# §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 $0\leq h\leq 1$ 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 $\sum_{j=a}^{n-1}(-1)^{j}f(j+h)=\frac{1}{2}\sum_{k=0}^{m-1}\frac{E_{k}\left(h% \right)}{k!}\left((-1)^{n-1}f^{(k)}(n)+(-1)^{a}f^{(k)}(a)\right)+R_{m}(n),$

where

 24.17.2 $R_{m}(n)=\frac{1}{2(m-1)!}\int_{a}^{n}f^{(m)}(x)\widetilde{E}_{m-1}\left(h-x% \right)\,\mathrm{d}x.$

### 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 $\mathcal{S}_{n}$ denote the class of functions that have $n-1$ continuous derivatives on $\mathbb{R}$ and are polynomials of degree at most $n$ in each interval $(k,k+1)$, $k\in\mathbb{Z}$. The members of $\mathcal{S}_{n}$ are called cardinal spline functions. The functions

 24.17.3 $S_{n}(x)=\frac{\widetilde{E}_{n}\left(x+\tfrac{1}{2}n+\tfrac{1}{2}\right)}{% \widetilde{E}_{n}\left(\tfrac{1}{2}n+\tfrac{1}{2}\right)},$ $n=0,1,\dots$, ⓘ Symbols: $\widetilde{E}_{\NVar{n}}\left(\NVar{x}\right)$: periodic Euler functions, $n$: integer and $x$: real or complex Permalink: http://dlmf.nist.gov/24.17.E3 Encodings: TeX, pMML, png See also: Annotations for §24.17(ii), §24.17(ii), §24.17 and Ch.24

are called Euler splines of degree $n$. For each $n$, $S_{n}(x)$ is the unique bounded function such that $S_{n}(x)\in\mathcal{S}_{n}$ and

 24.17.4 $S_{n}(k)=(-1)^{k},$ $k\in\mathbb{Z}$. ⓘ Symbols: $\in$: element of, $\mathbb{Z}$: set of all integers, $k$: integer and $n$: integer Permalink: http://dlmf.nist.gov/24.17.E4 Encodings: TeX, pMML, png See also: Annotations for §24.17(ii), §24.17(ii), §24.17 and Ch.24

The function $S_{n}(x)$ is also optimal in a certain sense; see Schoenberg (1971).

### Bernoulli Monosplines

A function of the form $x^{n}-S(x)$, with $S(x)\in\mathcal{S}_{n-1}$ is called a cardinal monospline of degree $n$. Again with the notation of §24.2(iii) define

 24.17.5 $M_{n}(x)=\begin{cases}\widetilde{B}_{n}\left(x\right)-B_{n},&n\text{ even},\\ \widetilde{B}_{n}\left(x+\frac{1}{2}\right),&n\text{ odd}.\end{cases}$

$M_{n}(x)$ is a monospline of degree $n$, and it follows from (24.4.25) and (24.4.27) that

 24.17.6 $M_{n}(k)=0,$ $k\in\mathbb{Z}$. ⓘ Symbols: $\in$: element of, $\mathbb{Z}$: set of all integers, $k$: integer, $n$: integer and $M_{n}(x)$: function Referenced by: §24.17(ii) Permalink: http://dlmf.nist.gov/24.17.E6 Encodings: TeX, pMML, png See also: Annotations for §24.17(ii), §24.17(ii), §24.17 and Ch.24

For each $n=1,2,\dots$ the function $M_{n}(x)$ is also the unique cardinal monospline of degree $n$ satisfying (24.17.6), provided that

 24.17.7 $M_{n}(x)=O\left(|x|^{\gamma}\right),$ $x\to\pm\infty$, ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $n$: integer, $x$: real or complex and $M_{n}(x)$: function Permalink: http://dlmf.nist.gov/24.17.E7 Encodings: TeX, pMML, png See also: Annotations for §24.17(ii), §24.17(ii), §24.17 and Ch.24

for some positive constant $\gamma$.

For any $n\geq 2$ the function

 24.17.8 $F(x)=\widetilde{B}_{n}\left(x\right)-2^{-n}B_{n}$

is the unique cardinal monospline of degree $n$ having the least supremum norm $\|F\|_{\infty}$ on $\mathbb{R}$ (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)).