# Bernoulli polynomials of the second kind

The Bernoulli polynomials of the second kind ψn(x), also known as the Fontana-Bessel polynomials, are the polynomials defined by the following generating function: Bernoulli polynomials of the second kind_sentence_0

The first five polynomials are: Bernoulli polynomials of the second kind_sentence_1

Some authors define these polynomials slightly differently Bernoulli polynomials of the second kind_sentence_2

so that Bernoulli polynomials of the second kind_sentence_3

and may also use a different notation for them (the most used alternative notation is bn(x)). Bernoulli polynomials of the second kind_sentence_4

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. Bernoulli polynomials of the second kind_sentence_5

## Integral representations Bernoulli polynomials of the second kind_section_0

The Bernoulli polynomials of the second kind may be represented via these integrals Bernoulli polynomials of the second kind_sentence_6

as well as Bernoulli polynomials of the second kind_sentence_7

These polynomials are, therefore, up to a constant, the antiderivative of the binomial coefficient and also that of the falling factorial. Bernoulli polynomials of the second kind_sentence_8

## Explicit formula Bernoulli polynomials of the second kind_section_1

For an arbitrary n, these polynomials may be computed explicitly via the following summation formula Bernoulli polynomials of the second kind_sentence_9

where s(n,l) are the signed Stirling numbers of the first kind and Gn are the Gregory coefficients. Bernoulli polynomials of the second kind_sentence_10

## Recurrence formula Bernoulli polynomials of the second kind_section_2

The Bernoulli polynomials of the second kind satisfy the recurrence relation Bernoulli polynomials of the second kind_sentence_11

or equivalently Bernoulli polynomials of the second kind_sentence_12

The repeated difference produces Bernoulli polynomials of the second kind_sentence_13

## Symmetry property Bernoulli polynomials of the second kind_section_3

The main property of the symmetry reads Bernoulli polynomials of the second kind_sentence_14

## Some further properties and particular values Bernoulli polynomials of the second kind_section_4

Some properties and particular values of these polynomials include Bernoulli polynomials of the second kind_sentence_15

where Cn are the Cauchy numbers of the second kind and Mn are the central difference coefficients. Bernoulli polynomials of the second kind_sentence_16

## Expansion into a Newton series Bernoulli polynomials of the second kind_section_5

The expansion of the Bernoulli polynomials of the second kind into a Newton series reads Bernoulli polynomials of the second kind_sentence_17

## Some series involving the Bernoulli polynomials of the second kind Bernoulli polynomials of the second kind_section_6

The digamma function Ψ(x) may be expanded into a series with the Bernoulli polynomials of the second kind in the following way Bernoulli polynomials of the second kind_sentence_18

and hence Bernoulli polynomials of the second kind_sentence_19

and Bernoulli polynomials of the second kind_sentence_20

where γ is Euler's constant. Bernoulli polynomials of the second kind_sentence_21

Furthermore, we also have Bernoulli polynomials of the second kind_sentence_22

where Γ(x) is the gamma function. Bernoulli polynomials of the second kind_sentence_23

The Hurwitz and Riemann zeta functions may be expanded into these polynomials as follows Bernoulli polynomials of the second kind_sentence_24

and Bernoulli polynomials of the second kind_sentence_25

and also Bernoulli polynomials of the second kind_sentence_26

The Bernoulli polynomials of the second kind are also involved in the following relationship Bernoulli polynomials of the second kind_sentence_27

between the zeta functions, as well as in various formulas for the Stieltjes constants, e.g. Bernoulli polynomials of the second kind_sentence_28

and Bernoulli polynomials of the second kind_sentence_29