A q-PRODUCT TUTORIAL FOR A q-SERIES MAPLE PACKAGE

FRANK GARVAN

Abstract. This is a tutorial for using a new q-series Maple package. The package includes facilities for conversion between q-series and q-products and finding algebraic relations between q-series. Andrews found an algorithm for converting a q-series into a product. We provide an implementation. As an application we are able to effectively find finite q-product factorisations when they exist thus answering a question of Andrews. We provide other applications involving factorisations into theta-functions and etaproducts.

Dedicated to George E. Andrews on the occasion of his 60th Birthday

Contents

1. Introduction 2 1.1. Installation instructions 2 2. Basic functions 3 2.1. Finite q-products 3 2.2. Infinite products 3 3. Product Conversion 5

3.1. prodmake  5
3.2. qfactor  6
3.3. etamake  8
3.4. jacprodmake  10

4. The Search for Relations 12

4.1. findhom  12
4.2. findhomcombo  13
4.3. findnonhom  15
4.4. findnonhomcombo  16
4.5. findpoly  17

5. Sifting coefficients 19 6. Product Identities 20

Date: Tuesday, December 15, 1998. Revised Friday, February 12, 1999. 1991 Mathematics Subject Classification. Primary: 11P81, 68Q40; Secondary:

05A17, 11F20, 33D10.

Key words and phrases. symbolic computation, products, q-series, theta functions, eta-products, product identities, partitions.

This research was supported by the NSF under grant number DMS-9870052.

6.1. The Triple Product Identity 20 6.2. The Quintuple Product Identity 21 6.3. Winquist’s Identity 23 References 25

1. Introduction

In the study of q-series one is quite often interested in identifying generating functions as infinite products. The classic example is the Rogers-Ramanujan identity

∑_{n=0}^{∞} \frac{q^{n^{2}}}{(q; q)_{n}} = ∏_{n=1}^{∞} \frac{1}{(1 − q^{5n−1})(1 − q^{5n−4})}.

Here we have used the notation in (2.2). It can be shown that the left-side of this identity is the generating function for partitions whose parts differ by at least two. The identity is equivalent to saying the number of such partitions of n is equinumerous with partitions of n into parts congruent to ±1 (mod 5).

The main goals of the package are to provide facility for handling the following problems.

1. Conversion of a given q-series into an “infinite” product.

2. Factorization of a given rational function into a finite q-product if one exists.

3. Find algebraic relations (if they exist) among the q-series in a

given list. A q-product has the form

∏_{j=1}^{N}(1 − q^{j})^{b_{j}}, (1.1)

where the b_{j} are integers.

In [4, §10.7], George Andrews also considered Problems 1 and 2, and asked for an easily accessible implementation. We provide implementations as well as considering factorisations into theta-products and eta-products. The package provides some basic functions for computing q-series expansions of eta functions, theta functions, Gaussian polynomials and q-products. It also has a function for sifting out coefficients of a q-series. It also has the basic infinite product identities: the triple product identity, the quintuple product identity and Winquist’s identity.

1.1. Installation instructions. The qseries package can be downloaded via the WWW. First use your favorite browser to access the URL:

http://www.math.ufl.edu/∼frank/qmaple.html then follow the di-

rections on that page. There are two versions: one for UNIX and one for WINDOWS.

2. Basic functions

We describe the basic functions in the package which are used to build q-series.

2.1. Finite q-products.

2.1.1. Rising q-factorial. aqprod(a,q,n) returns the product

(a; q)_{n} = (1 − a)(1 − aq) ⋯ (1 − aq^{n−1}). (2.2)

We also use the notation

(a; q)_{∞} = ∏_{n=1}^{∞}(1 − aq^{n−1}).

2.1.2. Gaussian polynomials. When 0 ≤ m ≤ n, qbin(q,m,n) returns the Gaussian polynomial (or q-binomial coefficient)

[ \begin{matrix}n \\ m \end{matrix} ]_{q} = \frac{(q)_{n}}{(q)_{m}(q)_{n−m}} ,

otherwise it returns 0.

2.2. Infinite products.

2.2.1. Dedekind eta products. Suppose ℜτ > 0, and q = exp(2πiτ). The Dedekind eta function [27, p. 121] is defined by

η(τ) = exp(πiτ/12) ∏_{n=1}^{∞}(1 − exp(2πinτ))

= q^{1/24} ∏_{n=1}^{∞}(1 − q^{n}).

etaq(q,k,T) returns the q-series expansion (up to q^{T}) of the eta product

∏_{n=1}^{∞}(1 − q^{kn}).

This corresponds to the eta function η(kτ) except for a power of q. Eta products occur frequently in the study of q-series. For example, the generating function for p(n), the number of partitions of n, can be written as

∑_{n=0}^{∞} p(n)q^{n} = \frac{1}{∏_{n=1}^{∞}(1 − q^{n})}.

See [1, pp. 3–4]. The generating function for the number of partitions of n that are p-cores [19], a_{p}(n), can be written

∑_{n=0}^{∞} a_{p}(n)q^{n} = ∏_{n=1}^{∞} \frac{(1 − q^{pn})^{p}}{(1 − q^{n})} .

Recently, Granville and Ono [21] were able to prove a long-standing conjecture in group representation theory using elementary and functiontheoretic properties of the eta product above.

2.2.2. Theta functions. Jacobi [24, Vol I, pp. 497–538] defined four theta functions θ_{i}(z, q), i = 1, 2, 3, 4. See also [42, Ch. XXI]. Each theta function can be written in terms of the others using a simple change of variables. For this reason, it is common to define

θ(z, q) := ∑_{n=−∞}^{∞} z^{n}q^{n^{2}}.

theta(z,q,T) returns the truncated theta-series

∑_{i=−T}^{T} z^{i}q^{i^{2}}.

The case z = 1 of Jacobi’s theta functions occurs quite frequently. We define

θ_{2}(q) := ∑_{n=−∞}^{∞} q^{(n+1)^{2}/2}

θ_{3}(q) := ∑_{n=−∞}^{∞} q^{n^{2}}

θ_{4}(q) := ∑_{n=−∞}^{∞}(−1)^{n}q^{n^{2}}

theta2(q,t), theta3(q,t), theta4(q,t) (resp.) returns the q-series

expansion to order O (q^{T} ) of θ_{2}(q), θ_{3}(q), θ_{4}(q) respectively. Let a, and b be positive integers and suppose |q| < 1. Infinite products of the form

(q^{a}; q^{b})_{∞}(q^{b−a}; q^{b})_{∞}

occur quite frequently in the theory of partitions and q-series. For example the right side of the Rogers-Ramanujan identity is the reciprocal of the product with (a, b) = (1, 5). In (3.4) we will see how the function jacprodmake can be used to identify such products.

3. Product Conversion

In [1, p. 233], [4, §10.7] there is a very nice and useful algorithm for converting a q-series into an infinite product. Any given q-series may be written formally as an infinite product

1 + ∑_{n=1}^{∞} b_{n}q^{n} = ∏_{n=1}^{∞}(1 − q^{n})^{−a_{n}}.

Here we assume the b_{n} are integers. By taking the logarithmic derivative of both sides we can obtain the recurrence

nb_{n} = ∑_{j=1}^{n} b_{n−j} ∑_{d|j} da_{d}.

Letting a_{n} = 1 we obtain the well-known special case

np(n) = ∑_{j=1}^{n} p(n − j)σ(j).

We can also easily construct a recurrence for the a_{n} from the recurrence above.

The function prodmake is an implementation of Andrews’ algorithm. Other related functions are etamake and jacprodmake.

3.1. prodmake. prodmake(f,q,T) converts the q-series f into an infinite product that agrees with f to O(q^{T}). Let’s take a look at the left side of the Rogers-Ramanujan identity.

> with(qseries):
> x:=1:
> for n from 1 to 8 do
        x := x + q^{∧}(n*n)/aqprod(q,q,n):
   od:
> x := series(x,q,50);

x := 1 + q + q^{2} + q^{3} + 2 q^{4} + 2 q^{5} + 3 q^{6} + 3 q^{7} + 4 q^{8} + 5 q^{9} + 6 q^{10} +7 q^{11} + 9 q^{12} + 10 q^{13} + 12 q^{14} + 14 q^{15} + 17 q^{16} + 19 q^{17} + 23 q^{18} +26 q^{19} + 31 q^{20} + 35 q^{21} + 41 q^{22} + 46 q^{23} + 54 q^{24} + 61 q^{25} + 70 q^{26} + 79 q^{27} + 91 q^{28} + 102 q^{29} + 117 q^{30} + 131 q^{31} + 149 q^{32} + 167 q^{33} + 189 q^{34} + 211 q^{35} + 239 q^{36} + 266 q^{37} + 299 q^{38} + 333 q^{39} + 374 q^{40} + 415 q^{41} + 465 q^{42} + 515 q^{43} + 575 q^{44} + 637 q^{45} + 709 q^{46}

+783 q^{47} + 871 q^{48} + 961 q^{49} + O(q^{50})

> prodmake(x,q,40);

1/ ((1 − q)(1 − q^{4})(1 − q^{6})(1 − q^{9})(1 − q^{11})(1 − q^{14})(1 − q^{16})(1 − q^{19})

(1 − q^{21})(1 − q^{24})(1 − q^{26})(1 − q^{29})(1 − q^{31})(1 − q^{34})(1 − q^{36})(1 − q^{39}))

We have rediscovered the right side of the Rogers-Ramanujan identity!

Exercise 1. Find (and prove) a product form for the q-series

∑_{n=0}^{∞} \frac{q^{n^{2}}}{(q; q)_{2n}} .

The identity you find is originally due to Rogers [34](p.330). See also Andrews [2](pp.38–39) for a list of some related papers.

3.2. qfactor. The function qfactor is a version of prodmake. qfactor(f,T) attempts to write a rational function f in q as a qproduct, i.e., as a product of terms of the form (1 − q^{i}). The second argument T is optional. It specifies an an upper bound for the exponents of q that occur in the product. If T is not specified it is given a default value of 4d+3 where d is the maximum of the degree in q of the numerator and denominator. The algorithm is quite simple. First the function is factored as usual, and then it uses prodmake to do further factorisation into q-products. Thus even if only part of the function can be written as a q-product qfactor is able to find it.

As an example we consider some rational functions T (r, h) introduced by Andrews [4](p.14) to explain Rogers’s [34] first proof of the Rogers-Ramanujan identities. The T (r, n) are defined recursively as follows:

(3.3)  T (r, 0) = 1,
(3.4)  T (r, 1) = 0,
(3.5)  T (r, N) = − ∑_{1≤2j≤N} [ \begin{matrix}r + 2j \\ j \end{matrix} ] T (r + 2j, N − 2j).

> T:=proc(r,j)
>  option remember;
>  local x,k;
>  x:=0;
>  if j=0 or j=1 then
>  RETURN((j-1)^{∧}2):
>  else
>  for k from 1 to floor(j/2) do
>  x:=x-qbin(q,k,r+2*k)*T(r+2*k,j-2*k);
>  od:
>  RETURN(expand(x));
>  fi:

QSERIES MAPLE TUTORIAL 7

> end:
> t8:=T(8,8);

t8 := 3 q^{9} + 21 q^{16} + 36 q^{25} + 9 q^{36} + q^{6} + q^{7} + 2 q^{8} + 5 q^{10} + 6 q^{11} + 9 q^{12} + 11 q^{13} + 15 q^{14} + 17 q^{15} + 33 q^{28} + 34 q^{27} + 37 q^{26}

+ 38 q^{24} + 36 q^{23} + 37 q^{22} + 34 q^{21} + 33 q^{20} + 29 q^{19} + 28 q^{18}
+ 23 q^{17} + 5 q^{38} + 6 q^{37} + 11 q^{35} + 15 q^{34} + 17 q^{33} + 21 q^{32}
+ 23 q^{31} + 28 q^{30} + 29 q^{29} + 3 q^{39} + q^{42} + q^{41} + 2 q^{40}

> factor(t8);

q^{6} (q^{4} + q^{3} + q^{2} + q + 1) (q^{4} − q^{3} + q^{2} − q + 1)

(q^{10} + q^{9} + q^{8} + q^{7} + q^{6} + q^{5} + q^{4} + q^{3} + q^{2} + q + 1) (q^{4} + 1)
(q^{6} + q^{3} + 1) (q^{8} + 1)

> qfactor(t8,20);

\frac{(1 − q^{9}) (1 − q^{10}) (1 − q^{11}) (1 − q^{16}) q^{6}}{(1 − q) (1 − q^{2}) (1 − q^{3}) (1 − q^{4})}

Observe how we used factor to factor t8 into cyclotomic polynomials. However, qfactor was able to factor t8 as a q-product. We see that

T (8, 8) = \frac{(q^{9}; q)_{3}(1 − q^{16})q^{6}}{(q; q)_{4}} .

Exercise 2. Use qfactor to factorize T (r, n) for different values of r and n. Then write T (r, n) (defined above) as a q-product for general r and n.

For our next example we examine the sum

∑_{k=−∞}^{∞}(−1)^{k}q^{k(3k+1)/2} [ \begin{matrix}b + c \\ c + k \end{matrix} ] [ \begin{matrix}c + a \\ a + k \end{matrix} ] [ \begin{matrix}a + b \\ b + k \end{matrix} ] .

> dixson:=proc(a,b,c,q)
>  local x,k,y;
>  x:=0:  y:=min(a,b,c):
>  for k from -y to y do
>  x:=x+(-1)^{∧}k*q^{∧}(k*(3*k+1)/2)*
>  qbin(q,c+k,b+c)*qbin(q,a+k,c+a)*qbin(q,b+k,a+b);
>  od:
>  RETURN(x):
> end:

> dx := expand(dixson(5,5,5,q)):
> qfactor(dx,20);

(1 − q^{6}) (1 − q^{7}) (1 − q^{8}) (1 − q^{9}) (1 − q^{10}) (1 − q^{11}) (1 − q^{12}) (1 − q^{13}) (1 − q^{14}) (1 − q^{15})/((1 − q)^{2} (1 − q^{2})^{2} (1 − q^{3})^{2} (1 − q^{4})^{2}

(1 − q^{5})^{2})

We find that

∑_{k=−∞}^{∞}(−1)^{k}q^{k(3k+1)/2} [ \begin{matrix}10 \\ 5 + k \end{matrix} ]^{3} = \frac{(q^{6}; q)_{10}}{(q; q)_{5}^{2}} . (3.6)

Exercise 3. Write the sum

∑_{k=−∞}^{∞}(−1)^{k}q^{k(3k+1)/2} [ \begin{matrix}2a \\ a + k \end{matrix} ]^{3}

as a q-product for general integral a. The identity you obtain is a special case of [4](Eq.(4.24), p.38).

3.3. etamake. Recall from (2.2.1) that etaq is the function to use for computing q-expansions of eta products. If one wants to apply the theory of modular forms to q-series it is quite useful to determine whether a given q-series is a product of eta functions. The function in the package for doing this conversion is etamake. etamake(f,q,T) will write the given q-series f as a product of eta functions which agrees with f up to q^{T}. As an example, let’s see how we can write the theta functions as eta products.

 > theta2(q,100)/q^{∧}(1/4);

2 q^{132} + 2 q^{110} + 2 q^{90} + 2 q^{72} + 2 q^{56} + 2 q^{42} + 2 q^{30} + 2 q^{20} + 2 q^{12} + 2 q^{6} + 2 q^{2} + 2 + q^{156}

> etamake(",q,100);

2 \frac{η( 4 τ )^{2}}{q^{1/4} η( 2 τ )}

> theta3(q,100);

2 q^{121} + 2 q^{100} + 2 q^{81} + 2 q^{64} + 2 q^{49} + 2 q^{36} + 2 q^{25} + 2 q^{16} + 2 q^{9} + 2 q^{4} + 2 q + 1

> etamake(",q,100);

\frac{η( 2 τ )^{5}}{η( 4 τ )^{2} η( τ )^{2}}

> theta4(q,100);

−2 q^{121} + 2 q^{100} − 2 q^{81} + 2 q^{64} − 2 q^{49} + 2 q^{36} − 2 q^{25} + 2 q^{16} − 2 q^{9} + 2 q^{4} − 2 q + 1

> etamake(",q,100);

\frac{η( τ )^{2}}{η( 2 τ )}

We are led to the well-known identities:

θ_{2}(q) = 2\frac{η(4τ)^{2}}{η(2τ)} ,
θ_{3}(q) = \frac{η(2τ)^{5}}{η(4τ)^{2}η(τ)^{2}} ,
θ_{4}(q) = \frac{η(τ)^{2}}{η(2τ)}.

The idea of the algorithm is quite simple. Given a q-series f (say with leading coefficient 1) one just keeps recursively multiplying by powers of the right eta function until the desired terms agree. For example, suppose we are given a q-series

f = 1 + cq^{k} + · · · .

Then the next step is to multiply by etaq(q,k,T)^{∧}(-c).

Exercise 4. Define the q-series

a(q) := ∑_{n=−∞}^{∞} ∑_{m=−∞}^{∞} q^{n^{2}+nm+m^{2}}

b(q) := ∑_{n=−∞}^{∞} ∑_{m=−∞}^{∞} ω^{n−m}q^{n^{2}+nm+m^{2}}

c(q) := ∑_{n=−∞}^{∞} ∑_{m=−∞}^{∞} q^{(n+1/3)^{2}+(n+1/3)(m+1/3)+(m+1/3)^{2}}

where ω = exp(2πi/3). Two of the three functions above can be written as eta products. Can you find them? Hint : It would be wise to define

    > omega := RootOf(z^{∧}2 + z + 1 = 0);

See [12] for the answer and much more.

3.4. jacprodmake. In (2.2.2) we observed that the right side of the Rogers-Ramanujan identity could be written in terms of a Jacobi product. The function jacprodmake converts a q-series into a Jacobi-type product if one exists. Given a q-series f, jacprodmake(f,q,T) attempts to convert f into a product of theta functions that agrees with f to order O(q^{T}). Each theta-function has the form JAC(a, b, ∞), where a, b are integers and 0 ≤ a < b. If 0 < a, then JAC(a, b, ∞) corresponds to the theta-product

(q^{a}; q^{b})_{∞}(q^{b−a}; q^{b})_{∞}(q^{b}; q^{b})_{∞}.

We call this a theta product because it is θ(−q^{(b−2a)/2}, q^{b/2}). The jacprodmake function is really a variant of prodmake. It involves using prodmake to compute the sequence of exponents and then searching for periodicity.

If a = 0, then JAC(0, b, ∞) corresponds to the eta-product

(q^{b}; q^{b})_{∞}.

We note that this product can also be thought of as a theta-product since it can be written

(q^{b}; q^{b})_{∞} = (q^{b}; q^{3b})_{∞}(q^{2b}; q^{3b})_{∞}(q^{3b}; q^{3b})_{∞}.

Let’s re-examine the Rogers-Ramanujan identity.

> with(qseries):
> x:=1:
> for n from 1 to 8 do
>  x:=x+q^{∧}(n*n)/aqprod(q,q,n):
> od:
> x:=series(x,q,50):
> y:=jacprodmake(x,q,40);

y := \frac{JAC (0, 5, ∞)}{JAC (1, 5, ∞)}

> z:=jac2prod(y);

z := \frac{1}{(q, q^{5})_{∞}(q^{4}, q^{5})_{∞}}

Note that we were able to observe that the left side of the Rogers-Ramanujan identity (at least up through q^{40}) can be written as a quotient of theta functions. We used the function jac2prod, to simplify the result and get it into a more recognizable form. The function jac2prod(jacexpr) converts a product of theta functions into q-product form; i.e., as a product of functions of the form (q^{a}; q^{b})_{∞}. Here jacexpr is a product (or quotient) of terms JAC(i, j, ∞), where i, j are integers and 0 ≤ i < j.

A related function is jac2series. This converts a Jacobi-type product into a form better for computing its q-series. It simply replaces each Jacobi-type product with its corresponding theta-series.

> with(qseries):
> x:=0:
> for n from 0 to 10 do
  x := x + q^{∧}(n*(n+1)/2)*aqprod(-q,q,n)/aqprod(q,q,2*n+1):
  od:
> x:=series(x,q,50):
> jacprodmake(x,q,50);

JAC(0, 14, ∞)^{6} /(JAC(1, 14, ∞)^{2} JAC(3, 14, ∞) JAC(4, 14, ∞)

JAC(5, 14, ∞) JAC(6, 14, ∞) \sqrt{\frac{JAC(7, 14, ∞)}{JAC(0, 14, ∞)}})

> jac2series(",500);

(q^{364} − q^{210} + q^{98} − q^{28} + 1 − q^{14} + q^{70} − q^{168} + q^{308} − q^{490})^{6} /((

−q^{621} + q^{496} − q^{385} + q^{288} − q^{205} + q^{136} − q^{81} + q^{40} − q^{13} + 1 − q + q^{16} − q^{45} + q^{88} − q^{145} + q^{216} − q^{301} + q^{400} − q^{513})^{2}(−q^{603} + q^{480} − q^{371} + q^{276} − q^{195} + q^{128} − q^{75} + q^{36} − q^{11} + 1 − q^{3} + q^{20} − q^{51} + q^{96} − q^{155} + q^{228} − q^{315} + q^{416} − q^{531})(−q^{594} + q^{472} − q^{364} + q^{270} − q^{190} + q^{124} − q^{72} + q^{34} − q^{10} + 1 − q^{4} + q^{22} − q^{54} + q^{100} − q^{160} + q^{234} − q^{322} + q^{424} − q^{540})(−q^{585} + q^{464} − q^{357} + q^{264} − q^{185} + q^{120} − q^{69} + q^{32} − q^{9} + 1 − q^{5} + q^{24} − q^{57} + q^{104} − q^{165} + q^{240} − q^{329} + q^{432} − q^{549})(−q^{576} + q^{456} − q^{350} + q^{258} − q^{180} + q^{116} − q^{66} + q^{30} − q^{8} + 1 − q^{6} + q^{26} − q^{60} + q^{108} − q^{170} + q^{246} − q^{336} + q^{440} − q^{558}) ((−2 q^{567} + 2 q^{448} − 2 q^{343} + 2 q^{252} − 2 q^{175} + 2 q^{112} − 2 q^{63} + 2 q^{28}

− 2 q^{7} + 1)/(

q^{364} − q^{210} + q^{98} − q^{28} + 1 − q^{14} + q^{70} − q^{168} + q^{308} − q^{490}))^{1/2})

It seems that the q-series

∑_{n≥0} \frac{(−q; q)_{n}q^{n(n+1)/2}}{(q; q)_{2n+1}}

can be written as Jacobi-type product. Assuming that this is the case we used jac2series to write this q-series in terms of theta-series at least up to q^{500}. This should provide an efficient method for computing the q-series expansion and also for computing the function at particular values of q.

Exercise 5. Use jacprodmake and jac2series to compute the q-series expansion of

∑_{n≥0} \frac{(−q; q)_{n}q^{n(3n+1)/2}}{(q; q)_{2n+1}}

up to q^{1000}, assuming it is Jacobi-type product. Can you identify the infinite product? This function occurs in Slater’s list [36](Eq.(46), p.156).

4. The Search for Relations

The functions for finding relations between q-series are findhom,

findhomcombo, findnonhom, findnonhomcombo, and findpoly.

4.1. findhom. If the q-series one is concerned with are modular forms of a particular weight, then theoretically these functions will satisfy homogeneous polynomial relations. See [18, p. 263], [9] for more details and examples. The function findhom(L,q,n,topshift) returns a set of potential homogeneous relations of order n among the q-series in the list L. The value of topshift is usually taken to be zero. However if it appears that spurious relations are being generated then a higher value of topshift should be taken.

The idea is to convert this into a linear algebra problem. This program generates a list of monomials of degree n of the functions in the given list of q-series L. The q-expansion (up to a certain point) of each monomial is found and converted into a row vector of a matrix. The set of relations is then found by computing the kernel of the transpose of this matrix. As an example, we now consider relations between the theta functions θ_{3}(q), θ_{4}(q), θ_{3}(q^{2}), and θ_{4}(q^{2}).

> with(qseries):
> findhom([theta3(q,100),theta4(q,100),theta3(q^{∧}2,100),
theta4(q^{∧}2,100)],q,1,0);
                     # of terms , 25
            -----RELATIONS-----of order---, 1

{{}}

> findhom([theta3(q,100),theta4(q,100),theta3(q^{∧}2,100),
theta4(q^{∧}2,100)],q,2,0);
                     # of terms , 31
            -----RELATIONS-----of order---, 2

{X_{1}^{2} + X_{2}^{2} − 2 X_{3}^{2}, −X_{1}X_{2} + X_{4}^{2}} From the session above we see that there is no linear relation between the functions θ_{3}(q), θ_{4}(q), θ_{3}(q^{2}) and θ_{4}(q^{2}). However, it appears that there are two quadratic relations:

θ_{3}(q^{2}) = (\frac{θ_{3}^{2}(q) + θ_{4}^{2}(q)}{2})^{1/2}

and

θ_{4}(q^{2}) = (θ_{3}^{2}(q)θ_{4}^{2}(q))^{1/2} .

This is Gauss’ parametrization of the arithmetic-geometric mean iteration. See [13, Ch. 2] for details.

Exercise 6. Define a(q), b(q) and c(q) as in Exercise 2. Find homogeneous relations between the functions a(q), b(q), c(q), a(q^{3}), b(q^{3}), c(q^{3}). In particular, try to get a(q^{3}) and b(q^{3}) in terms of a(q), b(q). See [12] for more details. These results lead to a cubic analog of the AGM due to Jon and Peter Borwein [10], [11].

4.2. findhomcombo. The function findhomcombo is a variant of findhom. Suppose f is a q-series and L is a list of q-series.

findhomcombo(f,L,q,n,topshift,etaoption) tries to express f as a

homogeneous polynomial in the members of L. If etaoption=yes then each monomial in the combination is “converted” into an eta-product

using etamake.

We illustrate this function with certain Eisenstein series. For p an odd prime define

χ(m) = (\frac{m}{p}) (the Legendre symbol).

Suppose k is an integer, k ≥ 2, and (p − 1)/2 ≡ k (mod 2). Define the Eisenstein series

U_{p,k}(q) := ∑_{m=1}^{∞} ∑_{n=1}^{∞} χ(m)n^{k−1}q^{mn}.

Then U_{p,k} is a modular form of weight k and character χ for the congruence subgroup Γ_{0}(p). See [28], [20] for more details. The classical result is the following identity found by Ramanujan [32, Eq. (1.52), p. 354]:

U_{5,2} = \frac{η(5τ)^{5}}{η(τ)} .

Kolberg [28] has found many relations between such Eisenstein series and certain eta products. The eta function η(τ) is a modular form of weight \frac{1}{2} [27, p. 121]. Hence the modular forms

B_{1} := \frac{η(5τ)^{5}}{η(τ)} , B_{2} := \frac{η(τ)^{5}}{η(5τ)}

are modular forms of weight \frac{(5−1)}{2} = 2. In fact, it can be shown that they are modular forms on Γ_{0}(5) with character (\frac{·}{5}). We might therefore expect that U_{5,6} can be written as a homogeneous cubic polynomial in B_{1}, B_{2}. We write a short maple program to compute the Eisenstein series U_{p,k}.

> with(numtheory):
> UE:=proc(q,k,p,trunk)
> local x,m,n:
> x:=0:
> for m from 1 to trunk do
>  for n from 1 to trunk/m do
>  x:=x + L(m,p)*n^{∧}(k-1)*q^{∧}(m*n):
>  od:
> od:
> end:

The function UE(q,k,p,trunk) returns the q-expansion of U_{p,k} up through qtrunk. We note that L(m,p) returns the Legendre symbol (\frac{m}{p}). We are now ready to study U_{5,6}.

> with(qseries):
> f := UE(q,6,5,50):
> B1 := etaq(q,1,50)^{∧}5/etaq(q,5,50):
> B2 := q*etaq(q,5,50)^{∧}5/etaq(q,1,50):
> findhomcombo(f,[B1,B2],q,3,0,yes);
                     # of terms , 25
   -----possible linear combinations of degree------, 3

\{η(5 τ)^{3}η(τ)^{9} + 40 η(5 τ)^{9}η(τ)^{3} + 335 \frac{η(5 τ)^{15}}{η(τ)^{3}} \}

{X_{1}^{2}X_{2} + 40 X_{1}X_{2}^{2} + 335 X_{2}^{3}}

It would appear that

U_{5,6} = η(5 τ)^{3}η(τ)^{9} + 40 η(5 τ)^{9}η(τ)^{3} + 335 \frac{η(5 τ)^{15}}{η(τ)^{3}} .

The proof is a straightforward exercise using the theory of modular forms.

Exercise 7. Define the following eta products:

C_{1} := \frac{η(7τ)^{7}}{η(τ)} ,  C_{2} := η(τ)^{3} η(7τ)^{3},  C_{3} := \frac{η(τ)^{7}}{η(7τ)}.

What is the weight of these modular forms?

Write U_{7,3} in terms of C_{1}, C_{2}, C_{3}.

The identity that you should find was originally due to Ramanujan. Also see Fine [15, p. 159] and [19, Eq. (5.4)].

If you are ambitious find U_{7,9} in terms of C_{1}, C_{2}, C_{3}.

4.3. findnonhom. In section 4.1 we introduced the function findhom to find homogeneous relations between q-series. The nonhomogeneous

analog is findnonhom.

The syntax of findnonhom is the same as findhom. Typically (but not necessarily) findhom is used to find relations between modular forms of a certain weight. To find relations between modular functions we would use findnonhom. We consider an example involving theta functions.

> with(qseries):
> F := q -> theta3(q,500)/theta3(q^{∧}5,100):
> U := 2*q*theta(q^{∧}10,q^{∧}25,5)/theta3(q^{∧}25,20);

U :=

\frac{2 q (q^{575} + q^{360} + q^{195} + q^{80} + q^{15} + 1 + q^{35} + q^{120} + q^{255} + q^{440} + q^{675})}{2 q^{625} + 2 q^{400} + 2 q^{225} + 2 q^{100} + 2 q^{25} + 1}

> EQNS := findnonhom([F(q),F(q^{∧}5),U],q,3,20);
                     # of terms , 61
            -----RELATIONS-----of order---, 3

EQNS := {−1 − X_{1}X_{2}X_{3} + X_{2}^{2} + X_{3}^{2} + X_{3}}

> ANS:=EQNS[1];

ANS := −1 − X_{1}X_{2}X_{3} + X_{2}^{2} + X_{3}^{2} + X_{3}

> CHECK := subs(X[1]=F(q),X[2]=F(q^{∧}5),X[3]=U,ANS):
> series(CHECK,q,500);

O (q^{500})

We define

F (q) := \frac{θ_{3}(q)}{θ_{3}(q^{5})},

and

U(q) := 2\frac{∑_{n=−∞}^{∞} q^{25n^{2}+10n+1}}{θ_{3}(q^{25})} .

We note that U(q) and F (q) are modular functions since they are ratios of theta series. From the session above we see that it appears that

1 + F (q)F (q^{5})U(q) = F (q^{5})^{2} + U(q)^{2} + U(q).

Observe how we were able to verify this equation to high order. When findnonhom returns a set of relations the variable X has been declared global. This is so we can manipulate the relations. It this way we were able to assign ANS to the relation found and then use subs and series to check it to order O (q^{500}).

4.4. findnonhomcombo. The syntax of findnonhomcombo is the same

as findhomcombo. We consider an example involving eta functions. First we define

ξ := \frac{η(49 τ)}{η(τ)},

and

T := (\frac{η(7 τ)}{η(τ)})^{4} ,

Using the theory of modular functions it can be shown that one must be able to write T^{2} in terms of T and powers of ξ. We now use findnonhomcombo to get T^{2} in terms of ξ and T.

> xi:=q^{∧}2*etaq(q,49,100)/etaq(q,1,100):
> T:=q*(etaq(q,7,100)/etaq(q,1,100))^{∧}4:
> findnonhomcombo(T^{∧}2,[T,xi],q,7,-15);
                     # of terms , 42
                  matrix is , 37, x, 42
   -----possible linear combinations of degree------, 7

{147 X_{2}^{5} + 343 X_{2}^{7} + 343 X_{2}^{6} + X_{2} + 49 X_{2}^{4} + 49 X_{1}X_{2}^{3} + 7 X_{1}X_{2}

+21 X_{2}^{3} + 7 X_{2}^{2} + 35 X_{1}X_{2}^{2}}

Then it seems that

T^{2} = (35 ξ^{2} + 49 ξ^{3} + 7 ξ) T
+ 343 ξ^{7} + 343 ξ^{6} + 147 ξ^{5} + 49 ξ^{4} + 21 ξ^{3} + 7 ξ^{2} + ξ.

This is the modular equation used by Watson[41] to prove Ramanujan’s partition congruences for powers of 7. Also see [5] and [26], and see [16] for an elementary treatment.

Exercise 8. Define

ξ := \frac{η(25 τ)}{η(τ)},

and

T := (\frac{η(5 τ)}{η(τ)})^{6} .

Use findnonhomcombo to express T as a polynomial in ξ of degree 5. The modular equation you find was used by Watson to prove Ramanujan’s partition congruences for powers of 5. See [23] for an elementary treatment.

Exercise 9. Define a(q) and c(q) as in Exercise 2. Define

x(q) := \begin{matrix}c(q) & 3 \\ a(q) & 3 \end{matrix} ,

and the classical Eisenstein series (usually called E_{3}; see [35, p. 93])

N(q) := 1 − 504 ∑_{n=1}^{∞} \frac{n^{5}q^{n}}{1 − q^{n}} .

Use findnonhomcombo to express N(q) in terms of a(q) and x(q). HINT : N(q) is a modular form of weight 6 and a(q) and c(q) are modular forms of weight 1. See [8] for this result and many others.

4.5. findpoly. The function findpoly is used to find a polynomial relation between two given q-series with degrees specified.

findpoly(x,y,q,deg1,deg2,check) returns a possible polynomial

in X, Y (with corresponding degrees deg1, deg2) which is satisfied by the q-series x and y.

If check is assigned then the relation is checked to O (qcheck). We illustrate this function with an example involving theta functions

and the function a(q) and c(q) encountered in Exercises 2 and 7. It

18 FRANK GARVAN

can be shown that

a(q) = θ_{3}(q)θ_{3}(q^{3}) + θ_{2}(q)θ_{2}(q^{3}).

See [12] for details. This equation provides a better way of computing the q-series expansion of a(q) than the definition. In Exercise 2 you would have found that

c(q) = 3 \frac{η^{3}(3 τ)}{η(τ)}.

See [12] for a proof. Define

y := \frac{c^{3}}{a^{3}} ,

and

x := (\frac{θ_{2}(q)}{θ_{2}(q^{3})})^{2} + (\frac{θ_{3}(q)}{θ_{3}(q^{3})})^{2} .

We use findpoly to find a polynomial relation between x and y.

> with(qseries):
> x1 := radsimp(theta2(q,100)^{∧}2/theta2(q^{∧}3,40)^{∧}2):
> x2 := theta3(q,100)^{∧}2/theta3(q^{∧}3,40)^{∧}2:
> x := x1+x2:
> c := q*etaq(q,3,100)^{∧}9/etaq(q,1,100)^{∧}3:
> a := radsimp(theta3(q,100)*theta3(q^{∧}3,40)+theta2(q,100)
  *theta2(q^{∧}3,40)):
> c := 3*q^{∧}(1/3)*etaq(q,3,100)^{∧}3/etaq(q,1,100):
> y := radsimp(c^{∧}3/a^{∧}3):
> P1:=findpoly(x,y,q,3,1,60);
WARNING: X,Y are global.
                       dims , 8, 18
                    The polynomial is

(X + 6)^{3} Y − 27 (X + 2)^{2}

                  Checking to order, 60

O (q^{59})

P1 := (X + 6)^{3} Y − 27 (X + 2)^{2}

It seems that x and y satisfy the equation

p(x, y) = (x + 6)^{3} y − 27(x + 2)^{2} = 0.

Therefore it would seem that

\frac{c^{3}}{a^{3}} = 27 \begin{matrix}(x + 2) & 2 \\ (x + 6) & 3 \end{matrix} .

See [8, pp. 4237–4240] for more details.

Exercise 10. Define

m := (\frac{θ_{3}(q)}{θ_{3}(q^{3})})^{2} .

Use polyfind to find y = \frac{c^{3}}{a^{3}} as a rational function in m. The answer is Eq.(12.8) in [8].

5. Sifting coefficients

Suppose we are given a q-series

A(q) = ∑_{n=0}^{∞} a_{n}q^{n}.

Occasionally it will turn out the generating function

∑_{n=0}^{∞} a_{mn+r}q^{n}

will have a very nice form. A famous example for p(n) is due to Ramanujan:

∑_{n=0}^{∞} p(5n + 4)q^{n} = 5 ∏_{n=1}^{∞} \frac{(1 − q^{5n})^{6}}{(1 − q^{n})^{5}} .

See [1, Cor. 10.6]. In fact, G.H. Hardy and Major MacMahon [31, p. xxxv] both agreed that this is Ramanujan’s most beautiful identity.

Suppose s is the q-series

∑ a_{i}q^{i} + O(q^{T})

then sift(s,q,n,k,T) returns the q-series

∑ a_{ni+k}q^{i} + O(q^{T/n}).

We illustrate this function with another example from the theory of partitions. Let pd(n) denote the number of partitions of n into distinct parts. Then it is well known that

∑_{n=0}^{∞} pd(n)q^{n} = ∏_{n=1}^{∞}(1 + q^{n}) = ∏_{n=1}^{∞} \frac{(1 − q^{2n})}{(1 − q^{n})}.

We now examine the generating function of pd(5n + 1) in maple.

> PD:=series(etaq(q,2,200)/etaq(q,1,200),q,200):

> sift(PD,q,5,1,199);

1 + 4 q + 5010688 q^{26} + 53250 q^{15} + 668 q^{7} + 12 q^{2} + 165 q^{5} + 12076 q^{12} + 1087744 q^{22} + 109420549 q^{35} + 76 q^{4} + 32 q^{3} + 340 q^{6} + 1260 q^{8} + 2304 q^{9} + 4097 q^{10} + 7108 q^{11} + 20132 q^{13} + 32992 q^{14} + 84756 q^{16} + 133184 q^{17} + 206848 q^{18} + 317788 q^{19}

+ 728260 q^{21} + 20792120 q^{30} + 2368800 q^{24} + 483330 q^{20}
+ 1611388 q^{23} + 3457027 q^{25} + 7215644 q^{27} + 10327156 q^{28}

+ 14694244 q^{29} + 29264960 q^{31} + 40982540 q^{32} + 57114844 q^{33}

+ 79229676 q^{34} + 150473568 q^{36} + 206084096 q^{37}
+ 281138048 q^{38} + 382075868 q^{39}

> PD1:=":
> etamake(PD1,q,38);

\frac{η( 5 τ )^{3} η( 2 τ )^{2}}{q^{5/24} η( 10 τ ) η( τ )^{4}}

So it would seem that

∑_{n=0}^{∞} pd(5n + 1)q^{n} = ∏_{n=1}^{∞} \frac{(1 − q^{5n})^{3}(1 − q^{2n})^{2}}{(1 − q^{10n})(1 − q^{n})^{4}} .

This result was found originally by Rødseth [33].

Exercise 11. Rødseth also found the generating functions for pd(5n + r) for r = 0, 1, 2, 3 and 4. For each r use sift and jacprodmake to identify these generating functions as infinite products.

6. Product Identities

At present, the package contains the Triple Product identity, the Quintuple Product identity and Winquist’s identity. These are the most commonly used of the Macdonald identities [30], [37], [38]. The Macdonald identities are the analogs of the Weyl denominator for affine roots systems. Hopefully, a later version of this package will include these more general identities.

6.1. The Triple Product Identity. The triple product identity is

∑_{n=−∞}^{∞}(−1)^{n}z^{n}q^{n(n−1)/2} = ∏_{n=1}^{∞}(1 − zq^{n−1})(1 − q^{n}/z)(1 − q^{n}), (6.7)

where z ≠ 0 and |q| < 1. The Triple Product Identity is originally due to Jacobi [24, Vol I]. The first combinatorial proof of the triple product identity is due to Sylvester [39]. Recently, Andrews [3] and Lewis [29] have found nice combinatorial proofs. The triple product occurs frequently in the theory of partitions. For instance, most proofs of the Rogers-Ramanujan identity crucially depend on the triple product identity.

tripleprod(z,q,T) returns the q-series expansion to order O(q^{T}) of Jacobi’s triple product (6.7). This expansion is found by simply truncating the right side of (6.7).

> tripleprod(z,q,10);

\frac{q^{21}}{z^{6}} − \frac{q^{15}}{z^{5}} + \frac{q^{10}}{z^{4}} − \frac{q^{6}}{z^{3}} + \frac{q^{3}}{z^{2}} − \frac{q}{z} + 1 − z + z^{2}q − z^{3}q^{3} + z^{4}q^{6} − z^{5}q^{10} + z^{6}q^{15}

> tripleprod(q,q^{∧}3,10);

q^{57} − q^{40} + q^{26} − q^{15} + q^{7} − q^{2} + 1 − q + q^{5} − q^{12} + q^{22} − q^{35} + q^{51}

The last calculation is an illustration of Euler’s Pentagonal Number Theorem [1, Cor. 1.7 p.11]:

∏_{n=1}^{∞}(1 − q^{n}) = ∏_{n=1}^{∞}(1 − q^{3n−1})(1 − q^{3n−2})(1 − q^{3n}) = ∑_{n=−∞}^{∞}(−1)^{n}q^{n(3n−1)/2}. (6.8)

6.2. The Quintuple Product Identity. The following identity is the Quintuple Product Identity:

(−z, q)_{∞}(−\frac{q}{z} , q)_{∞}(z^{2}q, q^{2})_{∞}(\frac{q}{z^{2}} , q^{2})_{∞}(q, q)_{∞} (6.9)

= ∑_{m=−∞}^{∞} ((−z)^{−3 m} − (−z)^{3 m−1}) q^{\frac{m(3 m+1)}{2}} .

Here |q| < 1 and z ≠ 0. This identity is the BC_{1} case of the Macdonald identities [30]. The quintuple product identity is usually attributed to Watson [40]. However it can be found in Ramanujan’s lost notebook [32, p. 207]. Also see [7] for more history and some proofs.

The function quinprod(z,q,T) returns the quintuple product identity in different forms:

(i) If T is a positive integer it returns the q-expansion of the right side of (6.9) to order O(q^{T}).

(ii) If T = prodid then quinprod(z,q,prodid) returns the quintuple

product identity in product form. (iii) If T = seriesid then quinprod(z,q,seriesid) returns the quin-

tuple product identity in series form.

22 FRANK GARVAN

> quinprod(z,q,prodid);

(−z, q)_{∞} (−\frac{q}{z} , q)_{∞} (z^{2} q, q^{2})_{∞} (\frac{q}{z^{2}} , q^{2})_{∞} (q, q)_{∞} =

(\frac{q^{2}}{z^{3}} , q^{3})_{∞} (q z^{3}, q^{3})_{∞} (q^{3}, q^{3})_{∞} + z (\frac{q}{z^{3}} , q^{3})_{∞} (q^{2} z^{3}, q^{3})_{∞} (q^{3}, q^{3})_{∞}

> quinprod(z,q,seriesid);

(−z, q)_{∞} (−\frac{q}{z} , q)_{∞} (z^{2} q, q^{2})_{∞} (\frac{q}{z^{2}} , q^{2})_{∞} (q, q)_{∞} =

∑_{m=−∞}^{∞} ((−z)^{(−3 m)} − (−z)^{(3 m−1)}) q^{(1/2 m (3 m+1))}

> quinprod(z,q,3);

(z^{12} + \frac{1}{z^{11}} ) q^{22} + (−z^{9} − \frac{1}{z^{8}} ) q^{12} + (z^{6} + \frac{1}{z^{5}} ) q^{5} + (−z^{3} − \frac{1}{z^{2}} ) q + 1 + z

+ (−\frac{1}{z^{3}} − z^{4}) q^{2} + (\frac{1}{z^{6}} + z^{7}) q^{7} + (−\frac{1}{z^{9}} − z^{10}) q^{15} + (\frac{1}{z^{12}} + z^{13}) q^{26}

Let’s examine a more interesting application. Euler’s infinite product may be dissected according to the residue of the exponent of q mod 5:

∏_{n=1}^{∞}(1 − q^{n}) = E_{0}(q) + qE_{1}(q^{5}) + q^{2}E_{2}(q^{5}) + q^{3}E_{3}(q^{5}) + q^{4}E_{2}(q^{5}).

By (6.8) we see that E_{3} = E_{4} = 0 since n(3n − 1)/2 ≡ 0, 1 or 2 mod 5. Let’s see if we can identify E_{0}.

> with(qseries):
> EULER:=etaq(q,1,500):
> E0:=sift(EULER,q,5,0,499);

E0 := 1 + q − q^{3} − q^{7} − q^{8} − q^{14} + q^{20} + q^{29} + q^{31} + q^{42} − q^{52} − q^{66}

− q^{69} − q^{85} + q^{99}

  > jacprodmake(E0,q,50);

\frac{JAC( 2, 5, ∞) JAC( 0, 5, ∞)}{JAC( 1, 5, ∞)}

> jac2prod(");

\begin{matrix}( q^{5}, q^{5} )_{∞} & ( q^{2}, q^{5} )_{∞} & ( q^{3}, q^{5} )_{∞} \\ ( q, q^{5} )_{∞} & ( q^{4}, q^{5} )_{∞} \end{matrix}

> quinprod(q,q^{∧}5,20):
> series(",q,100);

1+q−q^{3}−q^{7}−q^{8}−q^{14}+q^{20}+q^{29}+q^{31}+q^{42}−q^{52}−q^{66}−q^{69}−q^{85}+O (q^{99})

From our maple session it appears that

E_{0} = \frac{(q^{5}; q^{5})_{∞}(q^{2}; q^{5})_{∞}(q^{3}; q^{5})_{∞}}{(q; q^{5})_{∞}(q^{4}; q^{5})_{∞}} , (6.10)

and that this product can be gotten by replacing q by q^{5} and z by q in the product side of the quintuple product identity (6.9).

Exercise 12. (i) Use the quintuple product identity (6.9) and Euler’s pentagonal number theorem to prove (6.10) above. (ii) Use maple to identify and prove product expressions for E_{1} and E_{2}. (iii) This time see if you can repeat (i), (ii) but split the exponent mod 7. (iv) Can you generalize these results to arbitrary modulus? Atkin and Swinnerton-Dyer found a generalization. See Lemma 6 in [6].

6.3. Winquist’s Identity. Back in 1969, Lasse Winquist [43] discovered a remarkable identity

(a; q)_{∞}(q/a; q)_{∞}(b; q)_{∞}(q/b; q)_{∞}(ab; q)_{∞}(q/(ab); q)_{∞}(a/b; q)_{∞} (6.11)

(b/(aq); q)_{∞}(q; q)_{∞}^{2}

= ∑_{n=0}^{∞} ∑_{m=−∞}^{∞}(−1)^{n+j}((a^{−3n} − a^{3n+3})(b^{−3m} − b^{3m+1})

+ (a^{−3m+1} − a^{3m+2})(b^{3n+2} − b^{−3n−1}))q^{3n(n+1)/2+m(3m+1)/2}.

By dividing both sides by (1 − a)(1 − b) and letting a, b → 1 he was

able to express the product ∏_{n=1}^{∞}(1 − q^{n})^{10} as a double series and prove

Ramanujan’s partition congruence

p(11n + 6) ≡ 0 (mod 11).

This was probably the first truly elementary proof of Ramanujan’s congruence modulo 11. The interested reader should see Dyson’s article [14] for some fascinating history on identities for powers of the Dedekind eta function and how they led to the Macdonald identities. A new proof of Winquist’s identity has been found recently by S.-Y. Kang [25]. Mike Hirschhorn [22] has found a four-parameter generalization of Winquist’s identity.

24 FRANK GARVAN

The function winquist(a,b,q,T) returns the q-expansion of the right side of (6.11) to order O(q^{T}).

We close with an example. For 1 < k < 33 define

Q(k) = ∏_{n=1}^{∞}(1 − q^{k})(1 − q^{33−k})(1 − q^{33}).

Now define the following functions:

A_{0} = Q(15), A_{3} = Q(12), A_{7} = Q(6), A_{8} = Q(3), A_{9} = Q(9);

B_{0} = Q(16) − q^{2}Q(5),
B_{1} = Q(14) − qQ(8),
B_{2} = Q(13) − q^{3}Q(2),
B_{4} = Q(7) + qQ(4),
B_{7} = Q(10) + q^{3}Q(1).

These functions occur in Theorem 6.7 of [17] as well as the function A_{0}B_{2} − q^{2}A_{9}B_{4}.

> with(qseries):
> Q:=n->tripleprod(q^{∧}n,q^{∧}33,10):
> A0:=Q(15):   A3:=Q(12):   A7:=Q(6):
> A8:=Q(3):   A9:=Q(9):
> B2:=Q(13)-q^{∧}3*Q(2):   B4:=Q(7)+q*Q(4):
> IDG:=series( ( A0*B2-q^{∧}2*A9*B4),q,200):
> series(IDG,q,10);

1 − q^{2} − 2 q^{3} + q^{5} + q^{7} + q^{9} + O (q^{11})

> jacprodmake(IDG,q,50);

\frac{JAC( 2, 11, ∞) JAC( 3, 11, ∞)^{2} JAC( 5, 11, ∞)}{JAC( 0, 11, ∞)^{3}}

> jac2prod(");

( q^{2}, q^{11} )_{∞} ( q^{9}, q^{11} )_{∞} ( q^{11}, q^{11} )_{∞} ( q^{3}, q^{11} )_{∞}^{2} ( q^{8}, q^{11} )_{∞}^{2} ( q^{5}, q^{11} )_{∞}
( q^{6}, q^{11} )_{∞}

> series(winquist(q^{∧}5,q^{∧}3,q^{∧}11,10),q,20);

1 − q^{2} − 2 q^{3} + q^{5} + q^{7} + O( q^{9} )

> series(IDG-winquist(q^{∧}5,q^{∧}3,q^{∧}11,10),q,60);

O( q^{49})

From our maple session it seems that

A_{0}B_{2} − q^{2}A_{9}B_{4} = (q^{2}; q^{11})_{∞}(q^{9}; q^{11})_{∞}(q^{11}; q^{11})_{∞}(q^{3}; q^{11})_{∞}^{2}(q^{8}; q^{11})_{∞}^{2} (6.12)
(q^{5}; q^{11})_{∞}(q^{6}; q^{11})_{∞},

and that this product appears in Winquist’s identity on replacing q by q^{11} and letting a = q^{5} and b = q^{3}.

Exercise 13. (i) Prove (6.12) by using the triple product identity (6.7) to write the right side of Winquist’s identity (6.11) as a sum of two products. (ii) In a similar manner find and prove a product form for

A_{0}B_{0} − q^{3}A_{7}B_{4}.

References

[1] George E. Andrews. The theory of partitions. Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. Encyclopedia of Mathematics and its Applications, Vol. 2.

[2] George E. Andrews. Partitions: yesterday and today, With a foreword by J. C. Turner. New Zealand Math. Soc., Wellington, 1979.

[3] George E. Andrews. Generalized Frobenius partitions. Mem. Amer. Math. Soc., 49(301):iv+44, 1984.

[4] George E. Andrews. q-series: their development and application in analysis, number theory, combinatorics, physics, and computer algebra. Published for the Conference Board of the Mathematical Sciences, Washington, D.C., 1986.

[5] A. O. L. Atkin. Ramanujan congruences for p_{−k}(n). Canad. J. Math. 20 (1968), 67-78; corrigendum, ibid., 21:256, 1968.

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Department of Mathematics, University of Florida, Gainesville, Florida 32611

E-mail address: frank@math.ufl.edu