The MRB constant is the sum of the series https://oeis.org/A037077.
cf. https://mathworld.wolfram.com/MRBConstant.html
The MRB consstant (CMRB) $$=\sum _k^{\infty } \left((2 k)^{\frac{1}{2 k}}-(2 k-1)^{\frac{1}{2 k-1}}\right)=\sum _m^{\infty } (-1)^m \left(m^{1/m}-1\right)$$
@Dark Malthorp, found integrals for it here using the Abel-Plana formula. $$\begin{eqnarray} C_{MRB} &=& \sum_{n=0}^\infty (-1)^n\left(1 - (n+1)^{\frac{1}{n+1}}\right) \\&=& \frac{i}{2} \int_0^\infty \frac{(-i t+1)^{\frac{1}{-i t+1}} - (i t+1)^{\frac{1}{i t+1}}}{\sinh{\pi t}} dt \\&=& \int_0^\infty\frac{\Im\left((1+i t)^{\frac{1}{1+i t}}\right)}{\sinh \pi t} dt\\&=&\int_0^\infty \frac{(\sqrt{x^2+1})^{1/(x^2+1)} \exp\left(\frac{x\arctan{x}}{x^2+1}\right)\sin\left(\frac{\arctan(x) - x\log\sqrt{x^2+1}}{x^2+1}\right)}{\sinh \pi x}dx \end{eqnarray} $$
Hence my new question: is there a sum using the the Abel-Plana formula in reverse so-to speak for its integrated analog, M2 $$=\int_1^{\infty }e^{\pi i t} \left(t^{1/t}-1\right) \, dt?$$
$\lim_{x->\infty}\int_1^{2x }e^{\pi i t} \left(t^{1/t}\right) \, dt=M2+-21/\pi.$ And $|M2+-21/\pi| $ is the integral of https://oeis.org/A157852, where a few other people have added their toughts.
Background and motivation along with notes from of years of research can be found throughout this Wolfram Community discussion.
RAHUL said a power series expansion might be useful in this case, so I came up with $$M2=\sum _{n=1}^{\infty } \left(\frac{i}{\pi }\right)^{1-n}\left( G_{2,3}^{3,0}\left(-i \pi \left| \begin{array}{c} 1,1 \\ 0,0,0 \\ \end{array} \right.\right)-i \pi G_{3,4}^{4,0}\left(-i \pi \left| \begin{array}{c} 1,1,1 \\ -1,0,0,0 \\ \end{array} \right.\right)-\pi ^2 G_{4,5}^{5,0}\left(-i \pi \left| \begin{array}{c} 1,1,1,1 \\ -2,0,0,0,0 \\ \end{array} \right.\right)+i \pi ^3 G_{5,6}^{6,0}\left(-i \pi \left| \begin{array}{c} 1,1,1,1,1 \\ -3,0,0,0,0,0 \\ \end{array} \right.\right)+\pi ^4 G_{6,7}^{7,0}\left(-i \pi \left| \begin{array}{c} 1,1,1,1,1,1 \\ -4,0,0,0,0,0,0 \\ \end{array} \right.\right)-i \pi ^5 G_{7,8}^{8,0}\left(-i \pi \left| \begin{array}{c} 1,1,1,1,1,1,1 \\ -5,0,0,0,0,0,0,0 \\ \end{array} \right.\right)-...\right)$$
As found by this Wolfram Cloud notebook. As shown in this Wolfram Cloud notebook. As published in this OEIS document.
Concluding what I know,
In traditional form,
$$M2=\sum _{n=1}^{\infty } \left(\frac{i}{\pi }\right)^{1-n}\left(\text G_{n+1,n+2}^{n+2,0}\left(^{1;\underbrace {x,…,x}_n}_{1-n,0; \underbrace {x,…,x}_n}\bigg|-\pi i\right)\right).$$
So, I found one that way, but believe there are more, because there are dozens of forms of the MRB constant all proved at https://www.mapleprimes.com/posts/214522-Formulas-For-The-MRB-Constant
Please note that Tyma Gaidash provided a formula from a reliable source, although not certain if conditions met. If anyone can show it working or not, and offer a proof that it or a variant of it leads to a sum for $\int_1^{\infty }e^{\pi i t} \left(t^{1/t}-1\right) \, dt,$ I will set up another bounty to reward such a worthy answer. (The goal here is to find a formula for that integral that involves integration, to the least amount as possible. (Which is an improvement from defining MeigerG in terms of integrals.)
Please ask questions if you've got them!
Let’s try using Slater’s Theorem to evaluate the Meijer G function which does fit the necessary conditions for using the formula. The simplest example is this computation for the $\,^{3,0}_{2,3}$ case. Here is your general formula:
$$\text G_{n+1,n+2}^{n+2,0}\left(^{1;\underbrace {1,…,1}_n}_{1-n,0; \underbrace {0,…,0}_n}\bigg|-\pi i\right)=\sum_{h=1}^{n+2}\frac{\prod\limits_{h\ne j=1}^{n+2}\Gamma(b_j-b_h)\prod\limits_{j=1}^0\Gamma(b_h-a_j+1)(-\pi i)^{b_h}}{\prod\limits_{j=n+3}^{n+2}\Gamma(b_h-b_j+1)\prod\limits_{j=1}^{n+1}\Gamma(a_j-b_h)}\,_{n+1}\text F_{n+1}\left(b_h-a_{n+1}+1;b_h-b_{n+2}+1;\pi i\right),a_{n+1}=\{1-n,0, \underbrace {0,…,0}_n\},b_{n+2}=\{1, \underbrace {1,…,1}_n\}$$
where $b_h-b_{n+2}+1,h\ne n+2$ I will work on simplifying this result to see if anything interesting comes up. I am also new to Slater’s Theorem, for the $p=n+1,q=n+2,p<q$ version, so if this is the wrong formula, then please tell me. Please correct me and give me feedback!
Special cases with the Hypergeometric function, Euler-Mascheroni constant, and more:
$$\text G_{2,3}^{3,0}\left(^{1,1}_{0,0,0}\big|-\pi i\right)=\frac12\left(\gamma^2-i\gamma \pi-\frac{\pi^2}{12}+\ln^2(\pi)+2\gamma\ln(\pi)-i\pi\ln(\pi)\right)+\pi i\,_3\text F_3(1,1,1;2,2,2;\pi i)$$
Let’s try the next case:
$$\text G_{3,4}^{4,0}\left(^{1,1,1}_{-1,0,0,0}\big|\pi i\right)=\sum_{h=1}^3\frac{\prod\limits_{h\ne j=1}^{3}\Gamma(b_j-b_h)\prod\limits_{j=1}^0\Gamma(b_h-a_j+1)(-\pi i)^{b_h}}{\prod\limits_{j=5}^4\Gamma(b_h-b_j+1)\prod\limits_{j=1}^3\Gamma(a_j-b_h)}\,_3\text F_3(b_h-a_3+1,b_{j\ne4}-b_4+1),a_3=\{1,1,1\},b_4=\{-1,0,0,0\}\mathop=^? \sum_{h=1}^3\frac{\prod\limits_{h\ne j=1}^{3}\Gamma(b_j-b_h)(-\pi i)^{b_h}}{\prod\limits_{j=1}^3\Gamma(a_j-b_h)}\,_3\text F_3(b_h-a_3+1,b_{j\ne4}-b_4+1) $$
After some experimentation and learning the indices of the Meijer G function, here is the copyable code for the simplified $\,^{4,0}_{3,4}$ case where appears the Trigonometric Integral functions and Riemann Zeta function and note that “gamma” is the Euler Gamm/ Euler-Mascheroni Constant:
-i π _3 F_3(1, 1, 1;2, 2, 2;i π) + i π _4 F_4(1, 1, 1, 1;2, 2, 2, 2;i π) + Ci(π) + i Si(π) + ζ(3)/3 - gamma ^2/2 + gamma ^3/6 - i/π - (i π)/2 + (i gamma π)/2 - 1/4 i gamma ^2 π + π^2/24 - ( gamma π^2)/24 - (i π^3)/48 + (log^3(π))/6 - (log^2(π))/2 + 1/2 gamma log^2(π) - 1/4 i π log^2(π) - gamma log(π) + 1/2 gamma ^2 log(π) + 1/2 i π log(π) - 1/2 i gamma π log(π) - 1/24 π^2 log(π)Here is the simplified $\,^{5,0}_{4,5}$ case with copyable code:
7/8 i π _3 F_3(1, 1, 1;2, 2, 2;i π) - 3/4 i π _4 F_4(1, 1, 1, 1;2, 2, 2, 2;i π) + 1/2 i π _5 F_5(1, 1, 1, 1, 1;2, 2, 2, 2, 2;i π) - (15 Ci(π))/16 - (15 i Si(π))/16 + 1/12 ζ(3) (-3 + 2 gamma - i π + 2 log(π)) - 9/32 + (3 gamma )/16 + (7 gamma ^2)/16 - gamma ^3/8 + gamma ^4/48 + 1/(16 π^2) + (15 i)/(16 π) + (27 i π)/32 - (7 i gamma π)/16 + 3/16 i gamma ^2 π - 1/24 i gamma ^3 π - (7 π^2)/192 + ( gamma π^2)/32 - ( gamma ^2 π^2)/96 + (i π^3)/64 - 1/96 i gamma π^3 - π^4/1280 + (log^4(π))/48 - (log^3(π))/8 + 1/12 gamma log^3(π) - 1/24 i π log^3(π) + (7 log^2(π))/16 - 3/8 gamma log^2(π) + 1/8 gamma ^2 log^2(π) + 3/16 i π log^2(π) - 1/8 i gamma π log^2(π) - 1/96 π^2 log^2(π) + 3/32 (3 - 2 gamma + i π - 2 log(π)) + 15/32 (-log(π) - (i π)/2) + (9 log(π))/8 + 7/8 gamma log(π) - 3/8 gamma ^2 log(π) + 1/12 gamma ^3 log(π) - 7/16 i π log(π) + 3/8 i gamma π log(π) - 1/8 i gamma ^2 π log(π) + 1/32 π^2 log(π) - 1/48 gamma π^2 log(π) - 1/96 i π^3 log(π) - 15/32 (log(π) + (i π)/2)Here is the more complicated simplification for the $\,^{6,0}_{5,6}$ case with code:
-(575 i π _3 F_3(1, 1, 1;2, 2, 2;i π))/1296 + 85/216 i π _4 F_4(1, 1, 1, 1;2, 2, 2, 2;i π) - 11/36 i π _5 F_5(1, 1, 1, 1, 1;2, 2, 2, 2, 2;i π) + 1/6 i π _6 F_6(1, 1, 1, 1, 1, 1;2, 2, 2, 2, 2, 2;i π) + (3661 Ci(π))/7776 + (3661 i Si(π))/7776 + ζ(5)/30 + ζ(3) (85/648 - (11 gamma )/108 + gamma ^2/36 + (11 i π)/216 - (i gamma π)/36 - π^2/432 + (log^2(π))/36 - (11 log(π))/108 + 1/18 gamma log(π) - 1/36 i π log(π)) - (575 gamma ^2)/2592 + (85 gamma ^3)/1296 - (11 gamma ^4)/864 + gamma ^5/720 + i/(243 π^3) - 211/(7776 π^2) - (3661 i)/(7776 π) - (3661 i π)/7776 + (575 i gamma π)/2592 - 85/864 i gamma ^2 π + 11/432 i gamma ^3 π - 1/288 i gamma ^4 π + (575 π^2)/31104 - (85 gamma π^2)/5184 + (11 gamma ^2 π^2)/1728 - ( gamma ^3 π^2)/864 - (85 i π^3)/10368 + (11 i gamma π^3)/1728 - 1/576 i gamma ^2 π^3 + (11 π^4)/23040 - ( gamma π^4)/3840 - (19 i π^5)/69120 + (log^5(π))/720 - (11 log^4(π))/864 + 1/144 gamma log^4(π) - 1/288 i π log^4(π) + (85 log^3(π))/1296 - 11/216 gamma log^3(π) + 1/72 gamma ^2 log^3(π) + 11/432 i π log^3(π) - 1/72 i gamma π log^3(π) - 1/864 π^2 log^3(π) - (575 log^2(π))/2592 + 85/432 gamma log^2(π) - 11/144 gamma ^2 log^2(π) + 1/72 gamma ^3 log^2(π) - 85/864 i π log^2(π) + 11/144 i gamma π log^2(π) - 1/48 i gamma ^2 π log^2(π) + (11 π^2 log^2(π))/1728 - 1/288 gamma π^2 log^2(π) - 1/576 i π^3 log^2(π) - (3661 (-log(π) - (i π)/2))/15552 - (3661 log(π))/7776 - (575 gamma log(π))/1296 + 85/432 gamma ^2 log(π) - 11/216 gamma ^3 log(π) + 1/144 gamma ^4 log(π) + (575 i π log(π))/2592 - 85/432 i gamma π log(π) + 11/144 i gamma ^2 π log(π) - 1/72 i gamma ^3 π log(π) - (85 π^2 log(π))/5184 + 11/864 gamma π^2 log(π) - 1/288 gamma ^2 π^2 log(π) + (11 i π^3 log(π))/1728 - 1/288 i gamma π^3 log(π) - (π^4 log(π))/3840 + (3661 (log(π) + (i π)/2))/15552Here is the code for the $\,^{7,0}_{6,7}$ case:
(76111 gamma ^2)/995328 - (5845 gamma ^3)/248832 + (415 gamma ^4)/82944 - (5 gamma ^5)/6912 + gamma ^6/17280 - 1/(4096 π^4) - (3367 i)/(2985984 π^3) + 39193/(5971968 π^2) + (952525 i)/(5971968 π) + (952525 i π)/5971968 - (76111 i gamma π)/995328 + (5845 i gamma ^2 π)/165888 - (415 i gamma ^3 π)/41472 + (25 i gamma ^4 π)/13824 - (i gamma ^5 π)/5760 - (76111 π^2)/11943936 + (5845 gamma π^2)/995328 - (415 gamma ^2 π^2)/165888 + (25 gamma ^3 π^2)/41472 - ( gamma ^4 π^2)/13824 + (5845 i π^3)/1990656 - (415 i gamma π^3)/165888 + (25 i gamma ^2 π^3)/27648 - (i gamma ^3 π^3)/6912 - (83 π^4)/442368 + (5 gamma π^4)/36864 - ( gamma ^2 π^4)/30720 + (95 i π^5)/663552 - (19 i gamma π^5)/276480 - (79 π^6)/23224320 - (952525 Ci(π))/5971968 + (76111 i π _3 F_3(1, 1, 1;2, 2, 2;i π))/497664 - (5845 i π _4 F_4(1, 1, 1, 1;2, 2, 2, 2;i π))/41472 + (415 i π _5 F_5(1, 1, 1, 1, 1;2, 2, 2, 2, 2;i π))/3456 - 25/288 i π _6 F_6(1, 1, 1, 1, 1, 1;2, 2, 2, 2, 2, 2;i π) + 1/24 i π _7 F_7(1, 1, 1, 1, 1, 1, 1;2, 2, 2, 2, 2, 2, 2;i π) + (952525 (-(i π)/2 - log(π)))/11943936 + (952525 log(π))/5971968 + (76111 gamma log(π))/497664 - (5845 gamma ^2 log(π))/82944 + (415 gamma ^3 log(π))/20736 - (25 gamma ^4 log(π))/6912 + ( gamma ^5 log(π))/2880 - (76111 i π log(π))/995328 + (5845 i gamma π log(π))/82944 - (415 i gamma ^2 π log(π))/13824 + (25 i gamma ^3 π log(π))/3456 - (i gamma ^4 π log(π))/1152 + (5845 π^2 log(π))/995328 - (415 gamma π^2 log(π))/82944 + (25 gamma ^2 π^2 log(π))/13824 - ( gamma ^3 π^2 log(π))/3456 - (415 i π^3 log(π))/165888 + (25 i gamma π^3 log(π))/13824 - (i gamma ^2 π^3 log(π))/2304 + (5 π^4 log(π))/36864 - ( gamma π^4 log(π))/15360 - (19 i π^5 log(π))/276480 + (76111 log^2(π))/995328 - (5845 gamma log^2(π))/82944 + (415 gamma ^2 log^2(π))/13824 - (25 gamma ^3 log^2(π))/3456 + ( gamma ^4 log^2(π))/1152 + (5845 i π log^2(π))/165888 - (415 i gamma π log^2(π))/13824 + (25 i gamma ^2 π log^2(π))/2304 - 1/576 i gamma ^3 π log^2(π) - (415 π^2 log^2(π))/165888 + (25 gamma π^2 log^2(π))/13824 - ( gamma ^2 π^2 log^2(π))/2304 + (25 i π^3 log^2(π))/27648 - (i gamma π^3 log^2(π))/2304 - (π^4 log^2(π))/30720 - (5845 log^3(π))/248832 + (415 gamma log^3(π))/20736 - (25 gamma ^2 log^3(π))/3456 + 1/864 gamma ^3 log^3(π) - (415 i π log^3(π))/41472 + (25 i gamma π log^3(π))/3456 - 1/576 i gamma ^2 π log^3(π) + (25 π^2 log^3(π))/41472 - ( gamma π^2 log^3(π))/3456 - (i π^3 log^3(π))/6912 + (415 log^4(π))/82944 - (25 gamma log^4(π))/6912 + ( gamma ^2 log^4(π))/1152 + (25 i π log^4(π))/13824 - (i gamma π log^4(π))/1152 - (π^2 log^4(π))/13824 - (5 log^5(π))/6912 + ( gamma log^5(π))/2880 - (i π log^5(π))/5760 + (log^6(π))/17280 - (952525 ((i π)/2 + log(π)))/11943936 - (952525 i Si(π))/5971968 + (-5845/124416 + (415 gamma )/10368 - (25 gamma ^2)/1728 + gamma ^3/432 - (415 i π)/20736 + (25 i gamma π)/1728 - 1/288 i gamma ^2 π + (25 π^2)/20736 - ( gamma π^2)/1728 - (i π^3)/3456 + (415 log(π))/10368 - 25/864 gamma log(π) + 1/144 gamma ^2 log(π) + (25 i π log(π))/1728 - 1/144 i gamma π log(π) - (π^2 log(π))/1728 - (25 log^2(π))/1728 + 1/144 gamma log^2(π) - 1/288 i π log^2(π) + (log^3(π))/432) ζ(3) + ζ(3)^2/432 + (-5/288 + gamma /120 - (i π)/240 + log(π)/120) ζ(5)Here is the code for the $\,^{8,0}_{7,8}$:
8985658285433/6046617600000000 - (65588746609 (137/60 - gamma ))/100776960000000 - (65588746609 gamma )/100776960000000 - (3673451957 gamma ^2)/186624000000 + (58067611 gamma ^3)/9331200000 - (874853 gamma ^4)/622080000 + (12019 gamma ^5)/51840000 - (137 gamma ^6)/5184000 + gamma ^7/604800 - i/(78125 π^5) + 61741/(1280000000 π^4) + (487056529 i)/(2799360000000 π^3) - 6168915439/(5598720000000 π^2) - (226576032859 i)/(5598720000000 π) - (1618229687263 i π)/40310784000000 + (3673451957 i gamma π)/186624000000 - (58067611 i gamma ^2 π)/6220800000 + (874853 i gamma ^3 π)/311040000 - (12019 i gamma ^4 π)/20736000 + (137 i gamma ^5 π)/1728000 - (i gamma ^6 π)/172800 + (3673451957 π^2)/2239488000000 - (58067611 gamma π^2)/37324800000 + (874853 gamma ^2 π^2)/1244160000 - (12019 gamma ^3 π^2)/62208000 + (137 gamma ^4 π^2)/4147200 - ( gamma ^5 π^2)/345600 - (58067611 i π^3)/74649600000 + (874853 i gamma π^3)/1244160000 - (12019 i gamma ^2 π^3)/41472000 + (137 i gamma ^3 π^3)/2073600 - (i gamma ^4 π^3)/138240 + (874853 π^4)/16588800000 - (12019 gamma π^4)/276480000 + (137 gamma ^2 π^4)/9216000 - ( gamma ^3 π^4)/460800 - (228361 i π^5)/4976640000 + (2603 i gamma π^5)/82944000 - (19 i gamma ^2 π^5)/2764800 + (10823 π^6)/6967296000 - (79 gamma π^6)/116121600 - (11 i π^7)/9289728 + (226576032859 Ci(π))/5598720000000 - (3673451957 i π _3 F_3(1, 1, 1;2, 2, 2;i π))/93312000000 + (58067611 i π _4 F_4(1, 1, 1, 1;2, 2, 2, 2;i π))/1555200000 - (874853 i π _5 F_5(1, 1, 1, 1, 1;2, 2, 2, 2, 2;i π))/25920000 + (12019 i π _6 F_6(1, 1, 1, 1, 1, 1;2, 2, 2, 2, 2, 2;i π))/432000 - (137 i π _7 F_7(1, 1, 1, 1, 1, 1, 1;2, 2, 2, 2, 2, 2, 2;i π))/7200 + 1/120 i π _8 F_8(1, 1, 1, 1, 1, 1, 1, 1;2, 2, 2, 2, 2, 2, 2, 2;i π) - (226576032859 (-(i π)/2 - log(π)))/11197440000000 - (4143957338071 log(π))/100776960000000 - (3673451957 gamma log(π))/93312000000 + (58067611 gamma ^2 log(π))/3110400000 - (874853 gamma ^3 log(π))/155520000 + (12019 gamma ^4 log(π))/10368000 - (137 gamma ^5 log(π))/864000 + ( gamma ^6 log(π))/86400 + (3673451957 i π log(π))/186624000000 - (58067611 i gamma π log(π))/3110400000 + (874853 i gamma ^2 π log(π))/103680000 - (12019 i gamma ^3 π log(π))/5184000 + (137 i gamma ^4 π log(π))/345600 - (i gamma ^5 π log(π))/28800 - (58067611 π^2 log(π))/37324800000 + (874853 gamma π^2 log(π))/622080000 - (12019 gamma ^2 π^2 log(π))/20736000 + (137 gamma ^3 π^2 log(π))/1036800 - ( gamma ^4 π^2 log(π))/69120 + (874853 i π^3 log(π))/1244160000 - (12019 i gamma π^3 log(π))/20736000 + (137 i gamma ^2 π^3 log(π))/691200 - (i gamma ^3 π^3 log(π))/34560 - (12019 π^4 log(π))/276480000 + (137 gamma π^4 log(π))/4608000 - ( gamma ^2 π^4 log(π))/153600 + (2603 i π^5 log(π))/82944000 - (19 i gamma π^5 log(π))/1382400 - (79 π^6 log(π))/116121600 - (3673451957 log^2(π))/186624000000 + (58067611 gamma log^2(π))/3110400000 - (874853 gamma ^2 log^2(π))/103680000 + (12019 gamma ^3 log^2(π))/5184000 - (137 gamma ^4 log^2(π))/345600 + ( gamma ^5 log^2(π))/28800 - (58067611 i π log^2(π))/6220800000 + (874853 i gamma π log^2(π))/103680000 - (12019 i gamma ^2 π log^2(π))/3456000 + (137 i gamma ^3 π log^2(π))/172800 - (i gamma ^4 π log^2(π))/11520 + (874853 π^2 log^2(π))/1244160000 - (12019 gamma π^2 log^2(π))/20736000 + (137 gamma ^2 π^2 log^2(π))/691200 - ( gamma ^3 π^2 log^2(π))/34560 - (12019 i π^3 log^2(π))/41472000 + (137 i gamma π^3 log^2(π))/691200 - (i gamma ^2 π^3 log^2(π))/23040 + (137 π^4 log^2(π))/9216000 - ( gamma π^4 log^2(π))/153600 - (19 i π^5 log^2(π))/2764800 + (58067611 log^3(π))/9331200000 - (874853 gamma log^3(π))/155520000 + (12019 gamma ^2 log^3(π))/5184000 - (137 gamma ^3 log^3(π))/259200 + ( gamma ^4 log^3(π))/17280 + (874853 i π log^3(π))/311040000 - (12019 i gamma π log^3(π))/5184000 + (137 i gamma ^2 π log^3(π))/172800 - (i gamma ^3 π log^3(π))/8640 - (12019 π^2 log^3(π))/62208000 + (137 gamma π^2 log^3(π))/1036800 - ( gamma ^2 π^2 log^3(π))/34560 + (137 i π^3 log^3(π))/2073600 - (i gamma π^3 log^3(π))/34560 - (π^4 log^3(π))/460800 - (874853 log^4(π))/622080000 + (12019 gamma log^4(π))/10368000 - (137 gamma ^2 log^4(π))/345600 + ( gamma ^3 log^4(π))/17280 - (12019 i π log^4(π))/20736000 + (137 i gamma π log^4(π))/345600 - (i gamma ^2 π log^4(π))/11520 + (137 π^2 log^4(π))/4147200 - ( gamma π^2 log^4(π))/69120 - (i π^3 log^4(π))/138240 + (12019 log^5(π))/51840000 - (137 gamma log^5(π))/864000 + ( gamma ^2 log^5(π))/28800 + (137 i π log^5(π))/1728000 - (i gamma π log^5(π))/28800 - (π^2 log^5(π))/345600 - (137 log^6(π))/5184000 + ( gamma log^6(π))/86400 - (i π log^6(π))/172800 + (log^7(π))/604800 + (65588746609 (-(i π)/2 + log(π)))/100776960000000 + (226576032859 ((i π)/2 + log(π)))/11197440000000 + (226576032859 i Si(π))/5598720000000 + (69558361/6998400000 + (256103 (137/60 - gamma ))/233280000 - (296057 gamma )/29160000 + (12019 gamma ^2)/2592000 - (137 gamma ^3)/129600 + gamma ^4/8640 + (296057 i π)/58320000 - (12019 i gamma π)/2592000 + (137 i gamma ^2 π)/86400 - (i gamma ^3 π)/4320 - (12019 π^2)/31104000 + (137 gamma π^2)/518400 - ( gamma ^2 π^2)/17280 + (137 i π^3)/1036800 - (i gamma π^3)/17280 - π^4/230400 - (296057 log(π))/29160000 + (12019 gamma log(π))/1296000 - (137 gamma ^2 log(π))/43200 + ( gamma ^3 log(π))/2160 - (12019 i π log(π))/2592000 + (137 i gamma π log(π))/43200 - (i gamma ^2 π log(π))/1440 + (137 π^2 log(π))/518400 - ( gamma π^2 log(π))/8640 - (i π^3 log(π))/17280 + (12019 log^2(π))/2592000 - (137 gamma log^2(π))/43200 + ( gamma ^2 log^2(π))/1440 + (137 i π log^2(π))/86400 - (i gamma π log^2(π))/1440 - (π^2 log^2(π))/17280 - (137 log^3(π))/129600 + ( gamma log^3(π))/2160 - (i π log^3(π))/4320 + (log^4(π))/8640 - (256103 (-(i π)/2 + log(π)))/233280000) ζ(3) + ((-137/60 + gamma )/2160 + (-(i π)/2 + log(π))/2160) ζ(3)^2 + (12019/2160000 - (137 gamma )/36000 + gamma ^2/1200 + (137 i π)/72000 - (i gamma π)/1200 - π^2/14400 - (137 log(π))/36000 + 1/600 gamma log(π) - (i π log(π))/1200 + (log^2(π))/1200) ζ(5) + ζ(7)/840$\,^{9,0}_{8,9}$ code:
-11008015293121/26873856000000000 + (816073489 (49/20 - gamma )^2)/14929920000000 + (39987600961 gamma )/149299200000000 + (532909451657 gamma ^2)/134369280000000 - (483900263 gamma ^3)/373248000000 + (68165041 gamma ^4)/223948800000 - (336581 gamma ^5)/6220800000 + (13489 gamma ^6)/1866240000 - (7 gamma ^7)/10368000 + gamma ^8/29030400 + 1/(1679616 π^6) + (1288991 i)/(656100000000 π^5) - 1802927233/(335923200000000 π^4) - (6367091071 i)/(335923200000000 π^3) + 94576236161/(671846400000000 π^2) + (5497117366741 i)/(671846400000000 π) + (4325716211663 i π)/537477120000000 - (532909451657 i gamma π)/134369280000000 + (483900263 i gamma ^2 π)/248832000000 - (68165041 i gamma ^3 π)/111974400000 + (336581 i gamma ^4 π)/2488320000 - (13489 i gamma ^5 π)/622080000 + (49 i gamma ^6 π)/20736000 - (i gamma ^7 π)/7257600 - (503530806053 π^2)/1612431360000000 + (483900263 gamma π^2)/1492992000000 - (68165041 gamma ^2 π^2)/447897600000 + (336581 gamma ^3 π^2)/7464960000 - (13489 gamma ^4 π^2)/1492992000 + (49 gamma ^5 π^2)/41472000 - ( gamma ^6 π^2)/12441600 + (483900263 i π^3)/2985984000000 - (68165041 i gamma π^3)/447897600000 + (336581 i gamma ^2 π^3)/4976640000 - (13489 i gamma ^3 π^3)/746496000 + (49 i gamma ^4 π^3)/16588800 - (i gamma ^5 π^3)/4147200 - (68165041 π^4)/5971968000000 + (336581 gamma π^4)/33177600000 - (13489 gamma ^2 π^4)/3317760000 + (49 gamma ^3 π^4)/55296000 - ( gamma ^4 π^4)/11059200 + (6395039 i π^5)/597196800000 - (256291 i gamma π^5)/29859840000 + (931 i gamma ^2 π^5)/331776000 - (19 i gamma ^3 π^5)/49766400 - (152233 π^6)/358318080000 + (553 gamma π^6)/1990656000 - (79 gamma ^2 π^6)/1393459200 + (77 i π^7)/159252480 - (11 i gamma π^7)/55738368 - (2339 π^8)/334430208000 - (816073489 (-5369/3600 + π^2/6))/14929920000000 - (5497117366741 Ci(π))/671846400000000 + (270127056529 i π _3 F_3(1, 1, 1;2, 2, 2;i π))/33592320000000 - (483900263 i π _4 F_4(1, 1, 1, 1;2, 2, 2, 2;i π))/62208000000 + (68165041 i π _5 F_5(1, 1, 1, 1, 1;2, 2, 2, 2, 2;i π))/9331200000 - (336581 i π _6 F_6(1, 1, 1, 1, 1, 1;2, 2, 2, 2, 2, 2;i π))/51840000 + (13489 i π _7 F_7(1, 1, 1, 1, 1, 1, 1;2, 2, 2, 2, 2, 2, 2;i π))/2592000 - (49 i π _8 F_8(1, 1, 1, 1, 1, 1, 1, 1;2, 2, 2, 2, 2, 2, 2, 2;i π))/14400 + 1/720 i π _9 F_9(1, 1, 1, 1, 1, 1, 1, 1, 1;2, 2, 2, 2, 2, 2, 2, 2, 2;i π) + (5497117366741 (-(i π)/2 - log(π)))/1343692800000000 + (11354123142131 log(π))/1343692800000000 + (532909451657 gamma log(π))/67184640000000 - (483900263 gamma ^2 log(π))/124416000000 + (68165041 gamma ^3 log(π))/55987200000 - (336581 gamma ^4 log(π))/1244160000 + (13489 gamma ^5 log(π))/311040000 - (49 gamma ^6 log(π))/10368000 + ( gamma ^7 log(π))/3628800 - (532909451657 i π log(π))/134369280000000 + (483900263 i gamma π log(π))/124416000000 - (68165041 i gamma ^2 π log(π))/37324800000 + (336581 i gamma ^3 π log(π))/622080000 - (13489 i gamma ^4 π log(π))/124416000 + (49 i gamma ^5 π log(π))/3456000 - (i gamma ^6 π log(π))/1036800 + (483900263 π^2 log(π))/1492992000000 - (68165041 gamma π^2 log(π))/223948800000 + (336581 gamma ^2 π^2 log(π))/2488320000 - (13489 gamma ^3 π^2 log(π))/373248000 + (49 gamma ^4 π^2 log(π))/8294400 - ( gamma ^5 π^2 log(π))/2073600 - (68165041 i π^3 log(π))/447897600000 + (336581 i gamma π^3 log(π))/2488320000 - (13489 i gamma ^2 π^3 log(π))/248832000 + (49 i gamma ^3 π^3 log(π))/4147200 - (i gamma ^4 π^3 log(π))/829440 + (336581 π^4 log(π))/33177600000 - (13489 gamma π^4 log(π))/1658880000 + (49 gamma ^2 π^4 log(π))/18432000 - ( gamma ^3 π^4 log(π))/2764800 - (256291 i π^5 log(π))/29859840000 + (931 i gamma π^5 log(π))/165888000 - (19 i gamma ^2 π^5 log(π))/16588800 + (553 π^6 log(π))/1990656000 - (79 gamma π^6 log(π))/696729600 - (11 i π^7 log(π))/55738368 + (532909451657 log^2(π))/134369280000000 - (483900263 gamma log^2(π))/124416000000 + (68165041 gamma ^2 log^2(π))/37324800000 - (336581 gamma ^3 log^2(π))/622080000 + (13489 gamma ^4 log^2(π))/124416000 - (49 gamma ^5 log^2(π))/3456000 + ( gamma ^6 log^2(π))/1036800 + (483900263 i π log^2(π))/248832000000 - (68165041 i gamma π log^2(π))/37324800000 + (336581 i gamma ^2 π log^2(π))/414720000 - (13489 i gamma ^3 π log^2(π))/62208000 + (49 i gamma ^4 π log^2(π))/1382400 - (i gamma ^5 π log^2(π))/345600 - (68165041 π^2 log^2(π))/447897600000 + (336581 gamma π^2 log^2(π))/2488320000 - (13489 gamma ^2 π^2 log^2(π))/248832000 + (49 gamma ^3 π^2 log^2(π))/4147200 - ( gamma ^4 π^2 log^2(π))/829440 + (336581 i π^3 log^2(π))/4976640000 - (13489 i gamma π^3 log^2(π))/248832000 + (49 i gamma ^2 π^3 log^2(π))/2764800 - (i gamma ^3 π^3 log^2(π))/414720 - (13489 π^4 log^2(π))/3317760000 + (49 gamma π^4 log^2(π))/18432000 - ( gamma ^2 π^4 log^2(π))/1843200 + (931 i π^5 log^2(π))/331776000 - (19 i gamma π^5 log^2(π))/16588800 - (79 π^6 log^2(π))/1393459200 - (483900263 log^3(π))/373248000000 + (68165041 gamma log^3(π))/55987200000 - (336581 gamma ^2 log^3(π))/622080000 + (13489 gamma ^3 log^3(π))/93312000 - (49 gamma ^4 log^3(π))/2073600 + ( gamma ^5 log^3(π))/518400 - (68165041 i π log^3(π))/111974400000 + (336581 i gamma π log^3(π))/622080000 - (13489 i gamma ^2 π log^3(π))/62208000 + (49 i gamma ^3 π log^3(π))/1036800 - (i gamma ^4 π log^3(π))/207360 + (336581 π^2 log^3(π))/7464960000 - (13489 gamma π^2 log^3(π))/373248000 + (49 gamma ^2 π^2 log^3(π))/4147200 - ( gamma ^3 π^2 log^3(π))/622080 - (13489 i π^3 log^3(π))/746496000 + (49 i gamma π^3 log^3(π))/4147200 - (i gamma ^2 π^3 log^3(π))/414720 + (49 π^4 log^3(π))/55296000 - ( gamma π^4 log^3(π))/2764800 - (19 i π^5 log^3(π))/49766400 + (68165041 log^4(π))/223948800000 - (336581 gamma log^4(π))/1244160000 + (13489 gamma ^2 log^4(π))/124416000 - (49 gamma ^3 log^4(π))/2073600 + ( gamma ^4 log^4(π))/414720 + (336581 i π log^4(π))/2488320000 - (13489 i gamma π log^4(π))/124416000 + (49 i gamma ^2 π log^4(π))/1382400 - (i gamma ^3 π log^4(π))/207360 - (13489 π^2 log^4(π))/1492992000 + (49 gamma π^2 log^4(π))/8294400 - ( gamma ^2 π^2 log^4(π))/829440 + (49 i π^3 log^4(π))/16588800 - (i gamma π^3 log^4(π))/829440 - (π^4 log^4(π))/11059200 - (336581 log^5(π))/6220800000 + (13489 gamma log^5(π))/311040000 - (49 gamma ^2 log^5(π))/3456000 + ( gamma ^3 log^5(π))/518400 - (13489 i π log^5(π))/622080000 + (49 i gamma π log^5(π))/3456000 - (i gamma ^2 π log^5(π))/345600 + (49 π^2 log^5(π))/41472000 - ( gamma π^2 log^5(π))/2073600 - (i π^3 log^5(π))/4147200 + (13489 log^6(π))/1866240000 - (49 gamma log^6(π))/10368000 + ( gamma ^2 log^6(π))/1036800 + (49 i π log^6(π))/20736000 - (i gamma π log^6(π))/1036800 - (π^2 log^6(π))/12441600 - (7 log^7(π))/10368000 + ( gamma log^7(π))/3628800 - (i π log^7(π))/7257600 + (log^8(π))/29030400 - (816073489 (49/20 - gamma ) (-(i π)/2 + log(π)))/7464960000000 + (816073489 (-(i π)/2 + log(π))^2)/14929920000000 - (5497117366741 ((i π)/2 + log(π)))/1343692800000000 - (5497117366741 i Si(π))/671846400000000 + (-533180263/279936000000 - (28567 (49/20 - gamma )^2)/311040000 + (27783497 gamma )/13996800000 - (154007 gamma ^2)/155520000 + (13489 gamma ^3)/46656000 - (49 gamma ^4)/1036800 + gamma ^5/259200 - (27783497 i π)/27993600000 + (154007 i gamma π)/155520000 - (13489 i gamma ^2 π)/31104000 + (49 i gamma ^3 π)/518400 - (i gamma ^4 π)/103680 + (32291 π^2)/622080000 - (13489 gamma π^2)/186624000 + (49 gamma ^2 π^2)/2073600 - ( gamma ^3 π^2)/311040 - (13489 i π^3)/373248000 + (49 i gamma π^3)/2073600 - (i gamma ^2 π^3)/207360 + (49 π^4)/27648000 - ( gamma π^4)/1382400 - (19 i π^5)/24883200 + (28567 (-5369/3600 + π^2/6))/311040000 + (27783497 log(π))/13996800000 - (154007 gamma log(π))/77760000 + (13489 gamma ^2 log(π))/15552000 - (49 gamma ^3 log(π))/259200 + ( gamma ^4 log(π))/51840 + (154007 i π log(π))/155520000 - (13489 i gamma π log(π))/15552000 + (49 i gamma ^2 π log(π))/172800 - (i gamma ^3 π log(π))/25920 - (13489 π^2 log(π))/186624000 + (49 gamma π^2 log(π))/1036800 - ( gamma ^2 π^2 log(π))/103680 + (49 i π^3 log(π))/2073600 - (i gamma π^3 log(π))/103680 - (π^4 log(π))/1382400 - (154007 log^2(π))/155520000 + (13489 gamma log^2(π))/15552000 - (49 gamma ^2 log^2(π))/172800 + ( gamma ^3 log^2(π))/25920 - (13489 i π log^2(π))/31104000 + (49 i gamma π log^2(π))/172800 - (i gamma ^2 π log^2(π))/17280 + (49 π^2 log^2(π))/2073600 - ( gamma π^2 log^2(π))/103680 - (i π^3 log^2(π))/207360 + (13489 log^3(π))/46656000 - (49 gamma log^3(π))/259200 + ( gamma ^2 log^3(π))/25920 + (49 i π log^3(π))/518400 - (i gamma π log^3(π))/25920 - (π^2 log^3(π))/311040 - (49 log^4(π))/1036800 + ( gamma log^4(π))/51840 - (i π log^4(π))/103680 + (log^5(π))/259200 + (28567 (49/20 - gamma ) (-(i π)/2 + log(π)))/155520000 - (28567 (-(i π)/2 + log(π))^2)/311040000) ζ(3) + ((49/20 - gamma )^2/25920 + π^2/77760 + (5369/3600 - π^2/6)/25920 + ((-49/20 + gamma ) (-(i π)/2 + log(π)))/12960 + (-(i π)/2 + log(π))^2/25920) ζ(3)^2 + (-336581/259200000 + (13489 gamma )/12960000 - (49 gamma ^2)/144000 + gamma ^3/21600 - (13489 i π)/25920000 + (49 i gamma π)/144000 - (i gamma ^2 π)/14400 + (49 π^2)/1728000 - ( gamma π^2)/86400 - (i π^3)/172800 + (13489 log(π))/12960000 - (49 gamma log(π))/72000 + ( gamma ^2 log(π))/7200 + (49 i π log(π))/144000 - (i gamma π log(π))/7200 - (π^2 log(π))/86400 - (49 log^2(π))/144000 + ( gamma log^2(π))/7200 - (i π log^2(π))/14400 + (log^3(π))/21600) ζ(5) + (ζ(3) ζ(5))/10800 + (-7/14400 + gamma /5040 - (i π)/10080 + log(π)/5040) ζ(7)This is the limit of what my tablet can do, so I will stop here. Notice that the Meijer G functions really can be simplified into the following, so notice patterns above:
Let’s assume that products like $\prod\limits_{n=5}^4 f(n)=1$ based on this computation, but help is wanted in which convention should be used.
Codes:
MeijerG[{{Subscript[a, 1], \[Ellipsis], Subscript[a, n]}, {Subscript[a, n + 1], \[Ellipsis], Subscript[a, p]}}, {{Subscript[b, 1], \[Ellipsis], Subscript[b, m]}, {Subscript[b, m + 1], \[Ellipsis], Subscript[b, q]}}, z] == Sum[((Product[If[j == k, 1, Gamma[Subscript[b, j] - Subscript[b, k]]], {j, 1, m}] Product[Gamma[1 + Subscript[b, k] - Subscript[a, j]], {j, 1, n}])/(Product[Gamma[Subscript[a, j] - Subscript[b, k]], {j, n + 1, p}] Product[Gamma[1 - Subscript[b, j] + Subscript[b, k]], {j, m + 1, q}])) z^Subscript[b, k] HypergeometricPFQ[ {1 + Subscript[b, k] - Subscript[a, 1], \[Ellipsis], 1 + Subscript[b, k] - Subscript[a, p]}, {1 + Subscript[b, k] - Subscript[b, 1], \[Ellipsis], 1 + Subscript[b, k] - Subscript[b, k - 1], 1 + Subscript[b, k] - Subscript[b, k + 1], \[Ellipsis], 1 + Subscript[b, k] - Subscript[b, q]}, (-1)^(p - m - n) z], {k, 1, m}] /; (p < q || (p == q && m + n > p) || (p == q && m + n == p && Abs[z] < 1)) && ForAll[{j, k}, Element[{j, k}, Integers] && j != k && 1 <= j <= m && 1 <= k <= m, !Element[Subscript[b, j] - Subscript[b, k], Integers]]MeijerG[{{Subscript[a, 1], \[Ellipsis], Subscript[a, n]}, {Subscript[a, n + 1], \[Ellipsis], Subscript[a, p]}}, {{Subscript[b, 1], \[Ellipsis], Subscript[b, m]}, {Subscript[b, m + 1], \[Ellipsis], Subscript[b, q]}}, z] == Sum[((Product[If[j == k, 1, Gamma[Subscript[a, k] - Subscript[a, j]]], {j, 1, n}] Product[Gamma[1 + Subscript[b, j] - Subscript[a, k]], {j, 1, m}])/(Product[Gamma[Subscript[a, k] - Subscript[b, j]], {j, m + 1, q}] Product[Gamma[1 + Subscript[a, j] - Subscript[a, k]], {j, n + 1, p}])) z^(Subscript[a, k] - 1) HypergeometricPFQ[ {1 + Subscript[b, 1] - Subscript[a, k], \[Ellipsis], 1 + Subscript[b, q] - Subscript[a, k]}, {1 + Subscript[a, 1] - Subscript[a, k], \[Ellipsis], 1 + Subscript[a, k - 1] - Subscript[a, k], 1 + Subscript[a, k + 1] - Subscript[a, k], \[Ellipsis], 1 + Subscript[a, p] - Subscript[a, k]}, (-1)^(q - m - n)/z], {k, 1, n}] /; (p > q || (p == q && m + n == p + 1 && !IntervalMemberQ[Interval[{-1, 0}], z]) || (p == q && m + n > p + 1) || (p == q && m + n == p && Abs[z] > 1)) && ForAll[{j, k}, Element[{j, k}, Integers] && j != k && 1 <= j <= n && 1 <= k <= n, !Element[Subscript[a, j] - Subscript[a, k], Integers]]from this Wolfram Functions source.