Editing Gauss–Kuzmin–Wirsing operator (section)

==Relationship to the Riemann zeta function==
The GKW operator is related to the [[Riemann zeta function]]. Note that the zeta function can be written as

:<math>\zeta(s)=\frac{1}{s-1}-s\int_0^1 h(x) x^{s-1} \; dx</math>

which implies that

:<math>\zeta(s)=\frac{s}{s-1}-s\int_0^1 x \left[Gx^{s-1} \right]\, dx </math>

by change-of-variable.

===Matrix elements===
Consider the [[Taylor series]] expansions at ''x''&nbsp;=&nbsp;1 for a function ''f''(''x'') and <math>g(x)=[Gf](x)</math>.  That is, let

:<math>f(1-x)=\sum_{n=0}^\infty (-x)^n \frac{f^{(n)}(1)}{n!}</math>

and write likewise for ''g''(''x'').  The expansion is made about ''x''&nbsp;=&nbsp;1 because the GKW operator is poorly behaved at ''x''&nbsp;=&nbsp;0.  The expansion is made about 1&nbsp;−&nbsp;''x'' so that we can keep ''x'' a positive number, 0&nbsp;&le;&nbsp;''x''&nbsp;&le;&nbsp;1. Then the GKW operator acts on the Taylor coefficients as

:<math>(-1)^m \frac{g^{(m)}(1)}{m!} = \sum_{n=0}^\infty G_{mn} (-1)^n \frac{f^{(n)}(1)}{n!},</math>

where the matrix elements of the GKW operator are given by

:<math>G_{mn}=\sum_{k=0}^n (-1)^k {n \choose k} {k+m+1 \choose m} \left[ \zeta (k+m+2)- 1\right].</math>

This operator is extremely well formed, and thus very numerically tractable. The Gauss–Kuzmin constant is easily computed to high precision by numerically diagonalizing the upper-left ''n'' by ''n'' portion.  There is no known closed-form expression that diagonalizes this operator; that is, there are no closed-form expressions known for the eigenvectors.

===Riemann zeta===
The Riemann zeta can be written as

:<math>\zeta(s)=\frac{s}{s-1}-s \sum_{n=0}^\infty (-1)^n {s-1 \choose n} t_n</math>

where the <math>t_n</math> are given by the matrix elements above:

:<math>t_n=\sum_{m=0}^\infty \frac{G_{mn}} {(m+1)(m+2)}.</math>

Performing the summations, one gets:

:<math>t_n=1-\gamma + \sum_{k=1}^n (-1)^k {n \choose k} \left[ \frac{1}{k} - \frac {\zeta(k+1)} {k+1} \right]</math>

where <math>\gamma</math> is the [[Euler–Mascheroni constant]]. These <math>t_n</math> play the analog of the [[Stieltjes constants]], but for the [[falling factorial]] expansion. By writing

:<math>a_n=t_n - \frac{1}{2(n+1)}</math>

one gets: ''a''<sub>0</sub> = &minus;0.0772156... and ''a''<sub>1</sub> = &minus;0.00474863... and so on. The values get small quickly but are oscillatory.  Some explicit sums on these values can be performed.  They can be explicitly related to the Stieltjes constants by re-expressing the falling factorial as a polynomial with [[Stirling number]] coefficients, and then solving. More generally, the Riemann zeta can be re-expressed as an expansion in terms of [[Sheffer sequence]]s of polynomials.

This expansion of the Riemann zeta is investigated in the following references.<ref> {{cite journal |last1=Yeremin |first1=A. Yu. |last2=Kaporin |first2=I. E. |last3=Kerimov |first3=M. K. |year=1985 |title=The calculation of the Riemann zeta-function in the complex domain |journal=USSR Comput. Math. And Math. Phys. |volume=25 |issue=2 |pages=111–119 |doi=10.1016/0041-5553(85)90116-8 }}</ref><ref> {{cite journal |last1=Yeremin |first1=A. Yu. |last2=Kaporin |first2=I. E. |last3=Kerimov |first3=M. K. |year=1988 |title=Computation of the derivatives of the Riemann zeta-function in the complex domain |journal=USSR Comput. Math. And Math. Phys. |volume=28 |issue=4 |pages=115–124 |doi=10.1016/0041-5553(88)90121-8 }}</ref><ref> {{cite arXiv |last1=Báez-Duarte |first1=Luis |year=2003 |title=A new necessary and sufficient condition for the Riemann hypothesis |eprint=math.NT/0307215 }}</ref><ref> {{cite journal |last1=Báez-Duarte |first1=Luis |year=2005 |title=A sequential Riesz-like criterion for the Riemann hypothesis |journal=International Journal of Mathematics and Mathematical Sciences |volume=2005 |issue=21 |pages=3527–3537 |doi=10.1155/IJMMS.2005.3527 |doi-access=free }}</ref><ref> {{Cite journal  |last1=Flajolet |first1=Philippe  |last2=Vepstas|first2=Linas  |year=2006  |title=On Differences of Zeta Values  |journal=Journal of Computational and Applied Mathematics  |volume=220 |issue= 1–2|pages=58–73 |bibcode=2008JCoAM.220...58F  |arxiv=math/0611332 |doi=10.1016/j.cam.2007.07.040 |s2cid=15022096 }}</ref> The coefficients are decreasing as
:<math>\left(\frac{2n}{\pi}\right)^{1/4}e^{-\sqrt{4\pi n}}
\cos\left(\sqrt{4\pi n}-\frac{5\pi}{8}\right) +
\mathcal{O} \left(\frac{e^{-\sqrt{4\pi n}}}{n^{1/4}}\right).</math>