nested multiplication superfactorial (small gamma function)

...

7 doubles - how are they combined? 2 "stones" produce 2 houses; 3 form 6; 4 form 24; 5 form 120; 6 form 720; 7 form 5040; and beyond this their numbers increase so that the mouth can hardly utter them, nor the ear hear the number of them.

  —[Sefer Yetzirah 4.12]

Initially factorial was defined as n! = \(1 \times 2 \times 3 \times 4 ... \times n\) or simply written as \(n! = n\times(n-1)!\) where \(0!\) = 1, this factorial function is usually used to find permutations or combinations, well one thing that people don't pay attention to is that this factorial can be written with prod notation $\prod _{i=1}^{x}i$,

the factorial extension for complex numbers is called the gamma function (\(\Gamma(z)\)) which is defined as the following integral

$z!=\int _{0}^{\infty }x^{z-1}e^{-x}dx$

The integral above is an Improper integral and can sometimes produce \(\pi\) at the output.

Now define $\gamma (a,b)$is a 2 argument function that only accepts natural numbers until an integral or similar extension for complex numbers of this function is found.

If the variable \(b\) is equal to 1, then it will work like a regular factorial or $\prod _{i=1}^{x}i$ but remember that this function does not accept non-natural numbers

if the variable \(b\) is 2 then this function will be $\prod _{i=1}^{x}\prod _{i_{2}=1}^{i}i_{2}$,

or $\underbrace {1!\times 2!\times 3!\times ...\times x!} _{x}$

In fact, this function was created previously, namely the Barnes-G function.

This hyperfactorial is also related to the Glaisher–Kinkelin constant with the symbol \(A\) which is an irrational constant defined as $\lim _{n\rightarrow \infty }{\frac {\left(2\pi \right)^{\frac {n}{2}}n^{{\frac {n^{2}}{2}}-{\frac {1}{12}}}e^{-{\frac {3n^{2}}{4}}+{\frac {1}{12}}}}{G(n+1)}}$ , the more you increase n towards infinity the closer it will be to the Glaisher–Kinkelin constant, this constant appears in the duplication formula $G(x)G\left(x+{\frac {1}{2}}\right)^{2}G(x+1)=e^{\frac {1}{4}}A^{-3}2^{-2x^{2}+3x-{\frac {11}{12}}}\pi ^{x-{\frac {1}{2}}}G\left(2x\right)$ , this constant is about 1.28242712910 0622636...

if \(\gamma(a,1)\) is the Gamma function or \(\Gamma(a)\) and if \(\gamma(a,2)\) is the Barnes G function or \(G(a)\), there is no other function that I can get above the Barnes G function, maybe there is no complex number extension or I haven't found it yet, so after this the function \(\gamma(a,b)\) can only accept natural numbers and non-natural numbers are not accepted

with the given pattern \(\gamma(a,3)\) is $\prod _{i=1}^{a}\prod _{i_{2}=1}^{i}\prod _{i_{3}=1}^{i_{2}}i_{3}$

The following is the sum of \(\gamma(a,3)\) from 1 to 7

The graph of \(\prod_{n=1}^{x}\prod_{m=1}^{n}\prod_{a=1}^{m}a\) shown logarithmically, note the very rapid growth, the image was created in desmos

a	result
1	1
2	2
3	24
4	6912
5	238878720
6	5944066965504000
7	745453331864786829312000000

This function grows very fast, for example \(\gamma(10,3)

) is 46055492324773905212722208920097589966225904305970614833621406622679040000000000000000000000000, \(\gamma(100,3)

) is around \(10^212952\)

then next this if b = 3, if b = 4 according to the previous definition \(\gamma(a,4)\) is $\prod _{i=1}^{a}\prod _{i_{2}=1}^{i}\prod _{i_{3}=1}^{i_{2}}\prod _{i_{4}=1}^{i_{3}}i_{4}$

The following is the sum of \(\gamma(a,4)\) from 1 to 7,

function \(\prod{n=1}^{x}\prod _{k=1}^{n}\prod _{m=1}^{k}m!\) (red) compared to function \(10^x\) (green) but this system still cannot exceed \(10^{10^{x}}\)

a	result
1	1
2	2
3	48
4	331776
5	79254226206720
6	471092427871945743012986880000
7	351177419973413722592573060611594181593855426560000000000

and so on, then based on the pattern, it can be said that \(\gamma(a,5)\) is $\prod _{i=1}^{a}\prod _{i_{2}=1}^{i}\prod _{i_{3}=1}^{i_{2}}\prod _{i_{4}=1}^{i_{3}}\prod _{i_{5}=1}^{i_{4}}i_{5}$

\(\gamma(a,5)\) is $\prod _{i=1}^{a}\prod _{i_{2}=1}^{i}\prod _{i_{3}=1}^{i_{2}}\prod _{i_{4}=1}^{i_{3}}\prod _{i_{5}=1}^{i_{4}}\prod _{i_{6}=1}^{i_{5}}i_{6}$and so on

In conclusion, define \(\gamma(a,b)\) is a binary function that satisfies $\underbrace {\prod _{i=1}^{a}\prod _{i_{2}=1}^{i}\prod _{i_{3}=1}^{i_{2}}\prod _{i_{4}=1}^{i_{3}}...\prod _{i_{b}=1}^{i_{b-1}}i_{b}} _{b}$

this function can grow very fast, desmos itself cannot calculate \prod more than 4 and will ask to define the variable, if it has been defined it will not appear because of errors and so on

to end this blogpost, define \(\mathsf{\digamma(n)}\) (nested superfactorial digamma function) as \(\gamma(n,n)\) or

$\underbrace {\prod _{i=1}^{n}\prod _{i_{2}=1}^{i}\prod _{i_{3}=1}^{i_{2}}\prod _{i_{4}=1}^{i_{3}}...\prod _{i_{n}=1}^{i_{n-1}}i_{n}} _{n}$

This is the first value of 7 for \(\digamma(n)\),

n	result
1	1
2	2
3	24
4	331776
5	2524286414780230533120
6	18356962141505758798331790171539976807981714702571497465907439808868887035904000000
7	5101262518548258728945891181868950955955001607224762539748030927274644810006571505387259191811206793959788670295182572066866010362135771367947051132012526915711202702574141007954099155897521232723988907041528666295915651551212054155426312621842773666145180823822511666294137239768053841920000000000000000000000000000

\(\digamma(10)\) decimal expansion

21227913942103437465002517288503580758371635346658609372971537292006899560347005232762655271159324718955558984781289639415026266477586129479065129067335860688785903177036153105849078753477296202040035414672248239161434099241403651361659996104901985440296958667118509254375495198212808883322206693665261900842157031194608713862996028855579768587747338478160561525361416785541099768489271105789870128642842313838202315680477518137923823541859362964713030779677293505717657671580762349480959690292490626229790750958018347518473623153601473098945411658746254705467797600701512008252174421404831662115934066229813152183403593872007185680793724565443282645355876891126333636003359114898978652377472280107100391911408591805149744733759661009088340596166139920362883175030566489358261629148511635763683473065247098048498405783926045399989767986384577187308052259025606044355803261325187865546068789754652530547759356608649122030637999204718578438973485304555834001017956890616816082065141281832559449753631413807189593062520899444800213001492103959661903984008753723541783838720000000000000000000000000000000000000000000000000000000