(x/2)^ν/Γ(ν+1) を割った値を返すので、使用の際に注意が必要。
K=10;
my_besseli(n, x) = besseli(n, x)*(x/2)^n/n!;
for(k=0, K, print([k, besseli(k, 2*x)]));
print;
for(k=0, K, print([k, my_besseli(k, 2*x)]));
出力結果
[0, 1 + x^2 + 1/4*x^4 + 1/36*x^6 + 1/576*x^8 + 1/14400*x^10 + 1/518400*x^12 + 1/25401600*x^14 + 1/1625702400*x^16 + O(x^17)]
[1, 1 + 1/2*x^2 + 1/12*x^4 + 1/144*x^6 + 1/2880*x^8 + 1/86400*x^10 + 1/3628800*x^12 + 1/203212800*x^14 + 1/14631321600*x^16 + O(x^17)]
[2, 1 + 1/3*x^2 + 1/24*x^4 + 1/360*x^6 + 1/8640*x^8 + 1/302400*x^10 + 1/14515200*x^12 + 1/914457600*x^14 + 1/73156608000*x^16 + O(x^17)]
[3, 1 + 1/4*x^2 + 1/40*x^4 + 1/720*x^6 + 1/20160*x^8 + 1/806400*x^10 + 1/43545600*x^12 + 1/3048192000*x^14 + 1/268240896000*x^16 + O(x^17)]
[4, 1 + 1/5*x^2 + 1/60*x^4 + 1/1260*x^6 + 1/40320*x^8 + 1/1814400*x^10 + 1/108864000*x^12 + 1/8382528000*x^14 + 1/804722688000*x^16 + O(x^17)]
[5, 1 + 1/6*x^2 + 1/84*x^4 + 1/2016*x^6 + 1/72576*x^8 + 1/3628800*x^10 + 1/239500800*x^12 + 1/20118067200*x^14 + 1/2092278988800*x^16 + O(x^17)]
[6, 1 + 1/7*x^2 + 1/112*x^4 + 1/3024*x^6 + 1/120960*x^8 + 1/6652800*x^10 + 1/479001600*x^12 + 1/43589145600*x^14 + 1/4881984307200*x^16 + O(x^17)]
[7, 1 + 1/8*x^2 + 1/144*x^4 + 1/4320*x^6 + 1/190080*x^8 + 1/11404800*x^10 + 1/889574400*x^12 + 1/87178291200*x^14 + 1/10461394944000*x^16 + O(x^17)]
[8, 1 + 1/9*x^2 + 1/180*x^4 + 1/5940*x^6 + 1/285120*x^8 + 1/18532800*x^10 + 1/1556755200*x^12 + 1/163459296000*x^14 + 1/20922789888000*x^16 + O(x^17)]
[9, 1 + 1/10*x^2 + 1/220*x^4 + 1/7920*x^6 + 1/411840*x^8 + 1/28828800*x^10 + 1/2594592000*x^12 + 1/290594304000*x^14 + 1/39520825344000*x^16 + O(x^17)]
[10, 1 + 1/11*x^2 + 1/264*x^4 + 1/10296*x^6 + 1/576576*x^8 + 1/43243200*x^10 + 1/4151347200*x^12 + 1/494010316800*x^14 + 1/71137485619200*x^16 + O(x^17)]
[0, 1 + x^2 + 1/4*x^4 + 1/36*x^6 + 1/576*x^8 + 1/14400*x^10 + 1/518400*x^12 + 1/25401600*x^14 + 1/1625702400*x^16 + O(x^17)]
[1, x + 1/2*x^3 + 1/12*x^5 + 1/144*x^7 + 1/2880*x^9 + 1/86400*x^11 + 1/3628800*x^13 + 1/203212800*x^15 + 1/14631321600*x^17 + O(x^18)]
[2, 1/2*x^2 + 1/6*x^4 + 1/48*x^6 + 1/720*x^8 + 1/17280*x^10 + 1/604800*x^12 + 1/29030400*x^14 + 1/1828915200*x^16 + 1/146313216000*x^18 + O(x^19)]
[3, 1/6*x^3 + 1/24*x^5 + 1/240*x^7 + 1/4320*x^9 + 1/120960*x^11 + 1/4838400*x^13 + 1/261273600*x^15 + 1/18289152000*x^17 + 1/1609445376000*x^19 + O(x^20)]
[4, 1/24*x^4 + 1/120*x^6 + 1/1440*x^8 + 1/30240*x^10 + 1/967680*x^12 + 1/43545600*x^14 + 1/2612736000*x^16 + 1/201180672000*x^18 + 1/19313344512000*x^20 + O(x^21)]
[5, 1/120*x^5 + 1/720*x^7 + 1/10080*x^9 + 1/241920*x^11 + 1/8709120*x^13 + 1/435456000*x^15 + 1/28740096000*x^17 + 1/2414168064000*x^19 + 1/251073478656000*x^21 + O(x^22)]
[6, 1/720*x^6 + 1/5040*x^8 + 1/80640*x^10 + 1/2177280*x^12 + 1/87091200*x^14 + 1/4790016000*x^16 + 1/344881152000*x^18 + 1/31384184832000*x^20 + 1/3515028701184000*x^22 + O(x^23)]
[7, 1/5040*x^7 + 1/40320*x^9 + 1/725760*x^11 + 1/21772800*x^13 + 1/958003200*x^15 + 1/57480192000*x^17 + 1/4483454976000*x^19 + 1/439378587648000*x^21 + 1/52725430517760000*x^23 + O(x^24)]
[8, 1/40320*x^8 + 1/362880*x^10 + 1/7257600*x^12 + 1/239500800*x^14 + 1/11496038400*x^16 + 1/747242496000*x^18 + 1/62768369664000*x^20 + 1/6590678814720000*x^22 + 1/843606888284160000*x^24 + O(x^25)]
[9, 1/362880*x^9 + 1/3628800*x^11 + 1/79833600*x^13 + 1/2874009600*x^15 + 1/149448499200*x^17 + 1/10461394944000*x^19 + 1/941525544960000*x^21 + 1/105450861035520000*x^23 + 1/14341317100830720000*x^25 + O(x^26)]
[10, 1/3628800*x^10 + 1/39916800*x^12 + 1/958003200*x^14 + 1/37362124800*x^16 + 1/2092278988800*x^18 + 1/156920924160000*x^20 + 1/15064408719360000*x^22 + 1/1792664637603840000*x^24 + 1/258143707814952960000*x^26 + O(x^27)]
