Power Series Identity Research Articles

A power series identity and Bessel-type integrals over unitary groups

In 2008, Lehner, Wettig, Guhr, and Wei conjectured a power series identity and showed that it implied a determinantal formula for a Bessel-type integral over the unitary supergroup. The integral is the supersymmetric extension of Bessel-type integrals over the unitary group appearing as partition functions in quantum chromodynamics. The identity is proved by interpreting both sides as the same unitary integral, which can be computed using the Cartan decomposition. An equivalent identity of Schur functions is also given and interpreted probabilistically in terms of random partitions.

Lagrange inversion

We give a survey of the Lagrange inversion formula, including different versions and proofs, with applications to combinatorial and formal power series identities.

On [formula omitted]-series identities related to interval orders

We prove several power series identities involving the refined generating function of interval orders, as well as the refined generating function of the self-dual interval orders. These identities may be expressed as ∑n≥0(1p;1q)n=∑n≥0pqn(p;q)n(q;q)n and ∑n≥0(−1)n(1p;1q)n=∑n≥0pqn(p;q)n(−q;q)n=∑n≥0(qp)n(p;q2)n, where the equalities apply to the (purely formal) power series expansions of the above expressions at p=q=1, as well as at other suitable roots of unity.

A formal identity involving commuting triples of permutations

We prove a formal power series identity, relating the arithmetic sum-of-divisors function to commuting triples of permutations. This establishes a conjecture of Franklin T. Adams-Watters.

Rational points and cohomology of discriminant varieties

Let G be a finite unitary reflection group acting in a complex vector space V= C n . The discriminant variety X G of G is defined as the space of regular orbits of G on V. Classical examples include the varieties of complex polynomials of degree n with distinct (resp. non-zero distinct) roots. The normaliser G of G in GL( V) acts on X G ; in this work we determine the action of G/G on the cohomology of X G . In the classical cases this amounts to computing the cohomology of X G with certain local coefficient systems. Our methods are to compute equivariant weight polynomials by means of explicit counting of the rational points of certain varieties over finite fields, and then to exploit the weight purity of the relevant varieties. We obtain some power series identities as a byproduct.

Homology stability for classical regular semisimple varieties

We use techniques from homotopy theory, in particular the connection between configuration spaces and iterated loop spaces, to give geometric explanations of stability results for the cohomology of the varieties of regular semisimple elements in the simple complex Lie algebras of classical type A, B or C, as well as in the group \(GL_n({\Bbb C})\). We show that the cohomology spaces of stable versions of these varieties have an algebraic stucture, which identifies them as “free Poisson algebras” with suitable degree shifts. Using this, we are able to give explicit formulae for the corresponding Poincare series, which lead to power series identities by comparison with earlier work. The cases of type B and C involve ideas from equivariant homotopy theory. Our results may be interpreted in terms of the actions of a Weyl group on its coinvariant algebra (i.e. the coordinate ring of the affine space on which it acts, modulo the invariants of positive degree; this space coincides with the cohomology ring of the flag variety of the associated Lie group) and on the cohomology of its associated complex discriminant variety.

The Weyl-Kac character formula and power series identities

The Weyl-Kac character formula and power series identities

Uniform Approximation operators Generated by a Power Series Identity

The classical Bernstein polynomials $B_n (f;x)$ may be regarded as transforms of the sequences $\{ f({k / n})\} $ by the Euler summability matrix with parameter x. T. H. Gronwall has shown that the Euler matrix is generated by a certain formal power series identity. In this paper these ideas are combined to produce a wide class of approximation operators which generalize the Bernstein operators. A uniform convergence theorem, an order of convergence result and an asymptotic error formula are given.