If we take the Laurent expansion of the Riemann zeta function about s = 1 $$ \zeta(s) = {s-1} + ^\infty {n!}\gamma_n (s-1)^n $$ which defines γn, the Stieltjes constants, where γ₀ is the Euler-Mascheroni constant. Next perform a series reversion on this to give a series $$ \chi(s) = 1+{s}+^\infty {s^n} $$ which has expansion \chi(s) = 1 + {s} + {s^2} + {s^3} + {2s^4} + \cdots The coefficients κ(n) seem to decrease quite steadily, even up n being a few hundred, where the γn get large. n (n) ---- --------------- -- 0 1.000000000 1 1.000000000 2 0.5772156649 3 0.4059937693 4 0.3135616752 5 0.2556464523 6 0.2159181431 7 0.1869526867 8 0.1648872027 9 0.1475121704 10 0.1334717457 11 0.1218874671 12 0.1121649723 13 0.1038876396 14 0.09675470803 15 0.09054358346 : The first 16 coefficients of the inverse function. Letting $$ R_n=^n i_k $$ and $$ P_n=^n ki_k $$ and {i}n = {i₁, i₂, ⋯|P = n − 1}, I have observed the expression for κ(n) from series reversion to be $$ \kappa(n)=} (-1)^n\left[^{R-1}j-n\right]\left[^n {i_k!}\left({k!}\right)^{i_{k+1}}\right]\gamma_0^{i_1} $$ where we define κ(0)=1. Two examples $$ \kappa(3) = \gamma_0^2 - \gamma_1 = - \left[^{i_1+i_2+i_3-1} (j-n)\right]{1!}\right)^{i_2}\left({2!}\right)^{i_3}}{i_1!i_2!}\gamma_0^{i_1} $$ $$ \kappa(4) = \gamma_0^3 - 3 \gamma_0\gamma_1 + {2} = \left[^{i_1+i_2+i_3+i_4-1} (j-n)\right]{1!}\right)^{i_2}\left({2!}\right)^{i_3}\left({3!}\right)^{i_4}}{i_1!i_2!i_3!}\gamma_0^{i_1} $$ We can conjecture that $$ \kappa(n+1) < \kappa(n), \;\;\; n\in^{>0}? $$ Is this perhaps a more well behaved way to look at the Stieltjes constants?
Create a transform somewhat in analogy to the Mellin transform which to some extent extracts sequence coefficients. _s[f(x)](s) \approx \Gamma(s)\phi(-s) where f(x) = ^\infty {s!}\phi(s) x^s instead consider a transform ℐ[f(x)](s) such that [f(x)](s) \approx \Gamma(s)\chi(-s) where (x)}{x} = ^\infty {s!}\chi(s) x^s and example, the function f(x) = x + x^2 has inverse as series f^{-1}(x) = x - x^2 + 2 x^3 - 5 x^4 + \cdots (x)}{x} = 1 - x + 2 x^2 - 5 x^3 + \cdots = ^\infty {s!} C_s x^s = ^\infty {s!} {(s+1)!} x^s then \chi(s) = {\Gamma(2+s)} then [x+x^2](s) ={\Gamma(2-s)} x^{-1}[[x+x^2](s)](x) = f^{-1}(x) RESULTS Then it would seem that [2x^2 - ](s) ={\Gamma(1-s)} [W(x)](s) =\Gamma(s)(-1)^s \\ [-W(-x)](s) =\Gamma(s) \\ \left[{1-x}\right](s) =\Gamma(s)\Gamma(1-s) \\ \left[-2x}{2x}\right](s) =\Gamma(s)\Gamma(2-s) \\ \left[\log\left({x}\right)\right](s) =\Gamma(s-1)\\ \left[W\left({x}\right)\right](s) =\Gamma(s-2)\\ \left[e^x-1\right](s) ={\Gamma(2-s)} \\ \left[\log(x)\right](s) =(-1)^{1-s} \Gamma(s-1) \\ \left[{e^x-1}\right](s) ={1-s} \\ \left[-x-W(-xe^{-x})\right](s) =\Gamma(s)\zeta(s) \\ Some more generalised ones [\log(x^k)](s) = \left(-{k}\right)^{1-s} \Gamma(s-1)\\ \left[W\left({x^k}\right)\right](s) =k^{-1-1/k+s}\Gamma(s-1-{k})\\ \left[W\left(x^k\right)\right](s) =(-k)^{-1+1/k+s}\Gamma(s-1+{k})\\ \left[-{x+W(-e^{-x}x)}\right](s) = {s}\\ \left[-2W(-}{2})\right](s) = s^2 \Gamma(s)\\ \left[{\log(k/x)}\right](s) = k \Gamma(1-s)\\ \left[{1 - W(ex/k)}\right](s) = k s \Gamma(1-s)\\ \left[-x^k W(}{A})\right](s) = -A x^{k s}\Gamma(s)\\ As described in a previous article on here: It would appear that for the function f(x)=x^m+x, m>1 we get a series g(x)=^\infty {n}}{(m-1)n+1} these then have a set of consistent, hypergeometric series explainable as g(x)=_{(m-1)}F_{(m-2)}\left(\left\{{m},{m},\cdots,{m}\right\};\left\{{m-1},\cdots,{m-1},{m-1}\right\};-}{(m-1)^{m-1}}\right)\cdot x then {x}=_{(m-1)}F_{(m-2)}\left(\left\{{m},{m},\cdots,{m}\right\};\left\{{m-1},\cdots,{m-1},{m-1}\right\};-}{(m-1)^{m-1}}\right) which would give [x+x^m](s) = _x[\;_{(m-1)}F_{(m-2)}\left(\left\{{m},{m},\cdots,{m}\right\};\left\{{m-1},\cdots,{m-1},{m-1}\right\};-}{(m-1)^{m-1}}\right) ](s) which gives [x+x^2](s) = {\Gamma(2-s)}\\ [x+x^3](s) = {2})\Gamma({2})}{2 \Gamma(2-s)} \\ [x+x^4](s) = {2}}\pi \Gamma(-4s/3)\Gamma(s/3)}{\Gamma(2/3-s/3)\Gamma(4/3-s/3)\Gamma(-s/3)} =?={3 \Gamma(2-s)} it then seems like {m-1}\right)\Gamma\left({m-1}\right)}{(m-1)\Gamma(2-s)} = _x[x+x^m](s) and more generally {m-1}}\Gamma\left(1-{m-1}\right)\Gamma\left({m-1}\right)}{(m-1)\Gamma(2-s)} = _x[x+a x^m](s) FURTHER SMALL POLYNOMIALS There are some other small polynomials that give integer sequences upon reversion. Consider x − x² − x³. ^{-1}_s[\Gamma(a+b s)](x) = (-1)^b(b+a)^b W(-{b+a}}x^{{b+a}}}{b+a})^b