Can we find sums for $\int_1^{\infty }e^{\pi i t} \left(t^{1/t}-1\right) \, dt?$

Solution 1:

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))/15552

Here 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:

• $\ln^k(\pi),k\in\Bbb N$
• Riemann Zeta functions
• Increasingly complicated rational number coefficients
• Hypergeometric function terms in the form of $\,_p \text F_p(a,…,a;b,…,b;\pi i);a,b\in\Bbb N$
• $\pi^k,k\in\Bbb N$
• Arithmetic Operations
• The Euler-Mascheroni constant in the form of $γ^k,k\in\Bbb N$
• The Imaginary unit
• $\text{Ci}(\pi)$
• The Wilbraham-Gibbs constant: $\text G’$ $\text G’=\text{Si}(\pi)$

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.