Paramanand's Math Notes

Elementary Approach to Modular Equations: Ramanujan’s Theory 4

paramanands — Sun, 15 Jan 2012 10:30:26 +0000

Lambert Series

In this post we will focus our attention on series of the form:

which are more popularly known as Lambert Series. We will not deal with the general theorems concerning such series but will restrict ourselves to the Lambert series for the theta functions and study some identities involving these series.

To understand their origin let’s start with the following infinite product:

so that we arrive at

Taking logarithms we get

and a differentiation with respect to yields

Multiplying by we get

This represents a non-obvious identity between two Lambert series.

Theta Functions and Their Lambert series

Let’s recall the Fourier series for the elliptic functions:

where . As we shall see in this post these Fourier series become the basis of many identities involving Lambert series for theta functions.

Putting in the last equation we see that

Putting in the series for we get

The series for is bit complicated. We divide both sides by and take limits as to get

To summarize

i.e.

From the last equation we obtain

Again from equation

Hence we get

so that

Again starting with the Lambert series for we get

In the above manipulations of the Lambert series we were expressing the series as a double series and interchanging the order of summation. There are other general procedures to rearrange a double series one of which is called the Clausen’s procedure. This is given by the following formula:

m}^{\infty}(a_{mn} + a_{nm})\right)\,\,\,\,\cdots\,(10)' title='\displaystyle \sum_{m = 0}^{\infty}\sum_{n = 0}^{\infty}a_{mn} = \sum_{m = 0}^{\infty}\left(a_{mm} + \sum_{n > m}^{\infty}(a_{mn} + a_{nm})\right)\,\,\,\,\cdots\,(10)' class='latex' />

Applying this technique to the series for we get

n}^{\infty}q^{m(4n + 1)}\right) - \sum_{n = 0}^{\infty}\left(q^{(2n + 1)(2n + 2)} + 2q^{3n + 2}\sum_{m > n}^{\infty}q^{m(4n + 3)}\right)\\ &= \sum_{n = 0}^{\infty}\left(q^{2n(2n + 1)} + 2q^{n}\cdot\frac{q^{(n + 1)(4n + 1)}}{1 - q^{4n + 1}}\right) - \sum_{n = 0}^{\infty}\left(q^{(2n + 1)(2n + 2)} + 2q^{3n + 2}\cdot\frac{q^{(n + 1)(4n + 3)}}{1 - q^{4n + 3}}\right)\\ &= \sum_{n = 0}^{\infty}q^{2n(2n + 1)}\cdot\frac{1 + q^{4n + 1}}{1 - q^{4n + 1}} - \sum_{n = 0}^{\infty}q^{(2n + 1)(2n + 2)}\cdot\frac{1 + q^{4n + 3}}{1 - q^{4n + 3}}\end{aligned}' title='\displaystyle \begin{aligned}\psi^{2}(q) &= \sum_{n = 0}^{\infty}\frac{q^{n}}{1 - q^{4n + 1}} - \sum_{n = 0}^{\infty}\frac{q^{3n + 2}}{1 - q^{4n + 3}}\\ &= \sum_{n = 0}^{\infty}q^{n}\sum_{m = 0}^{\infty}q^{m(4n + 1)} - \sum_{n = 0}^{\infty}q^{3n + 2}\sum_{m = 0}^{\infty}q^{m(4n + 3)}\\ &= \sum_{n = 0}^{\infty}\sum_{m = 0}^{\infty}q^{n + m(4n + 1)} - \sum_{n = 0}^{\infty}\sum_{m = 0}^{\infty}q^{3n + 2 + m(4n + 3)}\\ &= \sum_{n = 0}^{\infty}\left(q^{2n(2n + 1)} + 2q^{n}\sum_{m > n}^{\infty}q^{m(4n + 1)}\right) - \sum_{n = 0}^{\infty}\left(q^{(2n + 1)(2n + 2)} + 2q^{3n + 2}\sum_{m > n}^{\infty}q^{m(4n + 3)}\right)\\ &= \sum_{n = 0}^{\infty}\left(q^{2n(2n + 1)} + 2q^{n}\cdot\frac{q^{(n + 1)(4n + 1)}}{1 - q^{4n + 1}}\right) - \sum_{n = 0}^{\infty}\left(q^{(2n + 1)(2n + 2)} + 2q^{3n + 2}\cdot\frac{q^{(n + 1)(4n + 3)}}{1 - q^{4n + 3}}\right)\\ &= \sum_{n = 0}^{\infty}q^{2n(2n + 1)}\cdot\frac{1 + q^{4n + 1}}{1 - q^{4n + 1}} - \sum_{n = 0}^{\infty}q^{(2n + 1)(2n + 2)}\cdot\frac{1 + q^{4n + 3}}{1 - q^{4n + 3}}\end{aligned}' class='latex' />

We can look back at the Fourier series of to derive further identities:

Replacing by and switching to Ramanujan’s function we get

Replacing with we get

Multiplying the last two equations we get

It is a surprise now that the RHS is indeed independent of . We note that the functions for are orthogonal on the interval of and hence we can multiply the series on the right and integrate the whole equation term by term with respect to on the interval to get the following:

On replacing by we get

Again putting in equation we get

From equation we get

Since this post has already grown quite long, it is time to conclude. By now the reader would have got a flavor of the identities relating theta functions to their Lambert series. In the next post we will continue our journey by establishing more identities of the similar form. Some of them would later be used to derive modular equations in the manner of Ramanujan.

Elementary Approach to Modular Equations: Ramanujan’s Theory 3

paramanands — Mon, 26 Dec 2011 12:19:11 +0000

Connection between Theta Functions and Hypergeometric Functions

Let’s recall the Gauss Transformation formula from an earlier post:

where is the hypergeometric function . Putting we get

If we set

where is one of the theta functions defined by Ramanujan (see this post) so that

and

Therefore we finally obtain:

The beauty of this formula is that it can be applied recursively. The hypergeometric function on the right is the same as that on the left except that the parameter is replaced by . Also the first factor on the right consists of parameter in numerator and in denominator so that the recursion can be utilized quite nicely to lead us to the following result:

When and therefore so that we obtain

We can use the Gauss transformation formula again by setting

and noting that

and

Thus we arrive at

By applying recursion we get

Dividing by and replacing by we get

Mutiplying by and taking exponentials on both sides we get

where is a positive integral power of and represents the Ramanujan’s function defined in previous post.

The same equation on taking reciprocals leads us to

The Inversion Fomula

Using the above result we can prove the fundamental inversion formula

Clearly if we set

then as and hence . Clearly then we have

and hence

Then we have

And thus we have and the inversion formula is established. In other words if

then

Using in we get

where

Using the Ramnujan’s notation we now have

Transformation Formula for

If we have a look at the definitions of and then we can see that

and

so that

Let be positive numbers such that and let so that . Then we have

In the classical notation this is

(see this post)

and therefore the above constitutes a proof of the transformation formula for theta functions without the use of Poisson Summation formula.

In the next post we will have a look various identities relating the theta functions of and and these will be finally transcribed in the form of modular equations. Most of the identities will be derived using Lambert series for the theta functions.

Elementary Approach to Modular Equations: Ramanujan’s Theory 2

paramanands — Sun, 18 Dec 2011 07:38:21 +0000

Ramanujan’s Theory of Elliptic Functions

Ramanujan used the letter in place of and studied the function in great detail and developed his theory of elliptic integrals and functions. Following Ramanujan let’s define

For Ramanujan elliptic function theory consisted of finding relations among . As can be easily seen from the above definitions corresponds to and corresponds to so that the nome is given by .

Ramanujan in fact studied the function and obtained the fundamental formulas connecting . In doing so Ramanujan went far ahead of his predecessors and developed the theory of theta functions to alternative bases. Using various identities in the theory of hypergeometric functions he studied the properties of for non-integral and the associated function (which is nome)

For this reduces to the classical theory. Also the factor is actually . This is the source of the constant in the formula for nome. Apart from the classical case Ramanujan developed the theories for

We will restrict ourselves however to the classical case when .

Let us then define along the lines of Ramanujan the function by

Behavior of Near

We first establish the important result that tends to positive constant value as . In other words we wish to prove that where is some positive constant. This is equivalent to proving

Since the denominator on the left tends to as it is sufficient to prove that

where is some constant. Replacing by we see that we need to prove

when . In the language of elliptic integrals this means that

Replacing by we see that

where or in other words

We will now establish that so that and

We proceed by starting with the definition of :

If we denote the integral on the right as we can see that by uniformity arguments as . And value of is easily calculated as:

Therefore we have the following relation

It now follows that

or in a grander form

This is the key result which is used in Ramanujan’s elliptic function theory. In the next post we use this result and the transformation fomulas of hypergeometric functions to develop Ramanujan’s theory.

Elementary Approach to Modular Equations: Ramanujan’s Theory 1

paramanands — Sun, 06 Nov 2011 07:49:23 +0000

Ramanujan developed his theory of modular equations using the theory of theta functions independently of Jacobi. A complete understanding of his approach is unfortunately not possible till now because he did not publish something like Fundamenta Nova containing detailed explanations of his approach. What we have today is his Notebooks edited by Bruce C. Berndt and his Collected Papers. His Notebooks are just statements of various mathematical formulas without any proof. A large part of these notebooks is concerned with modular equations and modern authors have not been able to discern his methods fully. Hence I will not be able to present a true picture of his approach. Rather I will try to present whatever I understand from his Collected Papers and his Notebooks and only focus on the elementary aspects.

Ramanujan’s Theta Functions

Ramanujan started his theory of modular equations from the relation and worked with the parameter . The modulus can be expressed in form of theta functions of parameter . Thus we have . From the equation we immediately see that . Thus an algebraic relation between and is actually a relation between theta functions of parameter and . Ramanujan was such an expert in formal manipulations of series, products and continued fractions that he readily obtained many relations between theta functions of and and then transformed them into the traditional relation between . Thus his modular equations are quite diverse in nature and sometimes they also contain more than 2 moduli.

Following Ramanujan we define the the function by

for all complex numbers with . This function has the following elementary properties

where is an integer

The first property is quite obvious and follows from the observation that changing the index of summation in definition of from to leads to exchange of and .

For the second property we have

Now so that and are of opposite signs and hence the sum cancels to zero.

To handle we have

Splitting the sum on right into even and odd indices we get

The last equation can be handled as follows:

If we put we obtain which gives the link between Ramanujan’s theta function and the classical Jacobi’s theta function.

The Jacobi’s Triple Product identity is given by

and this transforms to the following when we put

where we have the notation

Next Ramanujan defines his other theta functions in terms of :

Clearly from the definitions of and Jacobi’s Triple Product identity we can see that

Ramanujan changes one of the factors in a form so that the product formula for looks more symmetrical. Thus

Again we have

Again the series expansion for is obvious and by the triple product identity

Finally

Also it should be noted that the function is the equivalent of classical and we have . The functions and will be used later in the definitions Ramanujan’s invariants.

Theta Function Identities

The above formulas presented above provide the series as well as product expansions of these functions and it turns out that their product expansions are quite helpful in deriving an amazing number of results. Ramanujan uses this idea to the fullest extent as is shown by the following results:

To establish the first of these we can start with and therefore

and

Similarly we can establish that is also equal to and thereby the first result is established. The second set of identities is established in the following fashion

Since

it follows that

To get the series expansion requires some effort. We note the series and product expansions of given by

Dividing by and taking limits when we get

Replacing by we get

For the 3rd result we can see that

Since we have it follows that and therefore

Putting pieces together we get

Also note that the relation used above can be expressed as .

Finally

What we see in the proofs of the above results is that they are really elementary and depend upon formal manipulation of series and products. The only non-trivial identity we have used till now is the Jacobi’s Triple product which forms the basis of all the above product expansions and the relations between these Ramanujan theta functions.

Next we establish some identities connecting theta functions of and . These can be seen as equivalents of the classical theta function identities established here.

The first two identities follow by using the series expansions of and . The next one is handled as follows:

Similarly we can prove the next two identities. Next one follows from the application of the first and second:

The second last identity requires us to analyze the series expansions:

Clearly the terms in both sums cancel if is odd and add up when is even. Again if is even i.e then must be of same parity and hence . In that case . Thus we have finally

The last identity is obtained from the earlier ones as follows:

After these preliminary identities on theta functions it is time to relate them to the elliptic integral and the modulus . Ramanujan did this in a very ingenious way by studying the properties of hypergeometric function . This will be presented in the next post.

Elementary Approach to Modular Equations: Jacobi’s Transformation Theory 5

paramanands — Thu, 03 Nov 2011 05:47:30 +0000

Jacobi’s Second Real Transformation

The second transformation is obtained by taking so that . On the face of it the transformation thus involves imaginary quantities, but will be shown later to be a real transformation only. In the case of this transformation we will use in place of and in place of . Also we will keep the factor with the multiplier . We thus obtain the following by putting in the general formulas in :

Clearly the above formulas can be simplified by using the Jacobi’ imaginary transformations to get rid of the imaginary unit and thus we obtain:

and similar formulas for .

From the last equation we can see that the least positive value of which the right side vanishes is and the left side vanishes for and finally we get the relation .

Jacobi’s Second Complementary Transformation

From the first two equations above we can see that

Replacing by we get

We can see that the factors on the right side of the above equation which are independent of must lead to the value (as can be verified by taking limits as ). Therefore we have

From the above we can see that the least positive value of for which the right side vanishes is and the left side vanishes for so that we get .

From the equations and we get .

Finally from the Jacobi’s first and second real transformations (and their complementary forms) we see that corresponding to a there are two unique moduli such that such that

Moreover the relation between is algebraic and relation between is also algebraic. If we analyze the forms of the Jacobi’s transformations, we will see that the first complementary tranformation is analogous to the second transformation and second complementary transformation is analogous to the first transformation. From these considerations it follows that the the relation between is same as that between and the relation among is same as that among .

Combining both the first and second transformations we can obtain the multiplication formulas for elliptic functions. For example, since the relation between is same as that between we can replace by in the second transformation and let denote the corresponding multiplier. Then we have so that . The formula is now given by

Replacing by we get

Using the first Jacobi transformation we can replace the functions of modulus on the right side with functions of modulus and thereby we obtain the multiplication formula for the elliptic function . We can obtain the multiplication formula in another way by using the first transformation to switch from modulus to a larger modulus and then using the second transformation to switch from modulus to the smaller modulus .

To summarize Jacobi’s transformations:

Given a modulus and a positive prime we have two unique moduli such that and

Such a series of moduli may be called an ascending series of order . It follows that is also an ascending series of order . The relation between and is algebraic in nature.

Jacobi’s first transformation allows us to express elliptic functions of a given modulus in the form of elliptic functions of a greater modulus. Its complementary transformation provides a similar relation but in form of complementary moduli and therefore helps us to express an elliptic function of a given modulus in terms of elliptic functions of a smaller modulus.

Jacobi’s second transformation allows us to express elliptic functions of a given modulus in the form of elliptic functions of a smaller modulus. Its complementary transformation provides a similar relation but in form of complementary moduli and therefore helps us to express an elliptic function of a given modulus in terms of elliptic functions of a greater modulus.

From these considerations it is seen that the Jacobi’s first complementary transformation is analogous to Jacobi’s second transformation and Jacobi’s second complementary transformation is analogous to Jacobi’s first transformation.

Together both the Jacobi’s transformations allow us express the multiplication formulas for elliptic functions either by first switching to a higher modulus and then back to the original modulus or by first switching to a lower modulus and back to original modulus.

In terms of the multiplier, we see that the first transformation gives , the first complementary transformation gives , the second transformation gives , the second complementary transformation gives .

The above relations also hold when is not prime because then the transformations can be taken as composition of the transformations corresponding to prime factors of .

With these remarks we complete our presentation of the theory of transformation as developed by Jacobi in his masterpiece Fundamenta Nova. From the next post Ramanujan will take on the stage.

Elementary Approach to Modular Equations: Jacobi’s Transformation Theory 4

paramanands — Sun, 30 Oct 2011 07:45:54 +0000

Transformation of Elliptic Functions

The relation

and other variants of it

as described in previous post lead to the differential equation

when

and

Putting in the differential equation we see that . Hence the relation between and in effect expresses in terms of elliptic functions of a different but related modulus .

Thus we obtain

Using the expressions of it is easy to see that

In the last relation if we put so that we get the expression for as

and so

Putting in the expressions for we get the following results:

The above formulas remain unchanged if is replaced by and hence in the above formulas we can keep

Now the values of are the same as the values except for the order of terms and sign of each term. It is best to check this by putting some actual odd value of (like 3, 5 etc). But upon squaring these values we see that the problem of sign no longer exists and therefore in each of the above relations we can actually replace by (by putting and and noting that . Thus we arrive at the following:

In the development of these formulas so far we have assumed that where are relatively prime to each other, but we have not given any specific values to . It turns that by giving different values to and we arrive at different transformations. Out of all these transformations two are real i.e. the relation between and is expressed using real numbers as coefficients. We are going to study these two transformations next.

Jacobi’s First Real Transformation

By choosing we get which is a real number and from this we instantly see that all the numbers involved in the formulas mentioned above are real numbers. The corresponding formulas are as follows:

It is easy to see that the least positive value of for which the right side of vanishes is and the left side of this equation vanishes when . Equating these two values of we get the relation .

Jacobi’s First Complementary Transformation

Next we need to recast the above formulas in another form by using the identities:

On applying these identities to the formulas and using to replace we get

and using equation we get

and similarly

Dividing the equation by equation we get

Replacing by in the above equation and noting that we get

The factor on the right side which is independent of must be as can be easily seen when we divide both sides by and take limits as .

Thus we finally obtain

The only factor on the right side which can vanish is and the smallest value for which this vanishes is . The left side vanishes when and therefore upon equating these two values of we get .

We have thus derived the two relations and from which we obtain the fundamental relation . From this it also follows that the value of occurring in these equations is actually less than .

In the next post we will describe one more transformation which is also real and it leads to a value of which is greater than (which will be denoted by ) and in this case .

Elementary Approach to Modular Equations: Jacobi’s Transformation Theory 3

paramanands — Sat, 29 Oct 2011 08:59:15 +0000

Analytic Approach to Transformation Theory

Jacobi understood that the algebraic approach for obtaining modular equations could not be applied easily in case of higher degrees. Hence he followed an analytic approach. The idea he used was to express the relation in a form where each of and appears as a product of various factors. Essentially he examined the roots of and expressed them in form of a product where each factor corresponds to a given root. This approach was very useful for Jacobi as he used this relation to finally develop the theory of Theta Functions and their relation to elliptic functions. In fact the entire Fundamenta Nova is split into two sections, the first section dealing with transformation theory and the second section dealing with the expansion of elliptic functions into infinite series and products (which is basically the theory of theta functions).

The presentation here is based on the Calyley’s An Elementary Treatise on Elliptic Functions and our goal here will be to establish the existence of transformation of odd prime degree and to establish the fundamental relation which is induced by these transformations. We will not explore the development of theta functions based on this transformation theory as done in Jacobi’s or Cayley’s books.

To begin with lets fix the notation which is almost similar to that used by Jacobi/Cayley except that we use in place of and in place of . Let’s assume that and are some fixed integers relatively prime to each other not both zero and let

The transformation is given by and the resulting equations

i.e.

The polynomials are given by

and the polynomials are given by

The invariance of relation between and under the transformation is satisfied if we have

and

In order to prove that the above mentioned transformation indeed has all the properties desired takes some reasonable amount of algebraic manipulations and some analysis of the various equations involved. In what follows elliptic functions will be used heavily and therefore we will drop the modulus . If any other modulus is used for these elliptic elliptic functions it will written explicitly.

We start with the equation

and will show that this leads to all the other equations. Moreover from the form of the equations given earlier it follows that

Upon solving for we will obtain an expression , where are homogeneous polynomials of degree (this is because change their sign together and thus is a rational and even function).

Thus if we are able to show that vanishes for and that as then it will be obvious that has to be of the form:

Also from the relation between it is clear that when and therefore we must have

With this expression of we at once see that where are as mentioned above. Also if assume the invariance of this relation under the transformation we at once arrive at the value of described above. Again from equation we obtain

Looking at the degrees of and we can see that must be a constant multiple of and on putting we see that and so and hence it follows that and . Thus the expressions of are also verified. Using the invariance under we can then verify the expressions for .

Hence all the above expressions relating and are verified provided we establish that zeros and poles (points where a function tends to ) of are as described above. To that end we put in the expession for and obtain

Now we require some identities related to the elliptic functions which can be verified at once using addition formulas or otherwise:

and therefore

In the above relation we put successively and multiply the resulting equations and further multiply both sides by . Also while doing so we need to observe that . Thus we obtain the following:

From the above relation it is easy to see that the and hence is invariant when is replaced (in doing so each factor is changed into next factor in the product and the last factor is changed into the first so that the product itself remains invariant). If we put we have and hence . Therefore whenenver i.e. whenever . Since it follows that whenever or . Thus the zeroes of are verified.

It is easily seen that is a pole for and hence a pole for (clearly as ). Thus whenever i.e whenever and so the position of poles of are also verified.

Thus the tranformations of order described in the beginning of the post using complicated product expansions containing elliptic functions are verified. We will discuss the repercussions of these transformations in the next post.

Elementary Approach to Modular Equations: Jacobi’s Transformation Theory 2

paramanands — Thu, 27 Oct 2011 06:25:30 +0000

In this post we will apply the technique described in previous post to obtain modular equations of degree and .

Cubic Transformation

Since we have , the degrees of and are both and hence they both are constants. Since

it is clear that the function depends only on ratio and hence we can take and . Thus we have

The requirement of invariance under the transformation leads to the following conditions

and

Thus we have the following equations

and since at we have . Clearly we have then

Eliminating from the above equation is bit tricky. After some manipulations we have

Clearly the pattern is now obvious if we multiply the expressions for and (similarly for )

and thus we obtain the modular equation of degree :

Using the variable we can obtain an expression devoid of algebraic irrationalities. Clearly we have so that and since it follows that . Putting the value of in this equation we get

and thus finally

Quintic Transformation

Here we have , hence the degree of is and that of is zero. Thus we can take and (note that the degrees being talked of are degrees when are expressed as polynomials in ).

We have now

so that in this case

This relation is invariant under the transformation so we have

Like in the case of cubic transformation we have .

Obtaining a relation between by eliminating is reasonably cumbersome and it becomes manageable when we use the associated variables . Using these variables and equations (1) and (2) we have and putting these values in equation (4) we get

Again using equation (3) we get

Dividing (6) by (5) we get

Comparing this with (5) we get

and on simplifying further we have

The reader must have understood by now that this technique of finding modular equations is quite unsuitable for larger values of . In fact Jacobi treated only the cubic and quintic transformations in his Fundamenta Nova and Arthur Cayley extended this approach to in his An Elementary Treatise on Elliptic Functions. Our exposition here is based on Cayley’s book. For higher values of Jacobi provided the transformation in a form which contains elliptic functions of and thereby obtained equations from which follows the relation . This we study in the next post.

Elementary Approach to Modular Equations: Jacobi’s Transformation Theory 1

paramanands — Mon, 24 Oct 2011 09:11:39 +0000

To recapitulate the basics of elliptic integral theory (details here) we have

We have here and the complementary modulus or in other words . Also we denote and we have the Legendre’s Identity .

The Function

From the definition it is clear that is a strictly increasing function of and moreover as . Thus function maps interval to . Similarly is strictly decreasing and maps to . Therefore it follows that is a strictly decreasing function of and is a strictly increasing function of .

From the above arguments it follows that the function is a strictly decreasing function of . When , and as . Thus the function maps the interval to the interval . For any given positive number we have a unique positive such that . Let be any positive number and then is also positive and therefore there is a unique number such that . We traditionally denote . Therefore we can write

Again let’s think in the following way. Suppose a number and a positive number is given. From we can calculate and hence corresponding to which there is a unique number such that . In this fashion turns out to be a function of . Therefore the equation determines as a function of which is one-one and invertible. From the continuity of the function turns out to be continuous. To derive these results formally we need the fundamental formulas:

(for the proofs of these results read here)

Using these we have

and therefore

Thus we have finally another fundamental relation

which shows that is strictly decreasing. On differentiating the equation we get

and therefore is a strictly increasing function of and maps to . Also ) 1 \Rightarrow l > (=,<) k' title='p <(=,>) 1 \Rightarrow l > (=,<) k' class='latex' />. We also see that the ratio is a function of and we traditionally call it the multiplier and denote by . Thus we have

Also it should be noted that if ) 1 \Rightarrow M_{p}(l, k) > (=, <) 1' title='p < (=, >) 1 \Rightarrow M_{p}(l, k) > (=, <) 1' class='latex' />.

Jacobi’s Transformation Theory

Jacobi shows that the relation between and is algebraic when is rational. First of all as an example we can take . Then by Landen’s transformation we see that . Clearly here the relation between and is algebraic and is also given by . Also

and therefore

so that as should have been the case.

Next let’s understand that it is necessary only to study the case when is a positive prime. For, if for positive prime the relation is leading to an algebraic relation between and then we can show that the same kind of relation holds when is any positive rational number. First suppose is composite positive integer and where both are primes. Then can be written . Since the relation between and is algebraic by assumption, the relation between is algebraic. The same reasoning can be extended to the case when is a product of any number of primes. Thus the case of positive integers is handled. In case of where are positive integers we can write as . Since the relation between and is algebraic therefore the relation between is algebraic.

So from now on let’s assume that is a positive prime. The algebraic relation between implied by the equation is called a modular equation of degree . The way Jacobi obtained these modular equations is related to the transformation of elliptic integrals. Consider the differential

By suitable rational transformation where are polynomials it is expected to reduce the differential to the form

Thus in effect we wish to establish a relation of the form

where is a rational function of and is some constant. That such a transformation is possible is not so obvious. Jacobi started his Fundamenta Nova by describing this elegant theory of transformation and showed that such a transformation exists for every value of the positive prime which is called the order of the transformation. In what follows we shall assume that is an odd prime ( is covered by the famous Landen Transformations).

The constant is called multiplier and depends upon the values of and is related to the multiplier . To simplify the process of transformation the rational function is chosen to be of very specific form. With the requirement that vanishes with we take where are polynomials of degrees . Also are supposed to be even functions so that they are functions of and then we can write

where are homogeneous polynomials of degree . Another condition which we impose is that the relation between and does not change when is replaced by . That this is possible needs to be demonstrated. First of all we can see that

We can next choose such that where is a constant. Thus the coefficients of are same as those of , but in reverse order and multiplied by suitable powers of . It thus follows that

Replacing by we get

Multiplying the above two equations we get

Now let’s suppose that in the defining equation if we put in place of then the value of changes to . Thus

Clearly this reduces to provided .

Hence a transformation of the form with of degree and of degree with being odd function and being even function is possible which remains invariant under the change of variables .

Next we would like to have further restrictions on the form of and . Putting leads to

and hence

If it is possible that for some polynomials then we can see that

and by invariance of relation between and under the transformation we can see that we must also have the relations

Now we can see that if is a factor of then divides and hence divides . From the same logic it follows that each divide and since the degrees of and are same (equal to ) it follows that the ratio is a constant which we denote by . Thus we arrive at

The idea is now to choose polynomials in a suitable form. We keep such that are even functions of and the degree of is . In general if then the degrees of are and if , degree of is and that of is .

Thus we arrive at the following transformation

so that

and then we obtain

The actual methodology of finding the polynomials (or ) would be dealt with in next post by showing examples with and . From the theoretical investigation above it is clear that the process of finding these polynomials (i.e. their coefficients) is totally algebraical and hence the relation between and is algebraic (note that , and that itself would be an algebraic function of ). It should also be observed that the multiplier is also an algebraical function of and (calculation of is easy if we observe that ).

Elementary Approach to Modular Equations: Hypergeometric Series 2

paramanands — Sat, 22 Oct 2011 08:54:55 +0000

To continue our adventures (which started here) with the hypergeometric function we are going to establish the following identity

If is neither zero nor a negative integer and if and , then

A curious thing to observe here is that the term on the left does not change if we replace by , but the right term does change its value. If we restrict ourselves to real positive values of then we need to understand that in the right term the symbol should actually be replaced by minimum of and or we can say that the identity as it is holds only when .

Again to establish the identity we start with function which satisfies the differential equation

Putting we get

and

and after some symbolic manipulations we arrive at the following differential equation

This is clearly satisfied by and hence the identity follows. On putting we see that and therefore we obtain the formula

or in simpler and expanded form

and this is valid for .

Clausen’s Formula

It turns out we can square the above identity on both sides and get another series which is like the hypergeometric series but slightly more complex. In order to proceed further we introduce the generalized hypergeometric series defined by

The differential equation satisfied by is obtained in the similar way as done for the hypergeometric function and we have

This needs to be simplified by a laborious symbolic manipulation as follows:

and

Finally the differential equation turns out to be

Using this differential equation Thomas Clausen in 1828 found a way to express the series as a square of the usual hypergeometric function and this lead to some sufficient conditions under which a generalized hypergeometric series could be guaranteed to be positive. His result is famously called the Clausen’s Formula which is as follows:

The conditions for validity turn out to be and both should not be zero or a negative integer.

The equation satisfied by the series on the right side is

On the other hand the function satisfied the differential equation

Differentiating with respect to we get

Let’s put so that .

Now mulplying (1) by and (2) by and adding the equations we get

and finally

Thus satisfies the differential equation

This establishes the Clausen’s Formula. Immediate application of this formula in the context of elliptic integrals is obtained by putting as follows:

or in the more striking form as

and this is valid for .

This is a fundamental result which is used by Ramanujan to derive certain series for . With this we complete the background material in the theory of hypergeometric series which is needed to understand Ramanujan’s Modular Equations and Approximations to .

In the next post we will define a modular equation and also derive such equations as done by Jacobi in his Fundamenta Nova.