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A list of key readings about Tetration.

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Ethimology of tetration ^[1].

Tetration for base $b\!=\!2$ ^[2].

Tetration for base $b\!=\!\mathrm {e}$ ^[3]^[4]

Uniqueness of the tetration and arctetration at base $b\!>\!\exp(1/\mathrm {e} )$ ^[5]

Superexponentials (and the tetration) to base $b\!=\!\exp(1/\mathrm {e} )$ ^[6]

uperexponentials (and the tetration) for the case $1\!<\!b\!<\!\exp(1/\mathrm {e} )$ , and, in particular, for $b\!=\!{\sqrt {2}}$ ^[7]

Other solutions of equation $F(z+1)=\exp(F(z))$ : ^[8]

Application of tetration ^[9] ^[10] ^[11] ^[2].

Ackermann Function ^[11] ^[2].

About iterations: ^[12]

↑ R.L.Goodstein (1947). "Transfinite ordinals in recursive number theory". Journal of Symbolic Logic 12.
↑ ^2.0 ^2.1 ^2.2 D.Kouznetsov. Ackermann functions of complex argument. Preprint of the Institute for Laser Science, UEC, 2008. http://www.ils.uec.ac.jp/~dima/PAPERS/2008ackermann.pdf Cite error: Invalid <ref> tag; name "k2" defined multiple times with different content Cite error: Invalid <ref> tag; name "k2" defined multiple times with different content
↑ D.Kouznetsov. (2009). "Solutions of $F(z\!+\!1)=\exp(F(z))$ in the complex $z$ plane.". Mathematics of Computation, 78: 1647-1670. DOI:10.1090/S0025-5718-09-02188-7. Research Blogging. preprint: http://www.ils.uec.ac.jp/~dima/PAPERS/analuxp99.pdf
↑ D.Kouznetsov. (2009). "Superexponential as special function.". Vladikavkaz Mathematical Journal 12 (2): 31-45.
↑ H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. Aequationes Mathematicae, v.81, p.65-76 (2011) http://www.springerlink.com/content/u7327836m2850246/ http://tori.ils.uec.ac.jp/PAPERS/2011uniabel.pdf
↑ H.Trappmann, D.Kouznetsov. Computation of the Two Regular Super-Exponentials to base exp(1/e). Mathematics of computation, in preparation, 2011. http://tori.ils.uec.ac.jp/PAPERS/2011e1e.pdf
↑ D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756. http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html http://tori.ils.uec.ac.jp/PAPERS/2010sqrt2.pdf
↑ H.Kneser. “Reelle analytische L¨osungen der Gleichung $\varphi (\varphi (x))=e^{x}$ und verwandter Funktionalgleichungen”. Journal fur die reine und angewandte Mathematik, 187 (1950), 56-67.
↑ P.Walker. Infinitely differentiable generalized logarithmic and exponential functions. Mathematics of computation, 196 (1991), 723-733.
↑ M.H.Hooshmand. ”Ultra power and ultra exponential functions”. Integral Transforms and Special Functions 17 (8), 549-558 (2006)
↑ ^11.0 ^11.1 W.Ackermann. ”Zum Hilbertschen Aufbau der reellen Zahlen”. Mathematische Annalen 99(1928), 118-133
↑ A.Knoebel (1981). "Exponentials Reiterated". Amer. Math. Monthly 88: 235-252.

[good-1] R.L.Goodstein (1947). "Transfinite ordinals in recursive number theory". Journal of Symbolic Logic 12.

[k2-2] 2.0 ^2.1 ^2.2 D.Kouznetsov. Ackermann functions of complex argument. Preprint of the Institute for Laser Science, UEC, 2008. http://www.ils.uec.ac.jp/~dima/PAPERS/2008ackermann.pdf Cite error: Invalid <ref> tag; name "k2" defined multiple times with different content Cite error: Invalid <ref> tag; name "k2" defined multiple times with different content

[k-3] D.Kouznetsov. (2009). "Solutions of $F(z\!+\!1)=\exp(F(z))$ in the complex $z$ plane.". Mathematics of Computation, 78: 1647-1670. DOI:10.1090/S0025-5718-09-02188-7. Research Blogging. preprint: http://www.ils.uec.ac.jp/~dima/PAPERS/analuxp99.pdf

[vladie-4] D.Kouznetsov. (2009). "Superexponential as special function.". Vladikavkaz Mathematical Journal 12 (2): 31-45.

[uni-5] H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. Aequationes Mathematicae, v.81, p.65-76 (2011) http://www.springerlink.com/content/u7327836m2850246/ http://tori.ils.uec.ac.jp/PAPERS/2011uniabel.pdf

[e1e-6] H.Trappmann, D.Kouznetsov. Computation of the Two Regular Super-Exponentials to base exp(1/e). Mathematics of computation, in preparation, 2011. http://tori.ils.uec.ac.jp/PAPERS/2011e1e.pdf

[q2-7] D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756. http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html http://tori.ils.uec.ac.jp/PAPERS/2010sqrt2.pdf

[kneser-8] H.Kneser. “Reelle analytische L¨osungen der Gleichung $\varphi (\varphi (x))=e^{x}$ und verwandter Funktionalgleichungen”. Journal fur die reine und angewandte Mathematik, 187 (1950), 56-67.

[9] P.Walker. Infinitely differentiable generalized logarithmic and exponential functions. Mathematics of computation, 196 (1991), 723-733.

[uxp-10] M.H.Hooshmand. ”Ultra power and ultra exponential functions”. Integral Transforms and Special Functions 17 (8), 549-558 (2006)

[a-11] 11.0 ^11.1 W.Ackermann. ”Zum Hilbertschen Aufbau der reellen Zahlen”. Mathematische Annalen 99(1928), 118-133

[12] A.Knoebel (1981). "Exponentials Reiterated". Amer. Math. Monthly 88: 235-252.

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