24 Bernoulli and Euler Polynomials Applications 24.16 Generalizations 24.18 Physical Applications

§24.17 Mathematical Applications

Keywords:: Bernoulli polynomials, Euler polynomials
Permalink:: http://dlmf.nist.gov/24.17

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

§24.17(i) Summation

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

¶ Euler–Maclaurin Summation Formula

Keywords:: Euler–Maclaurin formula, summation formulas

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

¶ Boole Summation Formula

Keywords:: Boole summation formula, summation formulas

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

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, : : factorial, : integer, : integer, : integer, : parameter, : integer, : integer, $f^{{(m)}}(x)$ : integrable function and $R_{m}(n)$ : remainder
Permalink:: http://dlmf.nist.gov/24.17.E1
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where

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

Symbols:: : differential of , : : factorial, $\int$ : integral, $\mathop{\widetilde{E}_{{n}}\/}\nolimits\!\left(x\right)$ : 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
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¶ Calculus of Finite Differences

Keywords:: 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

Referenced by:: §3.11(vi)
Permalink:: http://dlmf.nist.gov/24.17.ii

¶ Euler Splines

Notes:: See Schumaker (1981, pp. 152–153), and Schoenberg (1973, pp. 40–41 and 101).
Keywords:: Euler splines, cardinal spline functions, monosplines, spline functions

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

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

Symbols:: $\mathop{\widetilde{E}_{{n}}\/}\nolimits\!\left(x\right)$ : periodic Euler functions, : integer, : real or complex and $S_{n}(x)$ : Euler spline
Permalink:: http://dlmf.nist.gov/24.17.E3
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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 $S_{n}(k)=(-1)^{k},$ $k\in\Integer$ .

Symbols:: $\in$ : element of, $\Integer$ : set of all integers, : integer, : integer and $S_{n}(x)$ : Euler spline
Permalink:: http://dlmf.nist.gov/24.17.E4
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The function $S_{n}(x)$ is also optimal in a certain sense; see Schoenberg (1971).

¶ Bernoulli Monosplines

Keywords:: Bernoulli monosplines, cardinal monosplines, monosplines, spline functions

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

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\widetilde{B}_{{n}}\/}\nolimits\!\left(x\right)$ : periodic Bernoulli functions, : integer, : real or complex and $M_{n}(x)$ : function
Permalink:: http://dlmf.nist.gov/24.17.E5
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$M_{n}(x)$ is a monospline of degree , and it follows from (24.4.25) and (24.4.27) that

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

Symbols:: $\in$ : element of, $\Integer$ : set of all integers, : integer, : integer and $M_{n}(x)$ : function
Referenced by:: ¶ ‣ §24.17(ii)
Permalink:: http://dlmf.nist.gov/24.17.E6
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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 $M_{n}(x)=\mathop{O\/}\nolimits\!\left(|x|^{\gamma}\right),$ $x\to\pm\infty$ ,

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, : integer, : real or complex and $M_{n}(x)$ : function
Permalink:: http://dlmf.nist.gov/24.17.E7
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for some positive constant $\gamma$ .

For any $n\geq 2$ the function

24.17.8 $F(x)=\mathop{\widetilde{B}_{{n}}\/}\nolimits\!\left(x\right)-2^{{-n}}\mathop{B% _{{n}}\/}\nolimits$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\widetilde{B}_{{n}}\/}\nolimits\!\left(x\right)$ : periodic Bernoulli functions, : integer and : real or complex
Permalink:: http://dlmf.nist.gov/24.17.E8
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is the unique cardinal monospline of degree having the least supremum norm $\|F\|_{\infty}$ on $\Real$ (minimality property).

§24.17(iii) Number Theory

Keywords:: Bernoulli polynomials, Euler polynomials, Fermat’s last theorem, Riemann zeta function, cyclotomic fields, number theory
Permalink:: http://dlmf.nist.gov/24.17.iii

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)).

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