In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Hall–Littlewood polynomials are symmetric function

Symmetric function

In algebra and in particular in algebraic combinatorics, the ring of symmetric functions, is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity...

s depending on a parameter t and a partition

Partition (number theory)

In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered to be the same partition; if order matters then the sum becomes a...

 λ. They are Schur functions when t is 0 and monomial symmetric functions when t is 1 and are special cases of Macdonald polynomial

Macdonald polynomial

In mathematics, Macdonald polynomials Pλ are a family of orthogonal polynomials in several variables, introduced by...

s.
They were first defined indirectly by Philip Hall

Philip Hall

Philip Hall FRS , was an English mathematician.His major work was on group theory, notably on finite groups and solvable groups.-Biography:...

 using the Hall algebra, and later defined directly by .

Definition

The Hall–Littlewood polynomial P is defined by
where λ is a partition of length at most n with elements λ_i, and m(i) elements equal to i, and S_n is the symmetric group

Symmetric group

In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...

of order n!.