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I've discussed a procedure for divergent summation using the matrix of Eulerian numbers occasionally in the last years (initially here, and here in MSE and MO but not in that generality and thus(?) without conclusive answers)); now I would like to formalize/make waterproof one heuristic.

Consider some (divergent) series $S = K + a + b + c + d + ... $. To assign a finite sum to it I'm trying to use the matrix of the Eulerian numbers (let's call it Eulermatrix for now) to do something in the spirit of Abel- or possibly(?) Borel-summation:

I define the function $$f(x) = K + ax + bx^2 + cx^3 + dx^4 + ... $$ with the goal to arrive at a valid/meaningful result for the case that $x \to 1$. The use of the Eulermatrix (see end of the post) implies then the introduction of a transform of $f(x)$ in the form $$ g(x) = K + ax + bx^2/2! + cx^3/3! + dx^4/4! + ... $$ If I now evaluate the sequence of partial sums according to the decomposition of the Eulerian numbers in combination with the reciprocal factorials I arrive at the form $$ s_n(x) = \sum_{k=0}^{n-1} (-1)^k ((n-k)x) ^k {g^{(k)}((n-k)x) \over k!} \qquad n=1 \ldots \infty \tag1$$ and if this converges to a finite value then I assume this as the (divergent) sum of the series and assume $$S = \lim _{n \to \infty} s_n(1)$$ I get meaningful results for series which have growth rates like geometric series with $q \lt 1$ and I think that the formalism of conversion into a double-sum and changing order of evaluation using the Eulermatrix gives also the range for $q$ as $- \infty \lt q \lt 1$ (because the $g(x)$ function is then entire). I can even evaluate the classical series $0! - 1! + 2! - 3! + \ldots - $ to which already L. Euler assigned the value of about $0.596...$ which is now called Gompertz constant.
So I'm confident, that my general reasoning using the Eulermatrix is valid.

But now: what I'm asking here is about the expression for $s_n(x)$ which reminds me of the Maclaurin-expression (which expresses a function by its derivatives at zero), but which I've not seen yet. Is that a valid transform? And up to which growth rate can this method handle series (I think, preferably alternating series)?


The Eulermatrix with a rowscaling by reciprocal factorials: $$ \small \begin{bmatrix} 1 & . & . & . & . & . \\ {1 \over 1!} & . & . & . & . & . \\ {1 \over 2!} & {1 \over 2!} & . & . & . & . \\ {1 \over 3!} & {4 \over 3!} & {1 \over 3!} & . & . & . \\ {1 \over 4!} & {11 \over 4!} & {11 \over 4!} & {1 \over 4!} & . & . \\ {1 \over 5!} & {26 \over 5!} & {66 \over 5!} & {26 \over 5!} & {1 \over 5!} & .\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{bmatrix} $$

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@wil:upps - that must have happened accidentally. Corrected (hopefully the best tag, I wanted something like matrix-transformations, Borel-transform (not the integral one) because I think this is related –  Gottfried Helms Feb 21 '13 at 14:19
    
np. I figured it was a typo. Except I just didn't know what you intended it to be. –  Willie Wong Feb 21 '13 at 14:24

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