Introduction to Ramanujan theta functions
Michael Somos 12 Oct 2019
michael.somos@gmail.com
In his work on elliptic functions Ramanujan used his own
version of theta functions defined by power series using his own
notation for the functions. Here is one way to motivate the form
of his particular definition of his functions.
To motivate the definition, recall the simplest converging
infinite series which is the geometric power series, the sum of all
nonnegative powers of x summing to 1/(1-x) . We would like a
multiplicative analog of this series. One possibility is to use
1-x^n as the factors in the infinite product, but starting at
n=1 because 1-x^0 = 1-1 = 0 would cause the product to vanish
immediately. Therefore we have the following definition.
Definition 1. The Ramanujan f function is defined by
f(-x) = (1-x)(1-x^2)(1-x^3)(1-x^4) ...
where |x|<1 is required for convergence. Often, we use formal power
series and so all we need is that x^n converges to zero as n goes
to infinity. The reason for the f(-x) instead of the simpler f(x)
comes from Ramanujan's general two variable theta function which is
Definition 2. The general two variable Ramanujan f function as
used in his notebooks is defined by him using the following series
f(a,b) = 1 +(a+b) +(ab)(a^2+b^2) +(ab)^3(a^3+b^3) +(ab)^6(a^4+b^4) + ...
where |ab|<1 is required for convergence. The idea behind Ramanujan's
function is that it is a two-way infinite sum of terms in which the
quotient of consecutive terms is a geometric progression. Thus we have
f(a,b) =... +a^6b^10 +a^3b^6 +ab^3 +b +1 +a +a^3b +a^6b^3 +a^10b^6 + ...
This can be written with summation notation using an index variable
n summing a^n(n+1)/2b^n(n-1)/2 over all integer n . Further,
and, surprisingly, it has an infinite product expression as follows
f(a,b) = (1+a)(1+b)(1-q)(1+aq)(1+bq)(1-q^2)(1+aq^2)(1+bq^2)(1-q^3) ...
where q=ab . This is equivalent to Jacobi's triple product identity,
thus f(a,b) is an example of a power series that has both infinite
sum and infinite product expressions.
In terms of f(a,b) Ramanujan's one variable theta function is
f(-x) = f(-x,-x^2) ,
which perhaps explains why Ramanujan used f(-x) instead of the more
obvious f(x) . The power series expansion of this theta function is
f(-x) = 1 - x - x^2 + x^5 + x^7 - x^12 - x^15 + x^22 + ... .
It appears that each term is a power of x with coefficients plus one
or minus one. The signs appear to alternate with two minus signs
followed by two plus signs and so on. The exponents of x are the
generalized pentagonal numbers. This power series is the ordinary
generating function of OEIS sequence A010815.
Ramanujan also defined theta function power series based on the
exponents of x being the square and triangular numbers. Thus define
Definition 3. The Ramanujan phi function is defined by
phi(x) := f(x,x) = 1 + 2 x + 2 x^4 + 2 x^9 + 2 x^16 + ... .
This power series is the ordinary generating function of OEIS sequence
A000122. This is closely related to another of his theta functions.
Definition 4. The Ramanujan psi function is defined by
psi(x) := f(x,x^3) = 1 + x + x^3 + x^6 + x^10 + x^15 + ... .
This power series is the ordinary generating function of OEIS sequence
A010054. The last power series is not actually a theta function, but is
used by Ramanujan in important ways but not as often as the phi and
psi theta functions. Its definition is as follows
Definition 5. The Ramanujan chi function is defined by
chi(x) := (1 + x)(1 + x^3)(1 + x^5) ... = 1 + x + x^3 + x^4 + ... .
This power series is the ordinary generating function of OEIS sequence
A000700. There seems to be no simple infinite sum expression for it.