2017/02/21 by Jósé L. Ramírez, Ramírez, José L., Miguel A. Méndez +2 · 1 citation
Mathematics · Physics and Astronomy · #05A15 #05A19 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #Combinatorics (math.CO) #FOS: Mathematics #Primary 11B83 #Secondary 11B73
paper · pdf · doi:10.48550/arxiv.1702.06519
openalex publication_date 2017/02/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the present article we introduce two new combinatorial interpretations of\nthe r-Whitney numbers of the second kind obtained from the combinatorics of\nthe differential operators associated to the grammar G:= y\→\nyxm, x\→ x . By specializing m=1 we obtain also a new\ncombinatorial interpretation of the r-Stirling numbers of the second kind.\nAgain, by specializing to the case r=0 we introduce a new generalization of\nthe Stirling number of the second kind and through them a binomial type family\nof polynomials that generalizes Touchard's. Moreover, we show several\nwell-known identities involving the r-Dowling polynomials and the r-Whitney\nnumbers using the combinatorial differential calculus. Finally we prove that\nthe r-Dowling polynomials are a Sheffer family relative to the generalized\nTouchard binomial family, study their umbral inverses, and introduce\n[m]-Stirling numbers of the first kind. From the relation between umbral\ncalculus and the Riordan matrices we give several new combinatorial identities\ninvolving the r-Whitney number of both kinds, Bernoulli and Euler\npolynomials.\n