The Bernoulli polynomials of the second kind
, also known as the Fontana-Bessel polynomials,
[arXiv]
/ref> are the polynomials defined by the following generating function:
:
The first five polynomials are:
:
Some authors define these polynomials slightly differently
:
so that
:
and may also use a different notation for them (the most used alternative notation is ).
The Bernoulli polynomials of the second kind were largely studied by the Hungarian mathematician Charles Jordan, but their history may also be traced back to the much earlier works.
Integral representations
The Bernoulli polynomials of the second kind may be represented via these integrals
:
as well as
:
These polynomials are, therefore, up to a constant, the antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
of the binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
and also that of the falling factorial
In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial
:\begin
(x)_n = x^\underline &= \overbrace^ \\
&= \prod_^n(x-k+1) = \prod_^(x-k) \,.
\e ...
.
Explicit formula
For an arbitrary , these polynomials may be computed explicitly via the following summation formula
:
where are the signed Stirling numbers of the first kind
In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according to their number of cycles (counting fixed poin ...
and are the Gregory coefficients Gregory coefficients , also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, or Cauchy numbers of the first kind,Ch. Jordan. ''The Calculus of Finite Differences'' Chelsea Publishing Company, USA, 1947.L. Comtet. ''Adva ...
.
Recurrence formula
The Bernoulli polynomials of the second kind satisfy the recurrence relation
:
or equivalently
:
The repeated difference produces
:
Symmetry property
The main property of the symmetry reads
:
Some further properties and particular values
Some properties and particular values of these polynomials include
:
where are the ''Cauchy numbers of the second kind'' and are the ''central difference coefficients''.
Expansion into a Newton series
The expansion of the Bernoulli polynomials of the second kind into a Newton series reads
:
Some series involving the Bernoulli polynomials of the second kind
The digamma function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
:\psi(x)=\frac\ln\big(\Gamma(x)\big)=\frac\sim\ln-\frac.
It is the first of the polygamma functions. It is strictly increasing and strictly ...
may be expanded into a series with the Bernoulli polynomials of the second kind
in the following way
:
and hence
and
:
where is Euler's constant
Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma ().
It is defined as the limiting difference between the harmonic series and the natural ...
. Furthermore, we also have
:
where is the gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
. The Hurwitz
Hurwitz is one of the variants of a surname of Ashkenazi Jewish origin (for historical background see the Horowitz page).
Notable people with the surname include:
*Adolf Hurwitz (1859–1919), German mathematician
** Hurwitz polynomial
**Hurwitz m ...
and Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
s may be expanded into these
polynomials as follows
:
and
:
and also
:
The Bernoulli polynomials of the second kind are also involved in the following relationship
:
between the zeta functions, as well as in various formulas for the Stieltjes constants
In mathematics, the Stieltjes constants are the numbers \gamma_k that occur in the Laurent series expansion of the Riemann zeta function:
:\zeta(s)=\frac+\sum_^\infty \frac \gamma_n (s-1)^n.
The constant \gamma_0 = \gamma = 0.577\dots is known a ...
, e.g.
:
and
:
which are both valid for and .
See also
* Bernoulli polynomials
In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler–MacLaurin formula.
These polynomials occur in ...
* Stirling polynomials
* Gregory coefficients Gregory coefficients , also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, or Cauchy numbers of the first kind,Ch. Jordan. ''The Calculus of Finite Differences'' Chelsea Publishing Company, USA, 1947.L. Comtet. ''Adva ...
* Bernoulli numbers
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
* Difference polynomials In mathematics, in the area of complex analysis, the general difference polynomials are a polynomial sequence, a certain subclass of the Sheffer polynomials, which include the Newton polynomials, Selberg's polynomials, and the Stirling interpolatio ...
* Poly-Bernoulli number In mathematics, poly-Bernoulli numbers, denoted as B_^, were defined by M. Kaneko as
:=\sum_^B_^
where ''Li'' is the polylogarithm. The B_^ are the usual Bernoulli numbers.
Moreover, the Generalization of Poly-Bernoulli numbers with a,b,c paramet ...
* Mittag-Leffler polynomials
In mathematics, the Mittag-Leffler polynomials are the polynomials ''g'n''(''x'') or ''M'n''(''x'') studied by .
''M'n''(''x'') is a special case of the Meixner polynomial ''M'n''(''x;b,c'') at ''b = 0, c = -1''.
Definition and exam ...
References
{{Reflist, 2
Mathematics
Polynomials
Number theory