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In mathematics, the Bernoulli polynomials occur in the study of many special functions and in particular the Riemann zeta function and the Hurwitz zeta function. This is in large part due to the fact that they are Sheffer sequences for the ordinary derivative operator. Unlike orthogonal polynomials, the Bernoulli polynomials are remarkable in that the number of crossings of the x-axis in the unit interval does not go up as the degree of the polynomials goes up. In the limit of large degree, the Bernoulli polynomials, appropriately scaled, approach the sine and cosine functions. Mathematics is the study of quantity, structure, space and change. ...
In mathematics, there is a theory or theories of special functions, particular functions such as the trigonometric functions that have useful or attractive properties, and which occur in different applications often enough to warrant a name and attention of their own. ...
In mathematics, the Riemann zeta function is a function which is of paramount importance in number theory, because of its relation to the distribution of prime numbers. ...
In mathematics, the Hurwitz zeta function is one of the many zeta functions. ...
In mathematics, a polynomial sequence, i. ...
In mathematics, the derivative of a function is one of the two central concepts of calculus. ...
In mathematics, two polynomials f and g are orthogonal to each other with respect to a nonnegative weight function w precisely if In other words, if polynomials are treated as vectors and the inner product of two polynomials f(x) and g(x) is defined as then the orthogonal polynomials...
In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ...
In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...
Generating functions
The generating function for the Bernoulli polynomials is In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ...
The generating function for the Euler polynomials is
The Bernoulli and Euler numbers The Bernoulli numbers are given by Bn = Bn(0). In mathematics, the Bernoulli numbers Bn were first discovered in connection with the closed forms of the sums for various fixed values of n. ...
The Euler numbers are given by En = 2nEn(1 / 2). The Euler numbers are a sequence En of integers defined by the following Taylor series expansion: (Note that e, the base of the natural logarithm, is also occasionally called Eulers number, as is the Euler characteristic. ...
Explicit expressions for low degrees The first few Bernoulli polynomials are: The first few Euler polynomials are
Differences The Bernoulli and Euler polynomials obey many relations from umbral calculus. In mathematics, before the 1970s, the term umbral calculus was understood to mean the surprising similarities between otherwise unrelated polynomial equations, and certain shadowy techniques that can be used to prove them. ...
- Bn(x + 1) − Bn(x) = nxn − 1
- En(x + 1) + En(x) = 2xn
Derivatives These polynomial sequences are Appel sequences: In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. ...
In mathematics, a polynomial sequence, i. ...
- Bn'(x) = nBn − 1(x)
- En'(x) = nEn − 1(x)
Translations These identities are also equivalent to saying that these polynomial sequences are Appel sequences. (Hermite polynomials are another example.) In mathematics, a polynomial sequence, i. ...
In mathematics, the Hermite polynomials, named in honor of Charles Hermite (pronounced air MEET), are a polynomial sequence defined either by (the probabilists Hermite polynomials), or sometimes by (the physicists Hermite polynomials). These two definitions are not exactly equivalent; either is a trivial rescaling of the other. ...
Symmetries - Bn(1 − x) = ( − )nBn(x)
- En(1 − x) = ( − )nEn(x)
- ( − )nBn( − x) = Bn(x) + nxn − 1
- ( − )nEn( − x) = − En(x) + 2xn
Fourier series The Fourier series of the Bernoulli polynomials is also a Dirichlet series and is a special case of the Hurwitz zeta function In mathematics, a Fourier series, named in honor of Joseph Fourier (1768-1830), is a representation of a periodic function with period 2π as a sum of periodic functions of the form which are the harmonics of ei x. ...
In mathematics, a Dirichlet series, one of a number of concepts named in honor of Johann Peter Gustav Lejeune Dirichlet, is a series of the form The most famous of Dirichlet series is which is the Riemann zeta function. ...
In mathematics, the Hurwitz zeta function is one of the many zeta functions. ...
Multiplication theorems
References - Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, (1972) Dover, New York. (See Chapter 23 (http://www.math.sfu.ca/~cbm/aands/page_804.htm))
- Tom M. Apostol Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. (See Chapter 12.11)
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