The Bernoulli polynomials of the second kind[1][2] ψn(x), also known as the Fontana–Bessel polynomials,[3] are the polynomials defined by the following generating function:
Some authors define these polynomials slightly differently[4][5]
and may also use a different notation for them (the most used alternative notation is bn(x)).
Under this convention, the polynomials form a Sheffer sequence.
The Bernoulli polynomials of the second kind were largely studied by the Hungarian mathematician Charles Jordan,[1][2] but their history may also be traced back to the much earlier works.
[3] The Bernoulli polynomials of the second kind may be represented via these integrals[1][2]
π cos π x − sin π x ln z
π cos π x − v sin π x
{\displaystyle {\begin{aligned}\psi _{n}(x)&={\frac {\left(-1\right)^{n+1}}{\pi }}\int _{0}^{\infty }{\frac {\pi \cos \pi x-\sin \pi x\ln z}{(1+z)^{n}}}\cdot {\frac {z^{x}dz}{\ln ^{2}z+\pi ^{2}}},\qquad -1\leq x\leq n-1\,\\[3mm]\psi _{n}(x)&={\frac {\left(-1\right)^{n+1}}{\pi }}\int _{-\infty }^{+\infty }{\frac {\pi \cos \pi x-v\sin \pi x}{\left(1+e^{v}\right)^{n}}}\cdot {\frac {e^{v(x+1)}}{v^{2}+\pi ^{2}}}\,dv,\qquad -1\leq x\leq n-1\,\end{aligned}}}
These polynomials are, therefore, up to a constant, the antiderivative of the binomial coefficient and also that of the falling factorial.
[1][2][3] For an arbitrary n, these polynomials may be computed explicitly via the following summation formula[1][2][3]
where s(n,l) are the signed Stirling numbers of the first kind and Gn are the Gregory coefficients.
The expansion of the Bernoulli polynomials of the second kind into a Newton series reads[1][2]
It can be shown using the second integral representation and Vandermonde's identity.
The Bernoulli polynomials of the second kind satisfy the recurrence relation[1][2]
The repeated difference produces[1][2]
The main property of the symmetry reads[2][4]
Some properties and particular values of these polynomials include
{\displaystyle {\begin{aligned}&\psi _{n}(0)=G_{n}\\[2mm]&\psi _{n}(1)=G_{n-1}+G_{n}\\[2mm]&\psi _{n}(-1)=\left(-1\right)^{n+1}\sum _{m=0}^{n}\left|G_{m}\right|=\left(-1\right)^{n}C_{n}\\[2mm]&\psi _{n}(n-2)=-\left|G_{n}\right|\\[2mm]&\psi _{n}(n-1)=\left(-1\right)^{n}\psi _{n}(-1)=1-\sum _{m=1}^{n}\left|G_{m}\right|\\[2mm]&\psi _{2n}(n-1)=M_{2n}\\[2mm]&\psi _{2n}(n-1+y)=\psi _{2n}(n-1-y)\\[2mm]&\psi _{2n+1}(n-{\tfrac {1}{2}}+y)=-\psi _{2n+1}(n-{\tfrac {1}{2}}-y)\\[2mm]&\psi _{2n+1}(n-{\tfrac {1}{2}})=0\end{aligned}}}
where Cn are the Cauchy numbers of the second kind and Mn are the central difference coefficients.
[1][2][3] The digamma function Ψ(x) may be expanded into a series with the Bernoulli polynomials of the second kind in the following way[3]
γ = − ln ( a + 1 ) −
γ =
where γ is Euler's constant.
ln ( 2 π ) −
where Γ(x) is the gamma function.
The Hurwitz and Riemann zeta functions may be expanded into these polynomials as follows[3]
The Bernoulli polynomials of the second kind are also involved in the following relationship[3]
between the zeta functions, as well as in various formulas for the Stieltjes constants, e.g.[3]
γ