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24 Bernoulli and Euler PolynomialsApplications

§24.17 Mathematical Applications

  1. §24.17(i) Summation
  2. §24.17(ii) Spline Functions
  3. §24.17(iii) Number Theory

§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=an1(1)jf(j+h)=12k=0m1Ek(h)k!((1)n1f(k)(n)+(1)af(k)(a))+Rm(n),


24.17.2 Rm(n)=12(m1)!anf(m)(x)E~m1(hx)dx.

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 n1 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 xnS(x), with S(x)𝒮n1 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)2nBn

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)).