicit formulas for some sequences satisfying particular cases of the general recurrence |nk|=(α(n-1) +βk+γ)|n-1/k|+(α′(n-1)+β′k+ γ′)|n-1k-1|+[n=k=0], which have been previously shown using other methods. Many interesting combinatorial sequences are special cases of this recurrence, such as binomial coefficients, both kinds of Stirling numbers, Lah numbers, and two types of Eulerian numbers. Among the cases we consider are α′=0, α=-β, and β=β′=0. We also provide combinatorial proofs of some prior identities satisfied by |nk| when α′=0 and when β=β′=0 as well as deduce some new ones in the former case. In addition, we introduce a polynomial generalization of |nk| when α′=0 which has among its special cases q-analogues of both kinds of Stirling numbers. Finally, we supply combinatorial proofs of two formulas relating binomial coefficients and the two kinds of Stirling numbers which were previously obtained by equating three different expressions for the solution of the aforementioned recurrence in the case when α′= β′=0 and all other weights are unity.
机译：我们为满足一般递归条件的某些序列提供明确公式的组合证明| nk | =（α（n-1）+βk+γ）| n-1 / k | +（α'（n-1）+ β'k+γ'）| n-1k-1 | + [n = k = 0]，先前已使用其他方法进行了显示。许多有趣的组合序列是这种复发的特殊情况，例如二项式系数，斯特林数，拉赫数和两种欧拉数。在我们考虑的情况中，α'= 0，α=-β和β=β'= 0。我们还提供| nk |满足的某些先验身份的组合证明。在前一种情况下，当α'= 0时和当β=β'= 0时，并推导出一些新的值。另外，我们介绍| nk |的多项式概括。当α'= 0时，在其特殊情况下具有两种斯特林数的q模拟。最后，我们提供了在α′=β′= 0且所有其他权重的情况下，通过将三个不同的表达式等式求解上述递归的公式，得到了两个与二项式系数和两种斯特林数有关的公式的组合证明团结一致

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