Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional calculus - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T06:53:50Z http://mathoverflow.net/feeds/question/111165 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/111165/riemann-zeta-function-at-positive-integers-and-an-appell-sequence-of-polynomials Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional calculus Tom Copeland 2012-11-01T15:48:45Z 2012-11-16T10:39:14Z <p>I was exploring some raising and lowering operators related to an infinitesimal generator for fractional integro-derivatives and found an <a href="http://en.wikipedia.org/wiki/Appell_sequence" rel="nofollow">Appell sequence</a> 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> <p>$p_{0}(x)=1$, $p_{1}(x)=x+\gamma$, and for $n>0$ $$p_{n+1}(x)=(x+\gamma)p_{n}(x)+\sum_{j=1}^{n}(-1)^j\binom{n}{j}j!\zeta (j+1)p_{n-j}(x)$$ </p> <p>where $\gamma=-\frac{\mathrm{d} }{\mathrm{d} \beta }\beta !\mid_{\beta =0 }$, the Euler-Mascheroni constant, and $\zeta(s)$ is the Riemann zeta function.</p> <p>They satisfy $$p_{n}(x)=\frac{\mathrm{d^n} }{\mathrm{d} \beta^n }\frac{\exp(\beta x)}{\beta !} \mid_{\beta =0 }.$$</p> <p>Explicitly,</p> <p>$$p_2(x)=(x+\gamma)^2-\zeta(2)$$ $$p_3(x)=(x+\gamma)^3-3\zeta(2)(x+\gamma)+2\zeta(3)$$ $$p_4(x)=(x+\gamma)^4-6\zeta(2)(x+\gamma)^2+8\zeta(3)(x+\gamma)+3[\zeta^2(2)-2\zeta(4)]$$ $$p_5=p_1^5-10\zeta(2)p_1^3+20\zeta(3)p_1^2+15[\zeta^2(2)-2\zeta(4)]p_1+4[-5\zeta(2)\zeta(3)+6\zeta(5)]$$</p> <p>Update: The coefficients appear related to <a href="http://oeis.org/A055137" rel="nofollow">OEIS-A055137</a>, coefficients of the characteristic polynomial of the adjacency matrix of the complete n-graph.</p> <p><strong>Can anyone provide a reference for these polynomials or point out an interesting combinatorial interpretation?</strong></p> <p><em><strong>Background: Rich associations with fractional calculus, <a href="http://en.wikipedia.org/wiki/Digamma_function" rel="nofollow">digamma</a> function, ladder operators</em></strong> </p> <p>The fractional integro-derivative can be represented as an exponentiated convolutional infinitesimal generator (cf. <a href="http://math.stackexchange.com/questions/125343/lie-group-heuristics-for-a-raising-operator-for-1n-fracdnd-betan-fra" rel="nofollow">MSE-Q125343</a>):</p> <p>$\displaystyle\frac{d^{-\beta}}{dx^{-\beta}}\frac{x^{\alpha}}{\alpha!}= \displaystyle\frac{x^{\alpha+\beta}}{(\alpha+\beta)!} = exp(-\beta R_x) \frac{x^{\alpha}}{\alpha!}$</p> <p>where </p> <p>$$R_xf(x)=\frac{1}{2\pi i}\displaystyle\oint_{|z-x|=|x|}\frac{-ln(z-x)+\lambda}{z-x}f(z)dz$$</p> <p>$$=(-ln(x)+\lambda)f(x)+\displaystyle\int_{0}^{x}\frac{f\left ( x\right )-f(u)}{x-u}du.$$</p> <p>with $\lambda=d\beta!/d\beta|_{\beta=0}$. (Note the integrand is related to the <a href="http://en.wikipedia.org/wiki/Q-derivative" rel="nofollow">q (Jackson) derivative</a>, and the <a href="http://en.wikipedia.org/wiki/Pincherle_derivative" rel="nofollow">Pincherle derivative</a> / commutator is $[R_x,x]=D_x^{-1}$.)</p> <p>Then $$exp(-\beta R_x) 1 =\displaystyle\frac{x^\beta}{\beta!} = exp(-\beta\psi_{.}(x)),$$</p> <p>with $(\psi_{.}(x))^n=\psi_n(x)$, which implies</p> <p>$$\psi_{n}(x)=(-1)^n \frac{d^n}{d\beta^n}\frac{x^\beta}{\beta!}|_{\beta=0},$$ $$L_x\psi_{n}(x)=n\psi_{n-1}(x)=-x\displaystyle\frac{d}{dx}\psi_{n}(x),$$ $$R_x\psi_{n}(x)=\psi_{n+1}(x).$$</p> <p>Let $x=e^z$ and $p_n(z)=(-1)^n \psi_{n}(e^z)$. Then</p> <p>$$exp(-\beta R_z) 1 =\displaystyle\frac{exp(\beta z)}{\beta!} = exp(\beta p_{.}(z)),$$</p> <p>$$L_z p_{n}(z)=n p_{n-1}(x)=\displaystyle\frac{d}{dz} p_{n}(z),$$ $$R_z p_{n}(z)= p_{n+1}(z)= (z+\gamma)p_n(z)-\displaystyle\int_{-\infty}^{z}\frac{p_n\left ( z\right )-p_n(u)}{e^z-e^u} e^u du$$</p> <p>with $\gamma=-d\beta!/d\beta|_{\beta=0}$, the Euler-Mascheroni constant. </p> <p>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 </p> <p>$$R_z p_{n}(z)= p_{n+1}(z)= (z+\gamma)p_n(z)-\displaystyle\int_{0}^{\infty}[p_n(z)-(p_{.}(z)-\omega)^n] \frac{1}{e^{\omega}-1}d\omega$$</p> <p>from which the recursion formula follows.</p> <p>In addition, using the operator formalism for <a href="http://mathworld.wolfram.com/ShefferSequence.html" rel="nofollow">Sheffer sequences</a>, of which the Appell is a special case, </p> <p>$$R_z=z-\frac{\mathrm{d} }{\mathrm{d} \beta}ln[\beta!]\mid _{\beta=\frac{\mathrm{d} }{\mathrm{d} z}=D_z}=z-\Psi(1+D_z)$$</p> <p>where $\Psi(x)$ is the <a href="http://en.wikipedia.org/wiki/Digamma_function" rel="nofollow">digamma or Psi function</a>.</p> <p><strong>UPDATE (Nov. 16, 2012)</strong>: Just found this exact sequence in the thesis "<a href="http://digital.library.adelaide.edu.au/dspace/bitstream/2440/50479/1/02whole.pdf" rel="nofollow">Regularized Equivariant Euler Classes and Gamma Functions</a>" by R. Lu with a discussion of the relationships to Chern and Pontrjagin classes.</p> http://mathoverflow.net/questions/111165/riemann-zeta-function-at-positive-integers-and-an-appell-sequence-of-polynomials/111254#111254 Answer by F. C. for Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional calculus F. C. 2012-11-02T07:40:10Z 2012-11-02T07:40:10Z <p>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 $S_n$.</p> http://mathoverflow.net/questions/111165/riemann-zeta-function-at-positive-integers-and-an-appell-sequence-of-polynomials/111368#111368 Answer by Tom Copeland for Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional calculus Tom Copeland 2012-11-03T11:55:23Z 2012-11-16T10:39:14Z <p><strong>Follow-up</strong> on Rupinski's and Chapoton's observations:</p> <p>To nail down the identification of the $p_n(x)$ with the <a href="http://en.wikipedia.org/wiki/Cycle_index" rel="nofollow">cycle index polynomials</a> for $S_n$ (or the partition polynomials of the refined Stirling numbers of the first kind <a href="https://oeis.org/A036039" rel="nofollow">A036039</a>), look at the Taylor series rep of the digamma operator for the raising / creation operator for the $p_n(z)$ basis </p> <p>$$R_z = z-\Psi(1+D_z) = z+\gamma + \sum_{n=1}^{\infty } (-1)^n\zeta (n+1)D_z^n.$$ </p> <p>This is precisely the raising operator for the cycle index polynomials as presented on page 23 of <a href="http://tcjpn.wordpress.com/2011/04/11/lagrange-a-la-lah/" rel="nofollow">Lagrange à la Lah Part I</a> with $c_1=z+\gamma=p_1(x)$ and $c_{n+1}=(-1)^n\zeta(n+1)$ for $n>0$</p> <p>$$D^{-1}_{c_1}= :\frac{c_{.}}{1-c_{.}D_{c_1}}: = c_1+\sum_{n=1}^{\infty } c_{n+1}D_{c_1}^n.$$</p> <p>Alternatively, the <a href="http://en.wikipedia.org/wiki/Newton%27s_identities" rel="nofollow">Newton identities</a> extrapolated to an entire function as an infinite order polynomial using the Weierstrass factorization maneuver can be applied to see the connections to the power and elementary symmetric polynomial formalism: </p> <p>$$\exp\left (-\beta p_{.}(z)\right )=\frac{\exp\left (-\beta z \right )}{\left (-\beta \right )!}=\exp\left (-\beta(z+\gamma) \right )\prod_{k=1}^{\infty }\left ( 1-\frac{\beta}{k} \right )\exp\left (\frac{\beta}{k} \right )$$</p> <p>$$=\exp\left [-(z+\gamma)\beta -\sum_{k=2}^{\infty } \frac{\zeta (k)\beta ^k}{k} \right ]=\exp\left [ :ln(1-a\beta ) :\right ]$$ where $a^1=a_{1}=(z+\gamma)$ and $a^k=a_k=\zeta(k)$ for $k>1$, but this is precisely the umbral form of the e.g.f. for the cycle index polynomials (mod signs).</p> <p>(Also there are connections to <a href="http://en.wikipedia.org/wiki/Rational_zeta_series" rel="nofollow">rational zeta series</a>.)</p> <p><strong>Update (Nov. 16, 2012)</strong>: The generating series appears on pg. 58 in "<a href="http://arxiv.org/abs/0806.0107" rel="nofollow">Hodge theoretic aspects of mirror symmetry</a>" by L. Katzarkov, M. Kontsevich, and T. Pantev (following Lu's references).</p>