Abstract
In this paper, we define new Daehee numbers, the degenerate Daehee numbers of the third kind, using the degenerate log function as generating function. We obtain some identities for the degenerate Daehee numbers of the third kind associated with the Daehee, degenerate Daehee, and degenerate Daehee numbers of the second kind. In addition, we derive a differential equation associated with the degenerate log function. We deduce some identities from the differential equation.
Highlights
After Carlitz [1,2], many mathematicians have studied degenerate functions and numbers.They mainly used (1 + λt) λ instead of et to degenerate polynomials and numbers
We have studied the degenerate Daehee numbers of the third kind
We obtained some relations between the degenerate Daehee numbers of the second kind and the third kind in Theorem 6 and Theorem 7
Summary
After Carlitz [1,2], many mathematicians have studied degenerate functions and numbers (see [3,4,5,6,7,8,9,10,11]). In the degenerate Cauchy numbers of the first and second kind, (1 + λt) λ was used instead of et We call this degenerate based on the exponential sense. In this case, the author used logλ (t) instead of log t for degenerating. The Daehee numbers, denoted by Dn , are defined by the generating function log(1 + t) tn. S. Kim et al presented degenerate Daehee polynomials and numbers of the second kind as follows [23]. When x = 0, Dλ, (n) = Dλ, (n, 0) are called the degenerate Daehee numbers of the second kind These degenerate numbers are based on the exponential sense. We define the degenerate Daehee numbers based on the log sense. We deduce a differential equation using the degenerate log function, and we derive some identities related to the degenerate Daehee numbers from this differential equation
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