Bernoulli polynomials

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In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. Bernoulli polynomials_sentence_0

They are used for series expansion of functions, and with the Euler–MacLaurin formula. Bernoulli polynomials_sentence_1

These polynomials occur in the study of many special functions and, in particular the Riemann zeta function and the Hurwitz zeta function. Bernoulli polynomials_sentence_2

They are an Appell sequence (i.e. a Sheffer sequence for the ordinary derivative operator). Bernoulli polynomials_sentence_3

For the Bernoulli polynomials, the number of crossings of the x-axis in the unit interval does not go up with the degree. Bernoulli polynomials_sentence_4

In the limit of large degree, they approach, when appropriately scaled, the sine and cosine functions. Bernoulli polynomials_sentence_5

A similar set of polynomials, based on a generating function, is the family of Euler polynomials. Bernoulli polynomials_sentence_6

Representations Bernoulli polynomials_section_0

The Bernoulli polynomials Bn can be defined by a generating function. Bernoulli polynomials_sentence_7

They also admit a variety of derived representations. Bernoulli polynomials_sentence_8

Generating functions Bernoulli polynomials_section_1

The generating function for the Bernoulli polynomials is Bernoulli polynomials_sentence_9

The generating function for the Euler polynomials is Bernoulli polynomials_sentence_10

Explicit formula Bernoulli polynomials_section_2

for n ≥ 0, where Bk are the Bernoulli numbers, and Ek are the Euler numbers. Bernoulli polynomials_sentence_11

Representation by a differential operator Bernoulli polynomials_section_3

The Bernoulli polynomials are also given by Bernoulli polynomials_sentence_12

where D = d/dx is differentiation with respect to x and the fraction is expanded as a formal power series. Bernoulli polynomials_sentence_13

It follows that Bernoulli polynomials_sentence_14

cf. Bernoulli polynomials_sentence_15

integrals below. Bernoulli polynomials_sentence_16

By the same token, the Euler polynomials are given by Bernoulli polynomials_sentence_17

Representation by an integral operator Bernoulli polynomials_section_4

The Bernoulli polynomials are also the unique polynomials determined by Bernoulli polynomials_sentence_18

The integral transform Bernoulli polynomials_sentence_19

on polynomials f, simply amounts to Bernoulli polynomials_sentence_20

This can be used to produce the inversion formulae below. Bernoulli polynomials_sentence_21

Another explicit formula Bernoulli polynomials_section_5

An explicit formula for the Bernoulli polynomials is given by Bernoulli polynomials_sentence_22

That is similar to the series expression for the Hurwitz zeta function in the complex plane. Bernoulli polynomials_sentence_23

Indeed, there is the relationship Bernoulli polynomials_sentence_24

where ζ(s, q) is the Hurwitz zeta function. Bernoulli polynomials_sentence_25

The latter generalizes the Bernoulli polynomials, allowing for non-integer values of n. Bernoulli polynomials_sentence_26

The inner sum may be understood to be the nth forward difference of x; that is, Bernoulli polynomials_sentence_27

where Δ is the forward difference operator. Bernoulli polynomials_sentence_28

Thus, one may write Bernoulli polynomials_sentence_29

This formula may be derived from an identity appearing above as follows. Bernoulli polynomials_sentence_30

Since the forward difference operator Δ equals Bernoulli polynomials_sentence_31

where D is differentiation with respect to x, we have, from the Mercator series, Bernoulli polynomials_sentence_32

As long as this operates on an mth-degree polynomial such as x, one may let n go from 0 only up to m. Bernoulli polynomials_sentence_33

An integral representation for the Bernoulli polynomials is given by the Nörlund–Rice integral, which follows from the expression as a finite difference. Bernoulli polynomials_sentence_34

An explicit formula for the Euler polynomials is given by Bernoulli polynomials_sentence_35

The above follows analogously, using the fact that Bernoulli polynomials_sentence_36

Sums of pth powers Bernoulli polynomials_section_6

(assuming 0 = 1). Bernoulli polynomials_sentence_37

See Faulhaber's formula for more on this. Bernoulli polynomials_sentence_38

The Bernoulli and Euler numbers Bernoulli polynomials_section_7

Explicit expressions for low degrees Bernoulli polynomials_section_8

The first few Bernoulli polynomials are: Bernoulli polynomials_sentence_39

The first few Euler polynomials are: Bernoulli polynomials_sentence_40

Maximum and minimum Bernoulli polynomials_section_9

At higher n, the amount of variation in Bn(x) between x = 0 and x = 1 gets large. Bernoulli polynomials_sentence_41

For instance, Bernoulli polynomials_sentence_42

which shows that the value at x = 0 (and at x = 1) is −3617/510 ≈ −7.09, while at x = 1/2, the value is 118518239/3342336 ≈ +7.09. Bernoulli polynomials_sentence_43

D.H. Bernoulli polynomials_sentence_44 Lehmer showed that the maximum value of Bn(x) between 0 and 1 obeys Bernoulli polynomials_sentence_45

unless n is 2 modulo 4, in which case Bernoulli polynomials_sentence_46

unless n is 0 modulo 4, in which case Bernoulli polynomials_sentence_47

These limits are quite close to the actual maximum and minimum, and Lehmer gives more accurate limits as well. Bernoulli polynomials_sentence_48

Differences and derivatives Bernoulli polynomials_section_10

The Bernoulli and Euler polynomials obey many relations from umbral calculus: Bernoulli polynomials_sentence_49

(Δ is the forward difference operator). Bernoulli polynomials_sentence_50

Also, Bernoulli polynomials_sentence_51

These polynomial sequences are Appell sequences: Bernoulli polynomials_sentence_52

Translations Bernoulli polynomials_section_11

These identities are also equivalent to saying that these polynomial sequences are Appell sequences. Bernoulli polynomials_sentence_53

(Hermite polynomials are another example.) Bernoulli polynomials_sentence_54

Symmetries Bernoulli polynomials_section_12

Zhi-Wei Sun and Hao Pan established the following surprising symmetry relation: If r + s + t = n and x + y + z = 1, then Bernoulli polynomials_sentence_55

where Bernoulli polynomials_sentence_56

Fourier series Bernoulli polynomials_section_13

The Fourier series of the Bernoulli polynomials is also a Dirichlet series, given by the expansion Bernoulli polynomials_sentence_57

Note the simple large n limit to suitably scaled trigonometric functions. Bernoulli polynomials_sentence_58

This is a special case of the analogous form for the Hurwitz zeta function Bernoulli polynomials_sentence_59

This expansion is valid only for 0 ≤ x ≤ 1 when n ≥ 2 and is valid for 0 < x < 1 when n = 1. Bernoulli polynomials_sentence_60

The Fourier series of the Euler polynomials may also be calculated. Bernoulli polynomials_sentence_61

Defining the functions Bernoulli polynomials_sentence_62

and Bernoulli polynomials_sentence_63

and Bernoulli polynomials_sentence_64

and Bernoulli polynomials_sentence_65

and Bernoulli polynomials_sentence_66

Inversion Bernoulli polynomials_section_14

The Bernoulli and Euler polynomials may be inverted to express the monomial in terms of the polynomials. Bernoulli polynomials_sentence_67

Specifically, evidently from the above section on integral operators, it follows that Bernoulli polynomials_sentence_68

and Bernoulli polynomials_sentence_69

Relation to falling factorial Bernoulli polynomials_section_15

denotes the Stirling number of the second kind. Bernoulli polynomials_sentence_70

The above may be inverted to express the falling factorial in terms of the Bernoulli polynomials: Bernoulli polynomials_sentence_71

where Bernoulli polynomials_sentence_72

denotes the Stirling number of the first kind. Bernoulli polynomials_sentence_73

Multiplication theorems Bernoulli polynomials_section_16

The multiplication theorems were given by Joseph Ludwig Raabe in 1851: Bernoulli polynomials_sentence_74

For a natural number m≥1, Bernoulli polynomials_sentence_75

Integrals Bernoulli polynomials_section_17

Two definite integrals relating the Bernoulli and Euler polynomials to the Bernoulli and Euler numbers are: Bernoulli polynomials_sentence_76

Periodic Bernoulli polynomials Bernoulli polynomials_section_18

A periodic Bernoulli polynomial Pn(x) is a Bernoulli polynomial evaluated at the fractional part of the argument x. Bernoulli polynomials_sentence_77

These functions are used to provide the remainder term in the Euler–Maclaurin formula relating sums to integrals. Bernoulli polynomials_sentence_78

The first polynomial is a sawtooth function. Bernoulli polynomials_sentence_79

Strictly these functions are not polynomials at all and more properly should be termed the periodic Bernoulli functions, and P0(x) is not even a function, being the derivative of a sawtooth and so a Dirac comb. Bernoulli polynomials_sentence_80

See also Bernoulli polynomials_section_19

Bernoulli polynomials_unordered_list_0


Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Bernoulli polynomials.