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.