Higher-order Bernoulli Polynomials Research Articles

Casimir effect around a cone.

The vacuum average of the energy density of a free, massless scalar field around a conical flux-tube singularity in d+1 space-time dimensions is calculated. A complex contour method is employed and the results involve higher-order Bernoulli polynomials. The formulas might have applications to cosmic strings.

Irreducibility of Bernoulli Polynomials of Higher Order

The Bernoulli polynomials of order k, where k is a positive integer, are defined byBm(k)(x) is a polynomial of degree m with rational coefficients, and the constant term of Bm(k)(x) is the mth Bernoulli number of order k, Bm(k). In a previous paper (3) we obtained some conditions, in terms of k and m, which imply that Bm(k)(x) is irreducible (all references to irreducibility will be with respect to the field of rational numbers). In particular, we obtained the following two results.

Some irreducibility theorems for Bernoulli polynomials of higher order

Some irreducibility theorems for Bernoulli polynomials of higher order

Systems of linear difference equations and expansions in series of exponential functions

Introduction. The principal purpose of the first part of this paper is to prove (?1.9) that the system (1.1) of linear non-homogeneous generalized difference equations has solutions gk(X), k =1, 2, , n, which are integral functions provided that the independent terms ck(x) are themselves integral functions and provided that the system has a certain non-singular character defined in ?1.3. In case the ck(x) are further restricted to be of exponential type (?1.5) then solutions of exponential type exist (?1.6) and indeed solutions of exponential type at most equal to q (called principal solutions) in case no 0,(x) is of higher type than q and at least one of them is of precisely this type. A useful symbolic notation (?1.2) is effective in carrying out the argument. In the second part of the paper we apply the results of the first part to the rather remarkable problem of the simultaneous expansion of n integral functions in composite power series, a problem which we have not seen treated elsewhere. The third part of the paper is devoted to the theory of a class of remarkable expansions in series of exponential functions, generalizing the theory of Fourier series. Whereas the basic region of convergence of Fourier series is a segment of a straight line, these new series, apart from certain particular cases, have certain polygons in the complex plane as their basic regions of convergence. The vertices of these polygons play the role of the end points of the segments in the case of Fourier series, while the remaining points of the polygon play the r6le of interior points of the segments. Several extensions of the theory are briefly indicated (?3.4) and an application is made (?3.5) to the expansion of Bernoulli polynomials of higher order in series of exponential functions.