q-Series

 

The q-series articles explain and illustrate concepts, identities, and theorems for:

  1. Inversions  

  2. q-Binomial Theorems

  3. Partitions

  4. Pentagonal Numbers, Ferrer's Diagrams, and Divisors 

  5. Triple and Quintuple Product Identities 

  6. Sums of Squares  

  7. Ramanujan’s Most Beautiful Identity

  8. Falls and Derangements

  9. q-Trigonometric Series

1.  Inversions

 

Two amazing results in this section are the connections between q-factorials and inversions and between the q-binomial coefficient and inversions.  For both, the exponents of the q terms are the number of inversions and the coefficients are the number of permutations with that number of inversions.

Connection between Inversions and q-Factorials

Connection between Inversions and q-Binomial Coefficients

2.  q-Binomial Theorems

 

This section introduces the q-shifted factorial, which is used extensively in q-series to generate q-polynomials.  Several identities use the q-binomial coefficient, and there are interesting connections among these identities.  There is an amazing identity by Euler and an identity by Chen, Chu, and Gu, both of which will be useful in future topics.

Connections among q-Binomial Theorems

Curious and Complicated q-Binomial Theorems

3.  Partitions

Whereas the first topic demonstrated connections between q-polynomials and inversions, this topic demonstrates the amazing connections between q-polynomials (as generated by q-shifted factorials) and partitions.  Various combinations of q-shifted factorials produce q-polynomials in which the coefficient of each qn is the number of partitions for the integer n. 

Partitions with Distinct and Repeated Parts

Partitions with Multiple Restrictions

Euler's Partition Identities

4.  Pentagonal Numbers, Ferrers Diagrams, and Divisors

Euler discovered that a simple q-shifted factorial generates a q-polynomial in which the pentagonal numbers occur as every second exponent.  Ferrers diagram can be used to demonstrate some identities for q-polynomials and partitions.  There are also interesting connections between q-polynomials and divisors where the coefficient of each qn is the number of divisors of n.

Pentagonal Numbers

Ferrers Diagram

Divisors

5.  Triple and Quintuple Product Identities

 

Several identities demonstrate that products of three and five q-shifted factorials can be represented by simple sums.  This provides our first experience with Ramanujan, who developed an elegant version of a triple product that can be related to the pentagonal numbers—another amazing connection.  Ramanujan also transformed a version of the quintuple product identity to produce q-polynomials in which the coefficients are odd integers except for integers that are multiples of 3.  Progressing from products to quotients of q-shifted factorials leads to q-hypergeometric series.

Triple Product Identities

Quintuple Product Identities

More Quintuple Product Identities

q-Hypergeometric Series

6.  Sums of Squares

 

There is a special type of q-hypergeometric series that is distinguished by having the same number of terms in the numerator as in the denominator.  Ramanujan discovered such a series and created an elegant identity between the sum and product term.  Cauchy had previously developed two identities that are special cases of Ramanujan’s identities.  One of these is used to derive a polynomial for which the coefficient of qm is the number of ways of writing m as a sum of two squares; the other is used to derive a polynomial for which the coefficient of qm is the number of ways of writing m as a sum of four squares.

Psi q-Hypergeometric Series

Sums of Squares

7.  Ramanujan’s Beautiful Identities

 

The highlight of this brief topic is Ramanujan’s “most beautiful” identity.  For the partitions of integers that are congruent to 4 mod 5, this identity makes it obvious that all the partitions for these integers are divisible by 5.   A second beautiful identity of Ramanujan's is for the partitions of integers that are congruent to 5 mod 7.  All the partitions for these integers are divisible by 7. 

Ramanujan's Beautiful Identities

8.  Falls and Derangements

 

A fall in a permutation is the position of an element that is larger than its next element, and the sum of the falls is the “major index” of the permutation.  When element k of a permutation is not in the kth position, the element is “deranged” and if all the elements of a permutation are deranged the permutation is called a “derangement.”  Amazingly, we can use the major index in the same way as inversions and can use derangements to calculate Euler’s number e.

Falls

Derangements

9.  q-Trigonometric Series

 

It is interesting to study plots of the q-analogues of the sine and cosine.

q-Trigonometric Series