Abstract
Let the numbers be defined by , where and are the exponential complete Bell polynomials. In this paper, by means of the methods of Riordan arrays, we establish general identities involving the numbers , binomial coefficients and inverse of binomial coefficients. From these identities, we deduce some identities involving binomial coefficients, Harmonic numbers and the Euler sum identities. Furthermore, we obtain the asymptotic values of some summations associated with the numbers by Darboux’s method.
Highlights
Introduction and PreliminariesLet Yr be the exponential complete Bell polynomials and In [1], Zave established the following series expansion: Pr ( x1, xr =( −1)r Yr −0! −1! x2 r 1)!
( ) of formal power series with h0 = h (0). It defines an infinite lower triangular array dn,k n,k∈N according to the rule dn,k = tn d (t )(h (t ))k
In Section, we obtain some for P (r, n, k ) and binomial coefficients by means of the Riordan arrays
Summary
Where ( ) Pr ( x1 , , xr ) =(−1)r Yr −0!x1 , −1!x2 , , − (r − 1)!xr and Yr are the exponential complete Bell polynomials. By means of the methods of Riordan arrays, we establish general identities involving the numbers P (r, n, k ) , binomial coefficients and inverse of binomial coefficients. From these identities, we deduce some identities involving binomial coefficients, Harmonic numbers and the Euler sum identities. We obtain the asymptotic values of some summations associated with the numbers P (r, n, k ) by Darboux’s method
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