#C26.txt;  April 23, 2018
#Having Maple do Polya's combintorial proof of the modularity of the theta function
Help:=proc():  print(`Polya(s,t,v)`): end:

Polya:=proc(s,t,v) local el,m,L,R,z,omega:


el:=sqrt(m*t):
z:=exp(s/el):
omega:=exp(2*Pi*I/el):


L:= ( ( ((omega^v*z)^(1/2) +  (omega^v*z)^(-1/2)))/2)^(2*m):
R:= el*binomial(2*m,m+el*v)*z^(el*v)/2^(2*m):

simplify(limit(L, m=infinity)), simplify(limit(R,m=infinity));


end: