Bernoulli polynomials of the second kind

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

${\frac {z(1+z)^{x}}{\ln(1+z)}}=\sum _{n=0}^{\infty }z^{n}\psi _{n}(x),\qquad |z|<1.$

The first five polynomials are:

${\begin{array}{l}\displaystyle \psi _{0}(x)=1\\[2mm]\displaystyle \psi _{1}(x)=x+{\frac {1}{2}}\\[2mm]\displaystyle \psi _{2}(x)={\frac {1}{2}}x^{2}-{\frac {1}{12}}\\[2mm]\displaystyle \psi _{3}(x)={\frac {1}{6}}x^{3}-{\frac {1}{4}}x^{2}+{\frac {1}{24}}\\[2mm]\displaystyle \psi _{4}(x)={\frac {1}{24}}x^{4}-{\frac {1}{6}}x^{3}+{\frac {1}{6}}x^{2}-{\frac {19}{720}}\end{array}}$

Some authors define these polynomials slightly differently

${\frac {z(1+z)^{x}}{\ln(1+z)}}=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}\psi _{n}^{*}(x),\qquad |z|<1,$

so that

$\psi _{n}^{*}(x)=\psi _{n}(x)\,n!$

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

$\psi _{n}(x)=\int \limits _{x}^{x+1}\!{\binom {u}{n}}\,du=\int \limits _{0}^{1}{\binom {x+u}{n}}\,du$

as well as

${\begin{array}{l}\displaystyle \psi _{n}(x)={\frac {(-1)^{n+1}}{\pi }}\int \limits _{0}^{\infty }{\frac {\pi \cos \pi x-\sin \pi x\ln z}{(1+z)^{n}}}\cdot {\frac {z^{x}dz}{\ln ^{2}z+\pi ^{2}}},\qquad -1\leq x\leq n-1\,\\[3mm]\displaystyle \psi _{n}(x)={\frac {(-1)^{n+1}}{\pi }}\int \limits _{-\infty }^{+\infty }{\frac {\pi \cos \pi x-v\sin \pi x}{\,(1+e^{v})^{n}}}\cdot {\frac {e^{v(x+1)}}{v^{2}+\pi ^{2}}}\,dv,\qquad -1\leq x\leq n-1\,\end{array}}$

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

Explicit formula

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

$\psi _{n}(x)={\frac {1}{(n-1)!}}\sum _{l=0}^{n-1}{\frac {s(n-1,l)}{l+1}}x^{l+1}+G_{n},\qquad n=1,2,3,\ldots$

where where s(n,l) are the signed Stirling numbers of the first kind and G_n are the Gregory coefficients.

Recurrence formula

The Bernoulli polynomials of the second kind satisfy the recurrence relation

$\psi _{n}(x+1)-\psi _{n}(x)=\psi _{n-1}(x)$

or equivalently

$\Delta \psi _{n}(x)=\psi _{n-1}(x)$

The repeated difference produces

$\Delta ^{m}\psi _{n}(x)=\psi _{n-m}(x)$

Symmetry property

The main property of the symmetry reads

$\psi _{n}({\tfrac {1}{2}}n-1+x)=(-1)^{n}\psi _{n}({\tfrac {1}{2}}n-1-x)$

Some further properties and particular values

Some properties and particular values of these polynomials include

${\begin{array}{l}\displaystyle \psi _{n}(0)=G_{n}\\[2mm]\displaystyle \psi _{n}(1)=G_{n-1}+G_{n}\\[2mm]\displaystyle \psi _{n}(-1)=(-1)^{n+1}\sum _{m=0}^{n}|G_{m}|=(-1)^{n}C_{n}\\[2mm]\displaystyle \psi _{n}(n-2)=-|G_{n}|\\[2mm]\displaystyle \psi _{n}(n-1)=(-1)^{n}\psi _{n}(-1)=1-\sum _{m=1}^{n}|G_{m}|\\[2mm]\displaystyle \psi _{2n}(n-1)=M_{2n}\\[2mm]\displaystyle \psi _{2n}(n-1+y)=\psi _{2n}(n-1-y)\\[2mm]\displaystyle \psi _{2n+1}(n-{\tfrac {1}{2}}+y)=-\psi _{2n+1}(n-{\tfrac {1}{2}}-y)\\[2mm]\displaystyle \psi _{2n+1}(n-{\tfrac {1}{2}})=0\end{array}}$

where C_n are the Cauchy numbers of the second kind and M_n 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

$\psi _{n}(x)=G_{0}{\binom {x}{n}}+G_{1}{\binom {x}{n-1}}+G_{2}{\binom {x}{n-2}}+\ldots +G_{n}$

Some series involving the Bernoulli polynomials of the second kind

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

$\Psi (v)=\ln(v+a)+\sum _{n=1}^{\infty }{\frac {(-1)^{n}\psi _{n}(a)\,(n-1)!}{(v)_{n}}},\qquad \Re (v)>-a,$

and hence

$\gamma =-\ln(a+1)-\sum _{n=1}^{\infty }{\frac {(-1)^{n}\psi _{n}(a)}{n}},\qquad \Re (a)>-1$

and

$\gamma =\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{2n}}{\Big \{}\psi _{n}(a)+\psi _{n}{\Big (}-{\frac {a}{1+a}}{\Big )}{\Big \}},\quad a>-1$

where γ is Euler's constant. Furthermore, we also have

$\Psi (v)={\frac {1}{v+a-{\tfrac {1}{2}}}}\left\{\ln \Gamma (v+a)+v-{\frac {1}{2}}\ln 2\pi -{\frac {1}{2}}+\sum _{n=1}^{\infty }{\frac {(-1)^{n}\psi _{n+1}(a)}{(v)_{n}}}(n-1)!\right\},\qquad \Re (v)>-a,$

where Γ(x) is the gamma function. The Hurwitz and Riemann zeta functions may be expanded into these polynomials as follows

$\zeta (s,v)={\frac {(v+a)^{1-s}}{s-1}}+\sum _{n=0}^{\infty }(-1)^{n}\psi _{n+1}(a)\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(k+v)^{-s}$

and

$\zeta (s)={\frac {(a+1)^{1-s}}{s-1}}+\sum _{n=0}^{\infty }(-1)^{n}\psi _{n+1}(a)\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(k+1)^{-s}$

and also

$\zeta (s)=1+{\frac {(a+2)^{1-s}}{s-1}}+\sum _{n=0}^{\infty }(-1)^{n}\psi _{n+1}(a)\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(k+2)^{-s}$

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

${\big (}v+a-{\tfrac {1}{2}}{\big )}\zeta (s,v)=-{\frac {\zeta (s-1,v+a)}{s-1}}+\zeta (s-1,v)+\sum _{n=0}^{\infty }(-1)^{n}\psi _{n+2}(a)\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(k+v)^{-s}$

between the zeta functions, as well as in various formulas for the Stieltjes constants, e.g.

$\gamma _{m}(v)=-{\frac {\ln ^{m+1}(v+a)}{m+1}}+\sum _{n=0}^{\infty }(-1)^{n}\psi _{n+1}(a)\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}{\frac {\ln ^{m}(k+v)}{k+v}}$

and

$\gamma _{m}(v)={\frac {1}{{\tfrac {1}{2}}-v-a}}\left\{{\frac {(-1)^{m}}{m+1}}\,\zeta ^{(m+1)}(0,v+a)-(-1)^{m}\zeta ^{(m)}(0,v)-\sum _{n=0}^{\infty }(-1)^{n}\psi _{n+2}(a)\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}{\frac {\ln ^{m}(k+v)}{k+v}}\right\}$

which are both valid for $\Re (a)>-1$ and $v\in \mathbb {C} \setminus \!\{0,-1,-2,\ldots \}$.

See also

• Bernoulli polynomials
• Stirling polynomials
• Gregory coefficients
• Bernoulli numbers
• Difference polynomials
• Poly-Bernoulli number
• Mittag-Leffler polynomials