0017: Article 7 (More Ramanujan continued fractions)

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Ramanujan’s Continued Fractions, Apéry’s Constant, and More

by Tito Piezas III

“In his favorite topics, like infinite series and continued fractions, he had no equal this century…” – G.H. Hardy on Ramanujan

I. Introduction

II. Euler’s Continued Fraction

III. Ramanujan and Apery’s Constant

IV. Constants

V. Between Two Continued Fractions

VI. Infinite Series

VII. Gamma Function

VIII. Generalized Hypergeometric Function

IX. Integrals

I. Introduction

In the article “Ramanujan’s Continued Fractions and the Platonic Solids” we discussed three kinds of beautiful continued fractions (for brevity, “cfrac”) which involve the argument q = e^2πiτ = exp(2πiτ) and their unexpected connections to geometry. Ramanujan, however, had many other kinds of cfracs up his sleeve. In “Chapter 12 of Ramanujan’s Second Notebook” [1], authors Berndt et al gave more than 30 others, many with several free variables. There are others in various places in his Notebooks but, for the moment, this author will include mostly those in [1].

The primary motivation for this article is to present Ramanujan’s cfracs in the original, visually-pleasing form. While not the most space-saving, in this modern era where information is measured in terabytes and we have convenient math formats like LaTex, it seems there is no reason not to do so. So without further ado…

II. Euler’s Continued Fraction Formula

Two very general results are Euler’s continued fraction (1748) and Gauss’ continued fraction (1813). (In the quote at the start of this article, Hardy was careful in specifying Ramanujan’s role as he was preceded by Euler and Gauss. It would have been interesting had they all met in the same century.)

Gauss’ results will be appropriate for the section on hypergeometric functions. Euler’s cfrac can be given as,

from which one can derive other forms, such as,

This automatically gives a continued fraction representation for the Riemann zeta function ζ(s), or the more general Hurwitz zeta function, though Ramanujan gave his own version in the next section.

III. Ramanujan and Apery’s constant

Entry 32iii: For any x > 0. Define v = 2(x²+x), then,

Form 1:

We can also express x in terms of v,

then for v ≥ 0,

with the case v = 1 specifically given by Ramanujan.

Form 2:

still with v = 2(x²+x), and the sequence P(m) = {1, 3, 7, 13,…} generated by P(m) = m²+m+1. This form, for x = v = 0, reduces to Euler’s cfrac for the Riemann zeta function ζ(s) at s = 3. This particular case naturally has,

(2m+1)(m²+m+1) = m³ + (m+1)³

or the sum of two consecutive cubes. (The sum of two consecutive squares will appear in the next section.) Apery would later give a similar formula,

where P(n) = 17n²+17n+5 yields the sequence {5, 39, 107, 209,…}, though it works only if x = y = 0. It is interesting that in Ramanujan’s version, there is this polynomial relationship between x and v, given by v = 2x²+2x, since there doesn’t seem to be anything analogous using Euler’s cfrac for the zeta function ζ(s) with odd exponent s > 3 of simple form,

v = P(x)

where v is a polynomial in x, though I didn't check the more complicated version,

P(v) = P(x)

where each side is a polynomial with rational coefficients.

The rest of this article will give a selection from Chap 12 with little commentary, starting with the simpler ones. I’ll add the more complicated results from other parts of his Notebooks in another article. The interested reader is encouraged to find generalizations, special cases, or simple proofs (if possible) for these beautiful mathematical objects.

IV. Constants

Entry 7: For x > 0,

While simple in form, this holds a hidden surprise. For x = 1, the convergents are {2/1, 4/5, 20/19, 100/101, 620/619,…} where the numerators N(n) are “weighted Fibonacci numbers” (sequence A153229) and denominators D(n) are alternating factorials with the formula for both as,

Entry 13: If a < b,

Entry 12: If x ≠ 0,

Entry 18: Let x be outside the real interval [-1,1], then,

V. Between Two Continued Fractions

Ramanujan gave several examples of two continued fractions converging to the same value (such as the one given for the Hurwitz zeta function in Part III). A few others are,

Entry 27: Let x,y > 0, then,

Notice that x and y just exchange places. A similar one is,

Entry 28: Let x > 0. Define,

One can just admire the artistry by which Ramanujan conjures up his continued fractions. The next (slightly modified by this author) is a transformation formula not in Chap 12, but in the Lost Notebook (p. 46). Let k ≥ 0, and,

u = (1+√(1+4k))/2

v = (1-√(1+4k))/2

For any |q| < 1, then,

One feature of this identity is the RHS accelerates the convergence of the LHS. Note that for k = 0, then both sides reduce to the Rogers-Ramanujan continued fraction without the factor q^1/5. Also, it is easily noticed that u and v are the roots of the quadratic equation x²-x-k = 0. A conjecture by this author is then,

Conjecture: “Given x²-ax-b = 0, a²+4b ≥ 0, and roots {x₁, x₂} where x₁ ≥ x₂. For any |q| < 1, then,”

For various {a,b} that satisfy the constraint a²+4b ≥ 0, one can easily check with Mathematica that the two cfracs apparently are converging on the same value. If indeed generally true, it should be interesting if this has a cubic version for x³-ax²+bx-c = 0 and its roots {x₁, x₂, x₃}.

VI. Infinite Series

(Note: Ramanujan’s infinite series Σ start with either k = 0 or 1, so care should be exercised.)

Entry 8:

Corollary: For x = 0, we get,

A similar form, but where the denominators of the infinite series involve only odd numbers is,

Entry 43: For x > 0, then,

For x = 1, this is faintly reminiscent of the previous entry. (Regarding this particular example, Kevin Brown of the Mathpages remarked, “Is there any other mathematician whose work is so instantly recognizable?”)

Entry 16: If {m,n} are not negative integers,

Corollary: For m = n = 0, this alternating series reduces to the Dirichlet eta function η(s) for s = 2, hence,

Entry 29: Let x > 0 and n² < 1.

Corollary: For n = 0, this reduces to a version of the Dirichlet eta function η(s) for s = 1,

This cfrac has the nice property that, depending on whether x is odd or even, then it evaluates to an expression involving either log(2) or π. For general x though,

Conjecture 1: For any x ≥ 1,

which, for integer x, then odd x involve log(2), while even x involve π. Specifically, f(1) = log(2), f(2) = (4-π)/2, f(3) = 1-log(2), and so on.

Entry 31: Let x > 0 and n² < 1. (Or |x| > 1 and n² is real.)

Corollary: As n → 0, then,

For integer x, just like for Entry 29, this involves two kinds of constants: 1) Catalan's constant C for even x, and 2) π for odd x > 1. Specifically, f(2) = 2-2C, f(3) = (12-π²)/24, and so on.

Entry 30: Let x > 0 and n² is real.

Corollary: As n → 0, then,

For positive integer x, this involves only one constant, namely f(1) = π²/12, f(2) = (-8+π²)/4, etc. The next one involves the square of the denominators of the previous entry.

Entry 38: Let |x| > 1 and n² is real. Define u = x²-n², then,

where the integer sequence P(m) = {1, 5, 13, 25,…} is the same as in Entry 40, or the sum of two consecutive squares, P(m) = m² + (m+1)².

Entry 32i & 32ii: If x > 0,

VII. Gamma Functions

To recall, the gamma function Γ(n) is an extension of the factorial n! to complex and real number arguments n.

Entry 25: If |x| > 1 and n² is real. (Or x > 0 and n² < 1.)

Corollary: If n = 0, then,

and for x an odd number, f(x) is a multiple of π, with the first x = 1 giving the well-known continued fraction,

Amazingly, there is also an expression for the square of entry 25,

Entry 26: If |x| > 1 and n² is real.

The next three entries are one of Ramanujan’s more fascinating results on gamma functions.

Entry 33: Given the (2² = 4) sign changes of,

Let p be the product of the gamma functions where the argument has an even number of minus signs, and q for an odd number of minus signs, namely,

p = Γ((x+a+b+1)/2) Γ((x-a-b+1)/2)

q = Γ((x-a+b+1)/2) Γ((x+a-b+1)/2)

then,

Entry 35: Given the (2³ = 8) sign changes of,

in contrast to Entry 33, let p be the product of the gamma functions where the argument has an odd number of minus signs, and q for an even number of minus signs. Suppose either a,b,c is a positive integer, then,

where,

u₁ = x²-a²-b²-c²+1

u₂ = x²-a²-b²-c²+5

u₃ = x²-a²-b²-c²+13

u₄ = x²-a²-b²-c²+25

and so on, with the sequence {1, 5, 13, 25,…) as P(m) = m² + (m+1)².

Entry 40: Given the (2⁴ = 16) sign changes of,

Like the previous entry, let p be the product of the gamma functions where the argument has an odd number of minus signs, and q for an even number of minus signs. Suppose at least one of {a,b,c,d} is a positive integer. Then,

where,

u₁ = 2(a⁴+b⁴+c⁴+d⁴+x⁴+1) - (a²+b²+c²+d²+x²-1)² - 2²

u₂ = 2(a⁴+b⁴+c⁴+d⁴+x⁴+1) - (a²+b²+c²+d²+x²-5)² - 6²

u₃ = 2(a⁴+b⁴+c⁴+d⁴+x⁴+1) - (a²+b²+c²+d²+x²-13)² - 14²

u₄ = 2(a⁴+b⁴+c⁴+d⁴+x⁴+1) - (a²+b²+c²+d²+x²-25)² - 26²

with the same sequence {1, 5, 13, 25,…} as P(m) = m² + (m+1)².

Incredible, isn’t it? As Berndt points out, Entry 40 is one of the crowning achievements of Ramanujan’s work on cfracs. There is only one proof for (by G. N. Watson, 1935) using certain assumptions. How Ramanujan knew in the first place that such an unusual continued fraction existed, or if it can be generalized further using 2ⁿ variables, is unknown.

VIII. Generalized Hypergeometric Functions

Define the ff functions,

where (p)_k is a Pochhammer symbol. One can see that the first two are just degenerate cases of the hypergeometric function ₂F₁ (which, historically, was the first one to be studied). For certain {a,b,c}, Gauss’ continued fraction involve the ratios of these functions. With sometimes different {a,b,c}, Ramanujan also gave cfracs with a simpler form,

Entry 19:

Entry 21 (Corollary 1):

Entry 22 (Limited case): If |x| < 1,

These are just particular cases of the more general entries which involve ratios. In this limited form, one can more easily see the “family resemblance” of the three. For Entry 21, the case n = x simplifies the cfrac, and an alternative form is,

Entry 42: If x > 0,

For x a positive integer, the function evaluates to an expression involving the xth power of e and a rational number,

In fact, for this case, the summation above and the cfrac are equivalent, as a positive integer x truncates the latter.

IX. Integrals

Ramanujan gave several cfracs that evaluate to an integral (one of which was given in Section III). Another is,

Entry 44: For x > 0,

For x = 1, this evaluates to the Gomperz constant. Also, compare to the similar,

which is derivable from Euler's cfrac, and perhaps may not be surprising since the log function satisfies the integral,

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