I was exploring some raising and lowering operators related to an infinitesimal generator for fractional integro-derivatives and found an [Appell sequence](http://en.wikipedia.org/wiki/Appell_sequence) of polynomials, i.e., an infinite sequence of polynomials for which $\frac{d}{dx}p\_n(x)=np\_{n-1}(x)$, that is defined by the following recursion relation:


$p\_{0}(x)=1$, $p\_{1}(x)=x+\gamma$, and for $n>0$
$$p\_{n+1}(x)=(x+\gamma)p\_{n}(x)+\sum\_{j=1}^{n}(-1)^j\binom{n}{j}j!\zeta (j+1)p\_{n-j}(x)$$


where $\gamma=-\frac{\mathrm{d} }{\mathrm{d} \beta }\beta !\mid\_{\beta =0 }$, the Euler-Mascheroni constant, and $\zeta(s)$ is the Riemann zeta function.


They satisfy $$p\_{n}(x)=\frac{\mathrm{d^n} }{\mathrm{d} \beta^n }\frac{\exp(\beta x)}{\beta !}
\mid\_{\beta =0 }.$$


Explicitly,


$$p\_2(x)=(x+\gamma)^2-\zeta(2)$$
$$p\_3(x)=(x+\gamma)^3-3\zeta(2)(x+\gamma)+2\zeta(3)$$
$$p\_4(x)=(x+\gamma)^4-6\zeta(2)(x+\gamma)^2+8\zeta(3)(x+\gamma)+3[\zeta^2(2)-2\zeta(4)]$$
$$p\_5=p\_1^5-10\zeta(2)p\_1^3+20\zeta(3)p\_1^2+15[\zeta^2(2)-2\zeta(4)]p\_1+4[-5\zeta(2)\zeta(3)+6\zeta(5)]$$


**Can anyone provide a reference for these polynomials or point out an interesting combinatorial interpretation?**


***Background: Rich associations with fractional calculus, [digamma](http://en.wikipedia.org/wiki/Digamma_function) function, ladder operators***


The fractional integro-derivative can be represented as an exponentiated convolutional infinitesimal generator (cf. [MSE-Q125343](https://math.stackexchange.com/questions/125343/lie-group-heuristics-for-a-raising-operator-for-1n-fracdnd-betan-fra)):


$\displaystyle\frac{d^{-\beta}}{dx^{-\beta}}\frac{x^{\alpha}}{\alpha!}= \displaystyle\frac{x^{\alpha+\beta}}{(\alpha+\beta)!} = exp(-\beta R\_x) \frac{x^{\alpha}}{\alpha!}$


where


$$R\_xf(x)=\frac{1}{2\pi i}\displaystyle\oint\_{|z-x|=|x|}\frac{-ln(z-x)+\lambda}{z-x}f(z)dz$$


$$=(-ln(x)+\lambda)f(x)+\displaystyle\int\_{0}^{x}\frac{f\left ( x\right )-f(u)}{x-u}du.$$


with $\lambda=d\beta!/d\beta|\_{\beta=0}$. (Note the integrand is related to the [q (Jackson) derivative](http://en.wikipedia.org/wiki/Q-derivative), and the [Pincherle derivative](http://en.wikipedia.org/wiki/Pincherle_derivative) / commutator is $[R\_x,x]=D\_x^{-1}$.)


Then
$$exp(-\beta R\_x) 1 =\displaystyle\frac{x^\beta}{\beta!} = exp(-\beta\psi\_{.}(x)), $$


with $(\psi\_{.}(x))^n=\psi\_n(x)$, which implies


$$\psi\_{n}(x)=(-1)^n \frac{d^n}{d\beta^n}\frac{x^\beta}{\beta!}|\_{\beta=0},$$
$$L\_x\psi\_{n}(x)=n\psi\_{n-1}(x)=-x\displaystyle\frac{d}{dx}\psi\_{n}(x),$$
$$R\_x\psi\_{n}(x)=\psi\_{n+1}(x).$$


Let $x=e^z$ and $p\_n(z)=(-1)^n \psi\_{n}(e^z)$. Then


$$exp(-\beta R\_z) 1 =\displaystyle\frac{exp(\beta z)}{\beta!} = exp(\beta p\_{.}(z)), $$


$$L\_z p\_{n}(z)=n p\_{n-1}(z)=\displaystyle\frac{d}{dz} p\_{n}(z),$$
$$R\_z p\_{n}(z)= p\_{n+1}(z)= (z+\gamma)p\_n(z)-\displaystyle\int\_{-\infty}^{z}\frac{p\_n\left ( z\right )-p\_n(u)}{e^z-e^u} e^u du$$


with $\gamma=-d\beta!/d\beta|\_{\beta=0}$, the Euler-Mascheroni constant.


Since $p\_n(z)$ is an Appell sequence and, consequently, $p\_n(x+y)=(p.(x)+y)^n$, umbrally, a change of integration variables $\omega=z-u$ gives


$$R\_z p\_{n}(z)= p\_{n+1}(z)= (z+\gamma)p\_n(z)-\displaystyle\int\_{0}^{\infty}[p\_n(z)-(p\_{.}(z)-\omega)^n] \frac{1}{e^{\omega}-1}d\omega$$


from which the recursion formula follows.


In addition, using the operator formalism for [Sheffer sequences](http://mathworld.wolfram.com/ShefferSequence.html), of which the Appell is a special case,


$$R\_z=z-\frac{\mathrm{d} }{\mathrm{d} \beta}ln[\beta!]\mid \_{\beta=\frac{\mathrm{d} }{\mathrm{d} z}=D\_z}=z-\Psi(1+D\_z)$$


where $\Psi(x)$ is the [digamma or Psi function](http://en.wikipedia.org/wiki/Digamma_function).


**UPDATE (Nov. 16, 2012)**: Just found this exact sequence in the thesis "[Regularized Equivariant Euler Classes and Gamma Functions](https://digital.library.adelaide.edu.au/dspace/bitstream/2440/50479/8/02whole.pdf)" by R. Lu with a discussion of the relationships to Chern and Pontrjagin classes.



Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional calculus
Let $P\_i$ be the power sum symmetric function. In your $p\_n$, Replace $x+\gamma$ by $P\_1$ and $\zeta(i)$ by $P\_i$. Then divide the result by $n!$. What you get looks like a well-known symmetric function, which corresponds to the sign representation of the symmetric group $S\_n$.


