24 Bernoulli and Euler Polynomials Applications 24.16 Generalizations 24.18 Physical Applications

§24.17 Mathematical Applications

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Keywords:: Bernoulli polynomials, Euler polynomials, applications, mathematical
Permalink:: http://dlmf.nist.gov/24.17
See also:: Annotations for Ch.24

§24.17(i) Summation
§24.17(ii) Spline Functions
§24.17(iii) Number Theory

§24.17(i) Summation

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Notes:: See Temme (1996b, pp. 17 and 18) or Nörlund (1924, pp. 29–36).
Permalink:: http://dlmf.nist.gov/24.17.i
See also:: Annotations for §24.17 and Ch.24

Euler–Maclaurin Summation Formula

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Keywords:: Euler–Maclaurin, Euler–Maclaurin formula, generalization, summation formulas
See also:: Annotations for §24.17(i), §24.17 and Ch.24

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

Boole Summation Formula

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Keywords:: Boole, Boole summation formula, summation formulas
See also:: Annotations for §24.17(i), §24.17 and Ch.24

Let $0\leq h\leq 1$ and , and be integers such that , , and $f^{(m)}(x)$ is absolutely integrable over . Then with the notation of §24.2(iii)

24.17.1 $\sum_{j=a}^{n-1}(-1)^{j}f(j+h)=\frac{1}{2}\sum_{k=0}^{m-1}\frac{E_{k}\left(h% \right)}{k!}\left((-1)^{n-1}f^{(k)}(n)+(-1)^{a}f^{(k)}(a)\right)+R_{m}(n),$

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Symbols:: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, : factorial (as in ), : integer, : integer, : integer, : parameter, : integer, : integer, $f^{(m)}(x)$ : integrable function and $R_{m}(n)$ : remainder
Permalink:: http://dlmf.nist.gov/24.17.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §24.17(i), §24.17(i), §24.17 and Ch.24

where

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

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , : factorial (as in ), $\int$ : integral, $\widetilde{E}_{\NVar{n}}\left(\NVar{x}\right)$ : periodic Euler functions, : integer, : real or complex, : parameter, : integer, : integer, $f^{(m)}(x)$ : integrable function and $R_{m}(n)$ : remainder
Permalink:: http://dlmf.nist.gov/24.17.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §24.17(i), §24.17(i), §24.17 and Ch.24

Calculus of Finite Differences

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Keywords:: calculus of finite differences
See also:: Annotations for §24.17(i), §24.17 and Ch.24

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

§24.17(ii) Spline Functions

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Referenced by:: §3.11(vi)
Permalink:: http://dlmf.nist.gov/24.17.ii
See also:: Annotations for §24.17 and Ch.24

Euler Splines

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Keywords:: Euler splines, cardinal, cardinal spline functions, cardinal splines, monosplines, spline functions
Notes:: See Schumaker (1981, pp. 152–153), and Schoenberg (1973, pp. 40–41 and 101).
See also:: Annotations for §24.17(ii), §24.17 and Ch.24

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

24.17.3 $S_{n}(x)=\frac{\widetilde{E}_{n}\left(x+\tfrac{1}{2}n+\tfrac{1}{2}\right)}{% \widetilde{E}_{n}\left(\tfrac{1}{2}n+\tfrac{1}{2}\right)},$ $n=0,1,\dots$ ,

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Symbols:: $\widetilde{E}_{\NVar{n}}\left(\NVar{x}\right)$ : periodic Euler functions, : integer and : 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 . For each , $S_{n}(x)$ is the unique bounded function such that $S_{n}(x)\in\mathcal{S}_{n}$ and

24.17.4 $S_{n}(k)=(-1)^{k},$ $k\in\mathbb{Z}$ .

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Symbols:: $\in$ : element of, $\mathbb{Z}$ : set of all integers, : integer and : 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

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Keywords:: Bernoulli, Bernoulli monosplines, cardinal monosplines, monosplines, spline functions
See also:: Annotations for §24.17(ii), §24.17 and Ch.24

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

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

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)$ : periodic Bernoulli functions, : integer, : real or complex and $M_{n}(x)$ : function
Permalink:: http://dlmf.nist.gov/24.17.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §24.17(ii), §24.17(ii), §24.17 and Ch.24

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

24.17.6 $M_{n}(k)=0,$ $k\in\mathbb{Z}$ .

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Symbols:: $\in$ : element of, $\mathbb{Z}$ : set of all integers, : integer, : 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 satisfying (24.17.6), provided that

24.17.7 $M_{n}(x)=O\left(|x|^{\gamma}\right),$ $x\to\pm\infty$ ,

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, : integer, : 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 $F(x)=\widetilde{B}_{n}\left(x\right)-2^{-n}B_{n}$

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)$ : periodic Bernoulli functions, : integer and : real or complex
Permalink:: http://dlmf.nist.gov/24.17.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §24.17(ii), §24.17(ii), §24.17 and Ch.24

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

§24.17(iii) Number Theory

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Keywords:: Bernoulli and Euler numbers and polynomials, Bernoulli and Euler polynomials, Bernoulli polynomials, Euler polynomials, Fermat’s last theorem, Riemann zeta function, applications, cyclotomic fields, mathematical, number theory, relations to other functions
Permalink:: http://dlmf.nist.gov/24.17.iii
See also:: Annotations for §24.17 and Ch.24

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 -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)); -adic analysis (Koblitz (1984, Chapter 2)).

24.16 Generalizations 24.18 Physical Applications

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