The Dimensional Fluxes of the Hypercylindrical Function

Table 2

Freedom degree 𝑘	∑^∞𝑛=0HY(𝑘,𝑛,𝑟)	∑^∞𝑛=0HY(𝑘,𝑛,1)	∫0^+∞HY(𝑘,𝑛,1)𝑑𝑛
0	2𝑒^𝑟/𝜋	1.730511958864530	1.846084380068597
1	2𝑒^𝑟	5.43656365691809	5.248099060246887
2	8𝑒^𝑟	21.7462546276723	22.46216193622815
3	12𝜋𝑒^𝑟	102.4768106720828	85.44633790197124
4	64𝜋(𝑒^𝑟−1)	345.4810604213614	237.5329239075405
5	120𝜋^2(𝑒^𝑟−𝑟−1)	850.6988994458283	508.7293133444046
6	384𝜋^2[2𝑒^𝑟−(𝑟+1)^2−1]	1654.544866434988	892.2064355912792
7	∑^∞𝑛=0(1680𝜋^3𝑟^𝑛+4/Γ(𝑛+5))	2688.661898887322	1338.143855891663
8	∑^∞𝑛=0(12288𝜋^3𝑟^𝑛+5/Γ(𝑛+6))	3790.427657262334	1768.586344896620
9	∑^∞𝑛=0(30240𝜋^4𝑟^𝑛+6/Γ(𝑛+7))	4757.702808149823	2104.228782270664
10	∑^∞𝑛=0(245760𝜋^4𝑟^𝑛+7/Γ(𝑛+8))	5416.805463135256	2289.343605902835
11	∑^∞𝑛=0(665280𝜋^5𝑟^𝑛+8/Γ(𝑛+9))	5672.025047777118	2304.879158127108
12	∑^∞𝑛=0(5898240𝜋^5𝑟^𝑛+9/Γ(𝑛+10))	5520.736405122306	2167.383511182325
13	∑^∞𝑛=0(17297280𝜋^6𝑟^𝑛+10/Γ(𝑛+11))	5036.815477026034	1917.751163982797
⋮	⋮	⋮	⋮
50	∑^∞𝑛=0HY(50,𝑛,𝑟)	0.000000117826838	0.000000029841827
⋮	⋮	⋮	⋮
∞	lim𝑘→∞∑^∞𝑛=0HY(𝑘,𝑛,𝑟)=0	0	lim𝑘→∞∫^∞0HY(𝑘,𝑛,𝑟)𝑑𝑛=0
—	∑𝑘∑𝑛	52812.977966365550	—