Miscellaneous formulae for the certain class of combinatorial sums and special numbers

Yilmaz Simsek

Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 46 (2021) no. 1.

Résumé

In a recent paper $[$Montes Taurus J. Pure Appl. Math. {\bf 3} {\rm(1) (2021), 38--61}$]$ we defined the class of combinatorial sums \[ y(n,ambda )=um_{j=0}^{n}\frac{(-1)^{n}}{(j+1)ambda ^{j+1}eft( ambda -1\right) ^{n+1-j}}. \] The purpose of this paper is to give some integral formulas, identities and combinatorial sums using the numbers $y(n,\lambda )$. The obtained results are related to the Bernoulli numbers and their interpolation functions, as well as the Pell numbers, the Harmonic numbers, the alternating Harmonic numbers, the Daehee numbers, and the Catalan-Qi numbers. Moreover, we give answers to some open problems involving the numbers $y(n,\lambda)$.