0019: Article 9 (More Pi Formulas)

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A Compilation of Ramanujan-Type Formulas for 1/π^m

by Tito Piezas III

I. Introduction

This is a compilation of recent developments by various authors for Ramunujan-type pi formulas of form,

for m > 1 where,

1) s_v is a finite product of Pochhammer symbols (x)_n for some rational x < 1

2) P(n) is a polynomial in n.

3) C is a rational constant

For m = 1, an infinite number of formulas are known (where C is algebraic) since there is a well-developed theory behind it. However, for m > 1, ONLY 12 are known so far (updated, Dec. 19, 2010), found by:

9 – Jésus Guillera (m = 2)

1 – Jésus Guillera, Gert Almkvist (m = 2)

1 – Boris Gourevitch (m = 3)

1 – Jim Cullen (m = 4)

Without a general guiding principle, the few examples have been found mostly by clever use of integer relations algorithms, with only four rigorously proven (by Guillera). By compiling these in one place and arranging them in a reasonable order, perhaps it can give – in one glance – an overview of what is known, and motivate others to find more of a similar kind. (In fact, four of the formulas have been discovered only in 2010. I’ll be updating this website each time there is a new one discovered.)

II. Type 1

Define the Pochhammer symbol products a_v as,

These are the basic combinations used by Ramanujan and a famous example is,

For a complete listing of the 36 formulas which use the a_v, rational C, and its associated e^π√d, refer to the article, “Pi Formulas and the Monster Group”.

III. Type 2

To generalize Ramanujan’s results, Guillera considered five Pochhammer symbols. There may be as much as eight different combinations that can be used in a pi formula. The first four are, define,

which are simply the a_v multiplied by ((1/2)_n/n!)², then,

Guillera’s early formulas (pre-2010) had a constant C that was a unit fraction (with a numerator N = 1). However, the third formula with C = (3/4)³ (from his “A New Ramanujan-Like Series for 1/π²”, 2010) proves that it need not be so, hence opens possibilities for b₄ and others. (Note: Guillera and Almkvist have searched hard for formulas using b₄, and subsequently by J. Cullen who extended the search, but they haven't found any yet.) <Update, 12/18/10>: They have also considered a variant of the general form which uses a ratio of polynomials A(n)/B(n). One by Almkvist is,

Compare it with the coeffiicients of the third formula. That is surely more than a coincidence. (Guillera has noticed a similar phenomenon between some of his formulas and Ramanujan's.) <End>

The next four Pochhammer products are, define,

then,

The third and fourth (both also found in 2010) also contains a C that is not a unit fraction. The latter, when translated into factorials, has the especially nice form,

Though not yet rigorously proven, this implies, as was discussed in “Ramanujan-Like Series for 1/π² and String Theory” (J. Guillera and Gert Almkvist, 2010), that any decimal digit of the number 1/π² can be computed without computing all the preceding decimals, similar to the Bailey-Borwein-Plouffe formula. (Note: This observation is courtesy of Wilfried Pigulla, since (1/3)(6n)!/n!⁶ is an integer for all n > 0.) The last one, b₈, involves Pochhammer symbols not used by Ramanujan and is the only one known so far.

IV. Type 3

This goes beyond five Pochhammer symbols and hence involves 1/pi^m for m > 2. Define,

then,

The first is by Boris Gourevitch (2002), while the second is by Jim Cullen (Dec, 2010). The third is a logical extension of b₈ where P(n) as a hypothetical 4th degree polynomial. (Though Almkvist has pointed out that the probability for c₃ is probably zero.)

(Update, 12/19/10): Consistent with Guillera’s observations (see Ramanujan’s Pi Formulas with a Twist), I noticed that we can transform Cullen's formula by letting,

then, for k = ½,

Almkvist simplied this by using binomial sums to find the Cullen-Almkvist formula,