A Maple Rendition of George Polya’s Elementary Proof^{α} of the Modularity of the Theta Function

Dr. Z.’s Experimental Math Spring 2018 class ^{β}

Trivial Fact 1^{γ}:

(z^{\frac{1}{2}} + z^{− \frac{1}{2}} )^{2m} = ∑_{v=−m}^{m} \binom{2m}{m + v} z^{v} .

Trivial Fact 2^{δ} : For any (Laurent) polynomial P (z) = ∑_{N_{1}<v<N_{2}} a_{v} z^{v} and any positive odd integer l, if ω = e^{2πi/l},

∑_{− \frac{l}{2} <v< \frac{l}{2}} P (ω^{v}z) = l ∑_{\frac{N_{1}}{l} <v< \frac{N_{2}}{l}} a_{lv}z^{lv} .

Combining (and dividing by 2^{2m}):

∑_{− \frac{l}{2} <v< \frac{l}{2}} \frac{1}{2^{2m}} ((ω^{v} z)^{1/2} + (ω^{v} z)^{−1/2})^{2 m} = l ∑_{− \frac{m}{l} <v< \frac{m}{l}} \frac{1}{2^{2m}} \binom{2m}{m + l v}z^{lv} .

Now let z = e^{s/l} and l = ⌊\sqrt{m t}⌋ , and take the limits as m → ∞. Go to maple, and copy-and-paste:

l:=sqrt(m*t):  z:=exp(s/l):  omega:=exp(2*Pi*I/l):  L:= ( ( ((omega**v*z)**(1/2) +
(omega**v*z)**(-1/2)))/2)**(2*m):  R:= l*binomial(2*m,m+l*v)*z**(l*v)/2**(2*m):
factor(limit(L, m=infinity)); simplify(limit(R,m=infinity));

and get, respectively,

e^{\frac{−(2 π v−is)^{2}}{4t}} , \begin{matrix}\sqrt{t} & e^{v(−tv+s)} \\ √ & \overline{π} \end{matrix} ,

yielding:

∑_{v=−∞}^{∞} e^{(s+2πiv)^{2}/(4t)} = (\frac{t}{π})^{1/2} ∑_{v=−∞}^{∞} e^{−tv^{2}+sv} .

April 24, 2018.

α G. Polya, Sitz. der Phys.-Math. Klasse, Berlin (1927), 158-161. See also R. Bellman, “A Brief Introduction to Theta

Functions” (1961), pp. 40-41.

β Yonah Biers-Ariel, Matthew P. Charnley, Amit Harel, George D. Hauser, Edna L. Jones, Ahsan Z. Khan, Brooke

Logan, Yukun Yao, Anthony Zaleski. Instructor: Doron Zeilberger.

γ Binomial Theorem.

δ Sum of a geometric series ∑_{−l/2<v<l/2} z^{v} = \frac{z^{l/2}−z^{−l/2}}{z^{1/2}−z^{−1/2}} .