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24 Bernoulli and Euler PolynomialsApplications

§24.17 Mathematical Applications


§24.17(i) Summation

Euler–Maclaurin Summation Formula

See §2.10(i). For a generalization see Olver (1997b, p. 284).

Boole Summation Formula

Let 0h1 and a,m, and n be integers such that n>a, m>0, and f(m)(x) is absolutely integrable over [a,n]. Then with the notation of §24.2(iii)

24.17.1 j=an-1(-1)jf(j+h)=12k=0m-1Ek(h)k!((-1)n-1f(k)(n)+(-1)af(k)(a))+Rm(n),


24.17.2 Rm(n)=12(m-1)!anf(m)(x)E~m-1(h-x)x.

Calculus of Finite Differences

See Milne-Thomson (1933), Nörlund (1924), or Jordan (1965). For a more modern perspective see Graham et al. (1994).

§24.17(ii) Spline Functions

Euler Splines

Let 𝒮n denote the class of functions that have n-1 continuous derivatives on and are polynomials of degree at most n in each interval (k,k+1), k. The members of 𝒮n are called cardinal spline functions. The functions

24.17.3 Sn(x)=E~n(x+12n+12)E~n(12n+12),

are called Euler splines of degree n. For each n, Sn(x) is the unique bounded function such that Sn(x)𝒮n and

24.17.4 Sn(k)=(-1)k,

The function Sn(x) is also optimal in a certain sense; see Schoenberg (1971).

Bernoulli Monosplines

A function of the form xn-S(x), with S(x)𝒮n-1 is called a cardinal monospline of degree n. Again with the notation of §24.2(iii) define

24.17.5 Mn(x)={B~n(x)-Bn,n even,B~n(x+12),n odd.

Mn(x) is a monospline of degree n, and it follows from (24.4.25) and (24.4.27) that

24.17.6 Mn(k)=0,

For each n=1,2, the function Mn(x) is also the unique cardinal monospline of degree n satisfying (24.17.6), provided that

24.17.7 Mn(x)=O(|x|γ),

for some positive constant γ.

For any n2 the function

24.17.8 F(x)=B~n(x)-2-nBn

is the unique cardinal monospline of degree n having the least supremum norm F on (minimality property).

§24.17(iii) Number Theory

Bernoulli and Euler numbers and polynomials occur in: number theory via (24.4.7), (24.4.8), and other identities involving sums of powers; the Riemann zeta function and L-series (§25.15, Apostol (1976), and Ireland and Rosen (1990)); arithmetic of cyclotomic fields and the classical theory of Fermat’s last theorem (Ribenboim (1979) and Washington (1997)); p-adic analysis (Koblitz (1984, Chapter 2)).