Gert Almkvist

Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, p, and the Ladies Diary

Paper 8: Gert Almkvist and Bruce Berndt, “Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, pi, and the Ladies Diary,” American Mathematical Monthly, vol. 95 (1988), pg. 585–608. Copyright 1988 Mathematical Association of America. All Rights Reserved.

Generalizations of Clausen's Formula and algebraic transformations of Calabi–Yau differential equations

We provide certain unusual generalizations of Clausens and Orrs theorems for solutions of fourth-order and fifth-order generalized hypergeometric equations. As an application, we present several examples of algebraic transformations of Calabi-Yau differential equations.

Hilbert series of fixed free algebras and noncommutative classical invariant theory

We compute an asymptotic formula for the number of invariants of a given degree, which compares nicely with the corresponding result in the classical commutative case. In Section 6 we show that if G is an infinite cyclic group generated by a unipotent matrix then

Asymptotic formulas and generalized Dedekind sums

We find asymptotic formulas as n → ∞ for the coefficients a(r, n) defined by (The case r = 1 gives the number of plane partitions of n.) Generalized Dedekind sums occur naturally and are studied using the Finite Fourier Transform. The methods used are unorthodox; many of the computations are not justified but the result is in many casesvery good numerically. The last section gives various formulas for Kinkelins constant.

Some new formulas for pi

We show how to find series expansions for π of the form π = Σ∞ n =0 S(n)/ an , where S(n) is some polynomial in n (depending on m, p, a). We prove that there exist such expansions for m = 8k, p = 4k, a = (−4)k, for any k, and give explicit examples for such expansions for small values of m, p, and a.

Some formulas in invariant theory

In an earlier paper 121 the author discussed the relationship between invariants of the group G = Z/pZ in characteristic p and the classical covariants of SL(2. k) in characteristic zero. This is more closely studied here in the corresponding representation rings. In characteristic zero the only indecomposable (1 irreducible) SL(2, k)-modules are the R,: sI where R, is the set of homogeneous forms of degree n of the polynomial ring R = k/.x, y]. In characteristic p there are p indecomposable G-modules I’,,..., VP. The corresponding representation rings are

A Hardy-Ramanujan formula for restricted partitions

Abstract In this paper, we extend the Hardy-Ramanujan-Rademacher formula for p ( n ), the number of partitions of n . In particular we provide such formulas for p ( j , n ), the number of partitions of j into at most n parts and for A ( j , n , r ), the number of partitions of j into at most n parts each ≤ r .

Borwein and Bradley's Apérv-Like Formulae for ζ(4n + 3)

We prove a formula for ζ(4n + 3) discovered by Borwein and Bradley (Experimental Mathematics 6:3 (1997), 181–194).

Exact asymptotic formulas for the coefficients of nonmodular functions

Exact asymptotic formulas for the number of partitions with various restrictions (e.g., into odd parts of size ≤ r, odd distinct parts ≤ r) are found. The idea is to introduce a differential operator into the formulas obtained from the Hardy-Ramanujan-Rademacher theory.

Representations of SL(2, C) and unimodal polynomials

On montre que les representations de SL(2, C) et les polynomes unimodaux sont la meme chose