Notes on degenerate numbers

Abstract

In this paper, using a theorem relating the potential polynomial F k ( z ) and the exponential Bell polynomial B n , j ( 0 , … , 0 , f r , f r + 1 , … ) , we obtain some explicit formulas for higher order degenerate Bernoulli numbers of the first and second kinds. We also prove new recurrence formulas for these numbers. Furthermore, we discuss other applications of the theorem, from which we deduce several formulas for degenerate Genocchi numbers, degenerate tangent numbers, and the coefficients of the higher order degenerate Euler polynomials. Finally, we examine the polynomials V ( k , j , z | λ ) and V 1 ( k , l , z | λ ) , and, in particular, we show how these polynomials are related to the degenerate Bernoulli, Genocchi, tangent, and van der Pol numbers, and the numbers generated by the reciprocal of ( 1 + λ x ) 1 / λ - x - 1 .