Abstract
Ther-Whitney numbers of the second kind are a generalization of all the Stirling-type numbers of the second kind which are in line with the unified generalization of Hsu and Shuie. In this paper, asymptotic formulas forr-Whitney numbers of the second kind with integer and real parameters are obtained and the range of validity of each formula is established.
Highlights
The r-Whitney numbers of the second kind, denoted by Wβ,r(n, m), have been introduced by Mezo [1] to obtain a new formula for Bernoulli polynomials
These numbers are equivalent to the numbers considered by Rucinski and Voigt [2] and the (r, β)-Stirling numbers [3]
They are considered as a generalization of all the Stirling-type numbers of the second kind which satisfy βm
Summary
The r-Whitney numbers of the second kind, denoted by Wβ,r(n, m), have been introduced by Mezo [1] to obtain a new formula for Bernoulli polynomials. The computed approximate values for m = 15, 30, 60, 80, 90 confirm this In this paper, another asymptotic formula for the rWhitney numbers of the second kind Wβ,r(n, n − m) with integral values of m and n is obtained using a similar analysis as that in [12], which is proved to be valid when m is in the range n − o(√n) ≤ n − m ≤ n. Another asymptotic formula for the rWhitney numbers of the second kind Wβ,r(n, n − m) with integral values of m and n is obtained using a similar analysis as that in [12], which is proved to be valid when m is in the range n − o(√n) ≤ n − m ≤ n This can be considered as the final range since it covers the right most tail of the interval 0 < m ≤ n. It is shown that the formula obtained is valid in the given range when n and m are real numbers
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