# Multiple zeta values related to fractional calculus and an Appell polynomial sequence

There is an Appell sequence of polynomials $$p_n(z)$$ related to an infinitesimal generator for one rep of the fractional calculus that have coefficients involving the Riemann zeta function values at integers greater than one. They can be generated several ways:

1) Through their exponential generating function

$$\frac{1}{t!} \exp(zt) = \exp[tp_.(z)]$$,

where $$(p.(z))^n = p_n(z)$$, or

2) $$\frac{1}{D_z!} z^n = p_n(z)$$

with the gamma function expressed in its Weierstrass factorization form and $$D_z=d/dz$$, or

3) through a recursion relation for the polynomials,

4) but most relevant to this question is through two reps of the raising operator incorporating the digamma function

$$R_z = z - \psi(1+D_x) = z + \gamma + \sum_{n>0} \zeta(n+1)D_z^n$$

$$= z + \gamma + \sum_{n>0} \frac{(-1)^n}{n}\binom{D_z}{n}$$

for which

$$R_z p_n(z) = p_{n+1}(z).$$ (These have real and complex contour integral reps as well.)

The first few polynomials are

$$p_0(z)=1$$

$$p_1(z)= z + \gamma$$

$$p_2(z)=(z+\gamma)^2-\zeta(2)$$

$$p_3(z)=(x+\gamma)^3-3\zeta(2)(z+\gamma)+2\zeta(3)$$

$$p_4(z)=(z+\gamma)^4-6\zeta(2)(z+\gamma)^2+8\zeta(3)(z+\gamma)+3[\zeta^2(2)-2\zeta(4)]$$

$$p_5=p_1^5-10\zeta(2)p_1^3+20\zeta(3)p_1^2+15[\zeta^2(2)-2\zeta(4)]p_1+4[-5\zeta(2)\zeta(3)+6\zeta(5)]$$.

Making a change of variable $$z=\ln(x)$$, the second raising op rep simplifies to

$$R_x= \ln(x) + \gamma + \sum_{n>0} \frac{(-1)^n}{n}\binom{xD_x}{n}$$

$$= \ln(x) + \gamma + \sum_{n>0} \frac{(-1)^n}{n}\frac{x^n}{n!}D_x^n.$$

Now the polynomials $$p_n(\ln(x))$$ can be regenerated as sums of products of the reciprocal integers similar to those for standard multiple zeta values, I believe.

Update: The references below give these coefficients as regularized MZVs since the formula for normal MZVs is divergent when naively applied in this case . So my revised question is

Updated Question: Are there some interesting summation identities similar to those for MZVs that emerge from using the raising operators?

References on regularized MZVs:

As happens often after I pose a question, I come across a new reference I hadn't found before. In "Analytic renormalization of multiple zeta functions. Geometry and combinatorics of the generalized Euler reflection formula for MZV" by Andrei Vieru, I found that the coefficients can indeed be expressed as MZVs. With $$z=0$$, we have the base numerical sequence for the Appell sequence expressed umbrally as $$p_n(z) = (z + p.(0))^n$$. The $$p_n(0)$$ are expressed as $$n! \zeta(1,1,..,1)$$ with $$n$$ ones. His approach is not clear to me yet. Any other references would be appreciated.

Further refs on the regularized MZVs are

A) "Regularized Equivariant Euler Classes and Gamma Functions" by Rongmin Lu

B) "Derivation and double shuffle relations for multiple zeta values" by Kentaro Ihara, Masanobu Kaneko and Don Zagier (Corollary 2 on pg 317)

C) "Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents" by Cartier (Eqn. 104 on pg. 155)

Compare these formula to the generating function for the cycle index polynomials of the symmetric groups, the refined Stirling numbers of the first kind OEIS A036039, as mentioned in the related MO-Q.

One way to see the connection to frac calc:

$$\frac{d}{dt} \exp[tp.(z)]= p.(z) \exp[tp.(z)] = R_z \exp[tp.(z)],$$

so

$$\exp[s\frac{d}{dt}] \exp[tp.(x)] = \exp[(s+t)p.(x)] = \exp[sR_z] \exp[tp.(x)],$$

and changing the variable

$$\exp[sR_x] \exp[tp.(\ln(x))]= \exp[sR_x] \frac{x^t}{t!} = \exp[(s+t)p.(\ln(x))] =\frac{x^{s+t}}{(s+t)!} = D_x^{-s} \frac{x^t}{t!}$$

if we identify

$$\exp[sR_x] = D_x^{-s}.$$

• Using the alternate expression of the raising op with no zetas, I'm getting a divergent expression for $p_3(x)$. – Tom Copeland Oct 7 at 2:01
• I think this will end up an exercise in how to extend a Newton series beyond its original range of convergence. Just hsve to remember how I did that years ago. Will delete this question and most likely put something on my blog. – Tom Copeland Oct 7 at 10:38
• Some background: "Multiple zeta valued and modular forms" about work by Matthes at maths.ox.ac.uk/node/33789 – Tom Copeland Oct 8 at 22:13
• Also "The Strange and Varied Appearances of Extended Multiple Zeta Values, or How a Pure Mathematician Sank so Low as to Write a Paper with Decimal Points" by Michael E. Hoffman – Tom Copeland Oct 14 at 2:01
• "Remark on divergent multizeta series (Microlocal Analysis and Asymptotic Analysis)" by Monvel – Tom Copeland Oct 15 at 16:44