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:

{\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^[4]^[5]

{\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!

and may also use a different notation for them (the most used alternative notation is $b n (x)$ ).

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].

Integral representations

The Bernoulli polynomials of the second kind may be represented via these integrals^[1]^[2]

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

as well as^[3]

{\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.^[1]^[2]^[3]

Explicit formula

For an arbitrary $n$ , these polynomials may be computed explicitly via the following summation formula^[1]^[2]^[3]

\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^[1]^[2]

\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^[1]^[2]

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

Symmetry property

The main property of the symmetry reads^[2]^[4]

\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.^[1]^[2]^[3]

Expansion into a Newton series

The expansion of the Bernoulli polynomials of the second kind into a Newton series reads^[1]^[2]

\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^[3]

\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]

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

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^[3]

\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^[3]

\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^[3]

{\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.^[3]

\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 $v\in \mathbb {C} \setminus \!\{0,-1,-2,\ldots \}$ .

References

1 2 3 4 5 6 7 8 9 Jordan, Charles (1928), "Sur des polynomes analogues aux polynomes de Bernoulli, et sur des formules de sommation analogues à celle de Maclaurin-Euler", Acta Sci. Math. (Szeged), 4: 130–150
1 2 3 4 5 6 7 8 9 10 Jordan, Charles (1965). The Calculus of Finite Differences (3rd Edition). Chelsea Publishing Company.
1 2 3 4 5 6 7 8 9 10 11 12 Blagouchine, Iaroslav V. (2018), "Three notes on Ser's and Hasse's representations for the zeta-functions" (PDF), Integers (Electronic Journal of Combinatorial Number Theory), 18A (#A3): 1–45 arXiv
1 2 Roman, S. (1984). The Umbral Calculus. New York: Academic Press.
↑ Weisstein, Eric W. Bernoulli Polynomial of the Second Kind. From MathWorld--A Wolfram Web Resource.

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[jordan2-1] 1 2 3 4 5 6 7 8 9 Jordan, Charles (1928), "Sur des polynomes analogues aux polynomes de Bernoulli, et sur des formules de sommation analogues à celle de Maclaurin-Euler", Acta Sci. Math. (Szeged), 4: 130–150

[jordan-2] 1 2 3 4 5 6 7 8 9 10 Jordan, Charles (1965). The Calculus of Finite Differences (3rd Edition). Chelsea Publishing Company.

[blagouchine-3] 1 2 3 4 5 6 7 8 9 10 11 12 Blagouchine, Iaroslav V. (2018), "Three notes on Ser's and Hasse's representations for the zeta-functions" (PDF), Integers (Electronic Journal of Combinatorial Number Theory), 18A (#A3): 1–45 arXiv

[roman-4] 1 2 Roman, S. (1984). The Umbral Calculus. New York: Academic Press.

[5] Weisstein, Eric W. Bernoulli Polynomial of the Second Kind. From MathWorld--A Wolfram Web Resource.