Hypergeometric modular equations

We record $$ \binom{42}2+\binom{23}2+\binom{13}2=1192 $$ functional identities that, apart from being amazingly amusing by themselves, find applications in derivation of Ramanujan-type formulas for $1/\pi$ and in computation of mathematical constants.


Introduction and statement of results
One of the best-known results of Ramanujan is his collection of series for 1/π in [23].One of the memorable achievements of Jonathan and Peter Borwein was proving the entries in Ramanujan's collection around 70 years later [7].As a representative example, we quote the series [23,Equation (29)] Here and below we use for the Pochhammer symbol (or shifted factorial) as well as the related notation for the generalized hypergeometric function.
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/),which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.c 2018 Australian Mathematical Publishing Association Inc.
A more recent investigation on interrelationships among Ramanujan's series in the works of several authors [4,11,20,24,27,28] suggests considering transformation formulas of the type 1, 1 ; 64x 3 (1 − x) (9 − 8x) 3 . (1.6) We should point out that algebraic transformations of hypergeometric functions, in particular, of modular origin, are related to the monodromy of the underlying linear differential equations.This is a reasonably popular topic, with Goursat's original 139-page contribution [18] as the starting point.See [2,21,25] for some recent developments.
The goal of this work is to systematically organize and classify identities of the type (1.3).In particular, we will show that the functions that occur in (1.4)- (1.6) are part of a single result that asserts that 42 functions are equal.Our results also encapsulate identities such as S. Cooper and W. Zudilin [3] where that is, f 6b (x) is the generating function for sums of squares of trinomial coefficients, while f 6c (x) is the generating function for sums of cubes of binomial coefficients.Results for Apéry, Domb and Almkvist-Zudilin numbers, as well as sums of the fourth powers of binomial coefficients, will also appear as special cases of our results.
Our results also include transformation formulas such as 1, 1 ; −64x 5 1 + x 1 − 4x that are part of a 13-function identity.This work is organized as follows.In the next section, some sequences and their generating functions are defined.
The main results are stated in Sections 3-5.Each section consists of a single theorem that asserts that a large number of functions are equal.
Short proofs, using differential equations, are given in Section 6. Alternative proofs using modular forms, that help put the results into context, are given in Section 7.
Several special cases are elucidated in Section 8.An application to Ramanujan's series for 1/π, using some of Aycock's ideas, is given in Section 9.

Definitions and background information
The series in (1.2) that defines the hypergeometric functions 2 F 1 and 3 F 2 converges for |x| < 1. Clausen's identity [3, page 116 It may be combined with the quadratic transformation formula [3, page 125] https://doi.org/10.1017/S144678871800037XPublished online by Cambridge University Press [4] Hypergeometric modular equations 341 We will be interested in the special case 2a = s, 2b = 1 − s, that is, where s assumes one of the values 1/6, 1/4, 1/3 or 1/2.For cosmetic reasons we introduce the following nicknames for these special instances of hypergeometric functions: where = s = 1, 2, 3, 4 and C s = 432, 64, 27, 16 for s = 1/6, 1/4, 1/3, 1/2, respectively.The arithmetic normalization constants C s are introduced in such a way that the series f (x) and F (x) all belong to the ring Z[[x]]; for example, The convergence domains of the series for f (x) and F (x) are then |x| < 1/C s and |x| ≤ 1/(4C s ), respectively.
Our further examples of the series from Z[[x]] are generating functions of so-called Apéry-like sequences.Let α, β and γ be fixed and consider the recurrence relations We assume that n is a nonnegative integer in each recurrence relation and use the single initial condition t(0) = T (0) = 1 to start each sequence.Define the generating functions It is known [2,10] that (2.4) When γ = 0, the first equality in (2.4) reduces to (2.1).
Formulas for the coefficients t(n) and T (n) that involve sums of binomial coefficients are known in the four special cases defined above, and these are listed in Table 1.The entries for t(n) come from a list that is originally due to Zagier [26,Section 4].It may also be mentioned that the numbers T (n) in the cases (α, β, γ) = (−17, −6, −72), (10, 3, −9) and (7, 2, 8) are called the Apéry numbers, Domb numbers and Almkvist-Zudilin numbers, respectively.Finally, let H(x) be defined by In our results below we use the labels 'Level 1', 'Level 2' etc to distinguish the appearance of different generating functions; the function H(x) is labeled 'Level 10' while the other labels can be extracted from the subscripts of the corresponding functions.Levels themselves, in particular, their origins and meaning, are discussed further in Section 7 in the context of modular forms.

Results: part 1
Our first meta-identity is the subject of the following theorem.Theorem 3.1.The following 42 functions are equal in a neighborhood of p = 0: (3.8) Level 2:

Results: part 2
The results in this section are consequences of the results in the previous section by taking square roots.For example, by (2.1) we find that the 3 F 2 hypergeometric function in (3.9) is related to the 2 F 1 function by The argument in the 2 F 1 function is obtained by noting that the solution of that satisfies x = 0 when p = 0 is given by By applying (4.1) and taking the square root of the expression in (3.9), we get an expression that involves the 2 F 1 function.This can be done, in principle, for all of the hypergeometric expressions in (3.1)- (3.24).In a similar way, the identity (2.4) can be used to take square roots of the expressions in (3.25)-(3.42).After taking square roots, the arguments of the resulting functions are not always rational functions of p as they are for the example (4.2) above; sometimes they are algebraic functions of p involving square roots that arise from solving quadratic equations.In Theorem 4.1 below, we list all of the functions obtained by taking square roots of the expressions in (3.1)-(3.42)for which the arguments are rational functions of p.The identities have been numbered so that the formula (4.x) in Theorem 4.1 below is the square root of the corresponding expression (3.x) in Theorem 3.1 above.Theorem 4.1.The following 23 functions are equal in a neighborhood of p = 0: (4.14) Level 3: (4.18) Level 4:

Results: part 3
The results in this section involve transformations of degrees 2, 5 and 10.
Theorem 5.1.The following 13 functions are equal in a neighborhood of p = 0: 3  (5.1) 3  (5.2) 3  (5.3) (5.4) Level 2: 4  (5.5) (5.6) Level 5, functions F: 2  (5.9) (5.10) Level 5, functions G: (5.12) Level 10, function H: (5.13) 6. Proofs of Theorems 3.1, 4.1 and 5.1 In this section, we will provide proofs of Theorems 3.1, 4.1 and 5.1.We begin with a proof of Theorem 4.1 because it is the simplest and therefore the explicit details are the easiest to write down.By changing variables, it can be shown that each function in (4.9)-(4.22)satisfies the differential equation In a similar way, the recurrence relation (2.3) implies that each of the functions f 6a , f 6b and f 6c satisfies a second-order linear differential equation of the form It may be emphasized that the proofs outlined above are conceptually simple, establish the correctness of the results and require very little mathematical knowledge.However, the proofs above are not illuminating: they give no insight into the structure of the identities, nor any reason for why the results exist or any clue as to how the identities were discovered.In the next section, the theory of modular forms will be used to give alternative proofs that also provide an explanation for how the identities were discovered.
In practice, the easiest way to show that each of the functions in (4.9)-(4.33)satisfies the differential equation (6.1) is to expand each function as a power series about p = 0 to a large number of terms and then use computer algebra to determine the differential equation.For example, using Maple, the differential equation satisfied by the function in (4.9) can be determined from the first 50 terms in the series expansion in powers of p by entering the commands which is equivalent to (6.1).The term ogf in the output stands for ordinary generating function.

Modular origins
We will now explain how the identities in Theorems 3.1, 4.1 and 5.1 were found.The explanation also puts the formulas into context and reveals why they exist.
https://doi.org/10.1017/S144678871800037XPublished online by Cambridge University Press A modular explanation for Theorem 3.1 requires the theory of modular forms for level 12 as developed in [17].Some of the details may be summarized as follows.Let h, p and z be defined by Ramanujan's Eisenstein series P and Q are defined by Dedekind's eta-function is defined for Im τ > 0 and q = exp(2πiτ) by We will require the following three lemmas, extracted from the literature.Lemma 7.1.Suppose that k and m are positive divisors of 12 and k > m.Then there are rational functions r k,m (h) such that kP(q k ) − mP(q m ) = z × r k,m (h).
Furthermore, there are rational functions s m (h) and t m (h) such that Q(q m ) = z 2 × s m (h) and η 24 (mτ) = z 6 × t m (h).Explicit formulas for the rational functions r k,m (h), s m (h) and t m (h) can be determined from the information in [17].Some examples will be given below, after Lemma 7.3.Lemma 7.2.For ∈ {1, 2, 3, 4}, let Z be defined by Then the following parameterizations of hypergeometric functions hold: where, as usual, q = exp(2πiτ) and the hypergeometric functions are as in (2.2).
https://doi.org/10.1017/S144678871800037XPublished online by Cambridge University Press Proof.These results may be found in [6,8,13] or they can be proved by putting together identities in those references and applying the special case of Clausen's identity given by (2.1).
The parameter in Lemma 7.2 is called the level.The next lemma gives the analogous results for level 6.
Lemma 7.3.The following parameterizations hold: Proof.This is explained in Zagier's work [26].Our functions f 6b and f 6c correspond to the functions f (z) in [26] in the cases C and A, respectively.The function in [26] in the cases C and A, respectively.The function f 6a corresponds to the function f (z) in [26] in the case F but with −q in place of q.
We are now ready to explain how the 42 functions in Theorem 3.1 arise and why they are equal.We will only focus on the level-3 functions in (3.15)-(3.18)as an illustration; the other functions can be obtained by a similar procedure by working with the other levels.
Proof of Theorem 3.1 using modular forms.By Lemma 7.1 and the explicit formulas in [17,Theorem 4.2], .
Substituting these in the result for Z 3 (q) in Lemma 7.2 gives 6  .
Proof.The recurrence relation for the coefficients can be deduced immediately by substituting the series expansion into the differential equations.This is a routine procedure, so we omit the details.
By the 'proof of Theorem 3.1 using modular forms' detailed above, and especially (7.2)-(7.4)and (7.8), all 42 functions in Theorem 3.1 are different expressions for Y.
By the change of variable p = h/(1 + h 2 ) and the formulas in [17], .
By Clausen's formula, the functions y in Theorem 4.1 are related to the functions Y in Theorem 3.1 by Y = y 2 and it follows that Moreover, by the formulas in [17], .
Finally, y satisfies the required differential equation with respect to p, by ( Theorem 5.1 can be proved in a similar way using the level-10 function (1 − q 10 j−9 )(1 − q 10 j−8 )(1 − q 10 j−2 )(1 − q 10 j−1 ) (1 − q 10 j−7 )(1 − q 10 j−6 )(1 − q 10 j−4 )(1 − q 10 j−3 ) and letting and then using the properties developed in [14] and [15] along with the results for level-5 modular forms that are summarized in [9].We omit most of the details, as they are similar to the proof of Theorem 3.1 given above.It is worth recording the modular parameterization.Proof.By (7.9) and [14, Theorem 3.5], Next, starting with (5.5) and using the formula for Z 2 in Lemma 7.2, Now use (7.9) to write p in terms of k and then use [14,Theorem 3.5] to express the resulting rational function of k in terms of eta-functions to get The proof may be completed by substituting the result of (7.10) into (7.11).
Alternative proofs of the identities in Example 8.5 may be given by taking Example 8.6.Part of this example was mentioned in the identity (1.7) as part of the introduction: Proof.Each of the three sets of identities can be proved by taking respectively, in (4.25), (4.28) and (4.31).
Example 8.7.Here we give representations for the functions in the previous example in terms of hypergeometric functions: Proof.The first set of identities may be proved by taking in (4.9), (4.11) and (4.28).To prove the second set of identities, take x = p(1 − 2p) in (4.15), (4.16) and (4.31).

Applications
In this section, we will show how the transformation formulas in Theorem 3.1 can be used to establish the equivalence of several of Ramanujan's series for 1/π.In the remainder of this section, we will use the binomial representation (2.2) of the related hypergeometric functions, so that the resulting formulas will be consistent with the data in [9, Tables 3-6].In order to prove Theorem 9.1, it will be convenient to make use of the following simple lemma.Lemma 9.2.Let x, y and r be analytic functions of a complex variable p and suppose that x(0) = y(0) = 0 and r(0) = 1.Suppose that a transformation formula of the form Then compute the required derivatives, let p have the same value as above and put λ = 3/40 in (9.5).We omit the details, as they are similar to the above.

Theorem 9 . 1 .
The following series identities are equivalent in the sense that any one can be obtained from the others by using the transformation formulas in Theorem 3.1:

Table 1 .
Solutions to recurrence relations.