Bernoulli polynomials of the second kind

The Bernoulli polynomials of the second kind^[1][2] ψ_n(x), also known as the Fontana–Bessel polynomials,^[3] are the polynomials defined by the following generating function: $${\displaystyle {\frac {z(1+z)^{x}}{\ln(1+z)}}=\sum _{n=0}^{\infty }z^{n}\psi _{n}(x),\qquad |z|<1.}$$

The first five polynomials are: $${\displaystyle {\begin{aligned}\psi _{0}(x)&=1\\[2mm]\psi _{1}(x)&=x+{\frac {1}{2}}\\[2mm]\psi _{2}(x)&={\frac {1}{2}}x^{2}-{\frac {1}{12}}\\[2mm]\psi _{3}(x)&={\frac {1}{6}}x^{3}-{\frac {1}{4}}x^{2}+{\frac {1}{24}}\\[2mm]\psi _{4}(x)&={\frac {1}{24}}x^{4}-{\frac {1}{6}}x^{3}+{\frac {1}{6}}x^{2}-{\frac {19}{720}}\end{aligned}}}$$

Some authors define these polynomials slightly differently^[4][5] $${\displaystyle {\frac {z\left(1+z\right)^{x}}{\ln(1+z)}}=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}\psi _{n}^{*}(x),\qquad |z|<1,}$$ so that ${\displaystyle \psi _{n}^{*}(x)=\psi _{n}(x)\,n!}$ and may also use a different notation for them (the most used alternative notation is b_n(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] $${\displaystyle \psi _{n}(x)=\int _{x}^{x+1}\!{\binom {u}{n}}\,du=\int _{0}^{1}{\binom {x+u}{n}}\,du}$$ as well as^[3] $${\displaystyle {\begin{aligned}\psi _{n}(x)&={\frac {\left(-1\right)^{n+1}}{\pi }}\int _{0}^{\infty }{\frac {\pi \cos \pi x-\sin \pi x\ln z}{(1+z)^{n}}}\cdot {\frac {z^{x}dz}{\ln ^{2}z+\pi ^{2}}},\qquad -1\leq x\leq n-1\,\\[3mm]\psi _{n}(x)&={\frac {\left(-1\right)^{n+1}}{\pi }}\int _{-\infty }^{+\infty }{\frac {\pi \cos \pi x-v\sin \pi x}{\left(1+e^{v}\right)^{n}}}\cdot {\frac {e^{v(x+1)}}{v^{2}+\pi ^{2}}}\,dv,\qquad -1\leq x\leq n-1\,\end{aligned}}}$$

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] $${\displaystyle \psi _{n}(x)={\frac {1}{(n-1)!}}\sum _{l=0}^{n-1}{\frac {s(n-1,l)}{l+1}}x^{l+1}+G_{n},\qquad n=1,2,3,\ldots }$$ where s(n,l) are the signed Stirling numbers of the first kind and G_n are the Gregory coefficients.

The expansion of the Bernoulli polynomials of the second kind into a Newton series reads^[1][2] $${\displaystyle \psi _{n}(x)=G_{0}{\binom {x}{n}}+G_{1}{\binom {x}{n-1}}+G_{2}{\binom {x}{n-2}}+\ldots +G_{n}}$$ 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] $${\displaystyle \psi _{n}(x+1)-\psi _{n}(x)=\psi _{n-1}(x)}$$ or equivalently $${\displaystyle \Delta \psi _{n}(x)=\psi _{n-1}(x)}$$

The repeated difference produces^[1][2] $${\displaystyle \Delta ^{m}\psi _{n}(x)=\psi _{n-m}(x)}$$

The main property of the symmetry reads^[2][4] $${\displaystyle \psi _{n}{\left({\tfrac {1}{2}}n-1+x\right)}=\left(-1\right)^{n}\psi _{n}{\left({\tfrac {1}{2}}n-1-x\right)}}$$

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 C_n are the Cauchy numbers of the second kind and M_n 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] $${\displaystyle \Psi (v)=\ln(v+a)+\sum _{n=1}^{\infty }{\frac {(-1)^{n}\psi _{n}(a)\,(n-1)!}{(v)_{n}}},\qquad \Re (v)>-a,}$$ and hence^[3] $${\displaystyle \gamma =-\ln(a+1)-\sum _{n=1}^{\infty }{\frac {(-1)^{n}\psi _{n}(a)}{n}},\qquad \Re (a)>-1}$$ and $${\displaystyle \gamma =\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{2n}}\left\{\psi _{n}(a)+\psi _{n}\left(-{\frac {a}{1+a}}\right)\right\},\quad a>-1}$$ where γ is Euler's constant. Furthermore, we also have^[3] $${\displaystyle \Psi (v)={\frac {1}{v+a-{\frac {1}{2}}}}\left\{\ln \Gamma (v+a)+v-{\frac {1}{2}}\ln(2\pi )-{\frac {1}{2}}+\sum _{n=1}^{\infty }{\frac {\left(-1\right)^{n}\psi _{n+1}(a)}{(v)_{n}}}\left(n-1\right)!\right\},\quad \Re (v)>-a,}$$ where Γ(x) is the gamma function. The Hurwitz and Riemann zeta functions may be expanded into these polynomials as follows^[3] $${\displaystyle \zeta (s,v)={\frac {(v+a)^{1-s}}{s-1}}+\sum _{n=0}^{\infty }(-1)^{n}\psi _{n+1}(a)\sum _{k=0}^{n}\left(-1\right)^{k}{\binom {n}{k}}(k+v)^{-s}}$$ and $${\displaystyle \zeta (s)={\frac {(a+1)^{1-s}}{s-1}}+\sum _{n=0}^{\infty }(-1)^{n}\psi _{n+1}(a)\sum _{k=0}^{n}\left(-1\right)^{k}{\binom {n}{k}}(k+1)^{-s}}$$ and also $${\displaystyle \zeta (s)=1+{\frac {(a+2)^{1-s}}{s-1}}+\sum _{n=0}^{\infty }(-1)^{n}\psi _{n+1}(a)\sum _{k=0}^{n}\left(-1\right)^{k}{\binom {n}{k}}(k+2)^{-s}}$$

The Bernoulli polynomials of the second kind are also involved in the following relationship^[3] $${\displaystyle {\big (}v+a-{\tfrac {1}{2}}{\big )}\zeta (s,v)=-{\frac {\zeta (s-1,v+a)}{s-1}}+\zeta (s-1,v)+\sum _{n=0}^{\infty }\left(-1\right)^{n}\psi _{n+2}(a)\sum _{k=0}^{n}\left(-1\right)^{k}{\binom {n}{k}}(k+v)^{-s}}$$ between the zeta functions, as well as in various formulas for the Stieltjes constants, e.g.^[3] $${\displaystyle \gamma _{m}(v)=-{\frac {\ln ^{m+1}(v+a)}{m+1}}+\sum _{n=0}^{\infty }(-1)^{n}\psi _{n+1}(a)\sum _{k=0}^{n}\left(-1\right)^{k}{\binom {n}{k}}{\frac {\ln ^{m}(k+v)}{k+v}}}$$ and $${\displaystyle \gamma _{m}(v)={\frac {1}{{\tfrac {1}{2}}-v-a}}\left\{{\frac {(-1)^{m}}{m+1}}\,\zeta ^{(m+1)}(0,v+a)-(-1)^{m}\zeta ^{(m)}(0,v)-\sum _{n=0}^{\infty }(-1)^{n}\psi _{n+2}(a)\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}{\frac {\ln ^{m}(k+v)}{k+v}}\right\}}$$ which are both valid for ${\displaystyle \Re (a)>-1}$ and ${\displaystyle v\in \mathbb {C} \setminus \!\{0,-1,-2,\ldots \}}$.