例如,C(x,q)=Sum_{n>=0}Sum_{k=0..n*(n-1)/2}T(n,k)*x^(2*n)*q^(2*k)/(2*n)!开始
C(x,q)=1+x^2/2!+(4*q^2+1)*x^4/4!+(24*q^6+16*q^4+20*q^2+1)*x^6/6!+(192*q^12+128*q^10+384*q^8+288*q^6+336*q^4+56*q^2+1)*x^8/8!+(1920*q^20+1280*q^18+3840*qq^16+5760*q*14+10176*q^12+588*q^10+12736*q^8+6448*q^6+2352*q^4+120*q^2+1)*x^10/10!+(23040*q^30+15360*q^28+46080*q^26+69120*q^24+164352*q^22+141056*q^20+341504*q^18+294912*qqu16+431616*q^14+385472*qq^12+472704*qqu10+214016*qqu8+93280*q^6+10032*q^4+220*qqu2+1)*x^12/12!+。。。
使得C(x,q)=cosh(积分C(q*x,q,dx)。
这个系数T(n,k)为x^(2*n)*q^(2*k)/(2*n)的不规则三角形!在C(x,q)中开始:
1;
1;
1, 4;
1, 20, 16, 24;
1, 56, 336, 288, 384, 128, 192;
1, 120, 2352, 6448, 12736, 5888, 10176, 5760, 3840, 1280, 1920;
1, 220, 10032, 93280, 214016, 472704, 385472, 431616, 294912, 341504, 141056, 164352, 69120, 46080, 15360, 23040;
1, 364, 32032, 740168, 4072640, 11702912, 18676672, 30112640, 23848704, 27599616, 17884032, 20958208, 13595136, 11074560, 5992448, 5945856, 2673664, 2300928, 967680, 645120, 215040, 322560;
1, 560, 84448, 3952832, 53301248, 230161152, 738249344, 1166436352, 1970874368, 2196244480, 2459786240, 1804101632, 2061498368, 1537437696, 1437724672, 989968384, 921092096, 487923712, 499621888, 282034176, 211599360, 117383168, 108036096, 42778624, 36814848, 15482880, 10321920, 3440640, 5160960; ...
相关系列。
S(x,q)=x+(q^2+1)*x^3/3!+(4*q^6+q^4+10*q^2+1)*x^5/5!+(24*q^12+16*q^10+20*q^8+85*q^6+91*q^4+35*qq^2+1)*x^7/7!+(192*q^20+128*q^18+384*q^16+288*q^14+1200*q^12+632*q^10+2273*q^8+1324*q^6+966*q^4+84*q^2+1)*x^9/9!+(1920*q^30+1280*q^28+3840*qq^26+5760*qqu24+10176*q^22+16448*q^20+19776*qqu18+27568*q^16+49872*q^14+69816*q^12+64329*q^10+50941*q^8+26818*q^6+5082*q^4+165*q^2+1)*x^11/11!+。。。
其中C(x,q)^2-S(x,q^2=1。