Asymptotic expansions of generalized Bernoulli polynomials

Abstract

Asymptotic expansions of certain generalized Bernoulli polynomials are obtained, some in terms of elementary functions and others in terms of gamma functions and their derivatives. The latter results can also be written in terms of elementary functions, by using the known asymptotic expansions of the gamma functions, and the leading terms are obtained in this way. The results obtained give the asymptotic form of the coefficients occurring in all the usual central-difference formulae of the calculus of finite differences.

Uniform Approximations of Bernoulli and Euler Polynomials in Terms of Hyperbolic Functions

“…we --1 , 2m c w 11 + (5) where f(w) is a meromorphic function with simple poles w1' w 2 , ••• and analytic at the origin. The contour C is a circle whose center is at the origin and which contains no poles off ( w) inside.…”

“…Vi= m + 1, m + 2, .... But, moreover, the polynomial Q~'(z) has been defined in (26) by using a function -hm (w) in the integral that has the same first 2m poles and the same residues as the function f(w) that defines Pn(z) in (5). The contour C in (5) and in (26) may be chosen to pass near the singularities of f(w) closest to the origin.…”

“…A very complete bibliography up to 1960 concerning tables and applications of the theory of Bernoulli and Euler polynomials can be found in [4]. Fore more recent works (after 1960), the reader is referred to [5][6][7][8][9][10][11] and references therein for a very complete survey of formulas involving these polynomials, zeros, asymptotic behavior, integral representations, and a number of other properties. Nevertheless, an extensive list of formulas involving Bernoulli and Euler polynomials can be found in ( [1], pp.…”

“…A more detailed study about these expansions concerned with the type of convergence is investigated in [7], where similar expansions for generalized Bernoulli polynomials are also obtained. In [5], asymptotic expansions of (generalized) Bernoulli polynomials are obtained in terms of elementary functions and also in terms of gamma functions. These expansions happen to fail when the argument z is allowed to grow arbitrarily.…”

Large degree asymptotics of generalized Bernoulli and Euler polynomials

“…This follows from (1.1). We first follow the approach given in [14], and observe that when μ is a negative integer or zero, say μ = −m (m = 0, 1, 2, . .…”