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Gauss–Kuzmin–Wirsing operator
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===Matrix elements=== Consider the [[Taylor series]] expansions at ''x'' = 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'' = 1 because the GKW operator is poorly behaved at ''x'' = 0. The expansion is made about 1 − ''x'' so that we can keep ''x'' a positive number, 0 ≤ ''x'' ≤ 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.
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