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Exponential factorial
#1
Hi.

I heard of a function called "exponential factorial" that works like this:






...


As one can see it is similar to tetration in that it involves a power tower, but it is not defined by iteration but by a different type of recurrence, similar to the factorial. Could there be a way to derive a smooth/analytic extension for this like there is with the factorial and gamma function and like how extensions have been proposed for tetration?
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#2
Andy has some info atthis post. How could we confirm that EF(0)=euler-mascheroni constant?
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#3
(10/02/2009, 03:40 AM)Base-Acid Tetration Wrote: Andy has some info atthis post. How could we confirm that EF(0)=euler-mascheroni constant?

I wouldn't know but I'm still trying to figure out how precisely he got those coefficients. It would seem a little good to be true wouldn't it, given how exotic this function is?
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#4
(10/06/2009, 12:05 AM)mike3 Wrote: I wouldn't know but I'm still trying to figure out how precisely he got those coefficients. It would seem a little good to be true wouldn't it, given how exotic this function is?

Essentially, by the method of undetermined coefficients.

Assume that the solution is a polynomial, say of the form , and substitute that in the functional equation gives

and expanding the left-hand-side of this equation about x=1, and truncating the result at 3 or so terms gives an approximately equal equation . Since this equation should hold for all (by assumption*), you can match each coefficient of x on each side to form a system of 3 equations in 3 unknowns . Even though this system of algebraic equations is simpler (in a way) than the single functional equation, they are not linear in the unknowns, so use your favorite nonlinear system method, and solve for .

Does this help?

Andrew Robbins

* Assuming the functional equation holds for all seems to lead to a contradiction, because and yet for all x, so you can't solve for x such that the functional equation holds.
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