Does $S = \sum_{t=1}^{\infty} \lambda^{\frac{t(t-1)}{2}}$ for $0<\lambda<1$ have a simple solution

Question

The difference equations $$S_t = 1 + x_t S_{t+1} $$ where $$x_1 = \lambda x_{t-1},\quad x_0 =1,\quad 0<\lambda <1 $$ occurs naturally in economic models. Iterating the first expression forwards gives the infinite sum $$ S_1 = 1 + \lambda + \lambda^3 + \lambda^6 + \lambda^{10} + \ldots $$ which has the general term $\lambda^{\frac{t(t-1)}{2}}$. Does anyone know if there is a simple closed form expression for $S_1$ as a function of $\lambda$?

Answer 1

I don't think there's a simple closed form for $S_1$. However, you can evaluate $S_1$ numerically.

Mathematica

s = Sum[λ^(t (t - 1)/2), {t, 0, ∞}, Assumptions -> 0 < λ < 1]

$$\frac{2 \sqrt[8]{\lambda }+\vartheta _2\left(0,\sqrt{\lambda }\right)}{2 \sqrt[8]{\lambda }}$$

Plot[s, {λ, 0, 1}, Frame -> True, PlotRange -> {{0, 1}, {0, 7}}, 
 FrameLabel -> (Style[#, Bold, 18] &) /@ {"λ", "S1"}]

R

library(elliptic)

lambda = c(1:95)/100
S1 = (2*lambda^(1/8) + theta2(0, q=lambda^(1/2), maxiter=100))/(2*lambda^(1/8))
plot(lambda, S1, type="l", xlab="lambda", ylab="S1", las=1, xlim=c(0,1), ylim=c(0,7))