Is this entire expression for tetration legal

[sheldonison]

....Being quick I'll write the glorious formula; \( \forall b \in \mathbb{R}\,\,;\,\, b > e^{\frac{1}{e}} \):

\( \frac{1}{^\omega b} = \sum_{N=0}^{\infty} \frac{\sum_{k=0}^{N} \frac{(-1)^{N-k}}{(N-k)!(^k b)}}{ \Gamma(\omega - N +1)} \)

Hey James, very interesting stuff, though its a little over my head. But I did try your formula for base e, for sexp(0.5), \( ^{0.5}e \)
\( \frac{1}{^z e} = \sum_{N=0}^{\infty} \frac{\sum_{k=0}^{N} \frac{(-1)^{N-k}}{(N-k)!(^k e)}}{ \Gamma(z - N +1)} \)

I generated the first hundred terms, and took the reciprocal and got the result for sexp(0.5), \( ^{0.5}e \) = -0.0889470119254744974, which is a strange result. Also, the summation also doesn't appear to be converging. Here are the first 100 terms for N=0 to N=99 in the outer loop summation. As you can see, the sum is slowly decreasing (larger negative number), which means the reciprocal is slowly getting smaller, and perhaps approaching zero. By the way, the inner loop converges very nicely with only k=0 to k=3 (four terms) required. For integer values of z, there is also the problem that gamma has a singularity for zero and negative integers, but if you somehow ignore the terms with the singularity, it seems to work.
- Sheldon

Code:

N=0 1.12837916709551257390
N=1 -0.356635834837458935278
N=2 -0.0558854027631410430498
N=3 -0.0206132530877876970179
N=4 -0.0141193897106172441830
N=5 -0.0148207070343604561859
N=6 -0.0178722393015305743419
N=7 -0.0217507480727162365390
N=8 -0.0258849705867867567488
N=9 -0.0300391250051902282142
N=10 -0.0341121984624033791621
N=11 -0.0380621797995234682754
N=12 -0.0418743225766593487278
N=13 -0.0455468128421149043589
N=14 -0.0490839363416516118949
N=15 -0.0524926981203164306902
N=16 -0.0557811147157649591746
N=17 -0.0589573488130329367398
N=18 -0.0620292803860181190169
N=19 -0.0650043080717605755730
N=20 -0.0678892727812415897521
N=21 -0.0706904456867871814295
N=22 -0.0734135490929702590636
N=23 -0.0760637929100568892563
N=24 -0.0786459172599043479163
N=25 -0.0811642360986323262444
N=26 -0.0836226791891616201340
N=27 -0.0860248311369194801384
N=28 -0.0883739669751719546777
N=29 -0.0906730842116677024391
N=30 -0.0929249314732365839175
N=31 -0.0951320339948591280177
N=32 -0.0972967162440782634464
N=33 -0.0994211219794611853738
N=34 -0.101507232030120356963
N=35 -0.103556880061972629931
N=36 -0.105571766571162804793
N=37 -0.107553471319117637288
N=38 -0.109503464398763159097
N=39 -0.111423116098393868509
N=40 -0.113313705708880295616
N=41 -0.115176429401406493529
N=42 -0.117012407286649168777
N=43 -0.118822689752075097934
N=44 -0.120608263161640512378
N=45 -0.122370054991416584391
N=46 -0.124108938465338699673
N=47 -0.125825736747199128971
N=48 -0.127521226738006875002
N=49 -0.129196142521777878991
N=50 -0.130851178497564809986
N=51 -0.132486992230976648134
N=52 -0.134104207054477737117
N=53 -0.135703414442310877300
N=54 -0.137285176182888040280
N=55 -0.138850026368874240602
N=56 -0.140398473222902578280
N=57 -0.141931000774856544457
N=58 -0.143448070404900858692
N=59 -0.144950122264901360038
N=60 -0.146437576589519465340
N=61 -0.147910834907073090058
N=62 -0.149370281159202759044
N=63 -0.150816282737450884553
N=64 -0.152249191444038307074
N=65 -0.153669344383391754809
N=66 -0.155077064790327269347
N=67 -0.156472662800217813649
N=68 -0.157856436165959503295
N=69 -0.159228670926092608054
N=70 -0.160589642028024068590
N=71 -0.161939613909932020797
N=72 -0.163278841044604707844
N=73 -0.164607568448171808999
N=74 -0.165926032156421792429
N=75 -0.167234459671161047302
N=76 -0.168533070378856319747
N=77 -0.169822075943608775267
N=78 -0.171101680676333554542
N=79 -0.172372081881860968129
N=80 -0.173633470185532711615
N=81 -0.174886029840737105782
N=82 -0.176129939018709985571
N=83 -0.177365370081821244524
N=84 -0.178592489841470093563
N=85 -0.179811459801623841895
N=86 -0.181022436388954587949
N=87 -0.182225571170454848279
N=88 -0.183421011059346164175
N=89 -0.184608898510033494726
N=90 -0.185789371702802181506
N=91 -0.186962564718902961836
N=92 -0.188128607706623473485
N=93 -0.189287627038901537828
N=94 -0.190439745462995875203
N=95 -0.191585082242693475689
N=96 -0.192723753293499332602
N=97 -0.193855871311223385361
N=98 -0.194981545894351078562
N=99 -0.196100883660557713855
SUM = -11.2426486101395293427