Burnside's Formula Research Articles

A double inequality related with Burnside's formula

where the constants a*=0.428844044... and a*=0.5 are the best possible. We believe that the method we used in the proof gives insight to undergraduate students to understand how simple inequalities can be established.

A short proof of Burnside's formula for the Gamma function

We present simple proofs for Burnside's asymptotic formula and for its extension to positive real numbers.

A general asymptotic formula of the gamma function based on the Burnside's formula

In this paper, based on the Burnside's formula and our early works, a general continued fraction approximation of the factorial function and some inequalities for the gamma function are established. Finally, for demonstrating the superiority of our new series over the Burnside's formula and our early formulas, some numerical computations are given.

A new asymptotic expansion and some inequalities for the gamma function

In this paper, based on the Burnside formula, an asymptotic expansion of the factorial function and some inequalities for the gamma function are established. Finally, for demonstrating the superiority of our new series over the Burnside formula, the classical Stirling series and the Mortici sequences, some numerical computations are given.

Sharp bounds for gamma and digamma function arising from Burnside's formula

The main aim of this paper is to improve the Burnside's formula for approximating the factorial function. We prove the complete monotonicity of a function involving the gamma function to establish new lower and upper sharp bounds for the gamma and digamma function.

Best estimates of the generalized Stirling formula

The aim of this paper is to introduce an approximations family of the factorial function that contains Stirling's formula, Burnside's formula and Gosper's formula. The parameters which provide the best approximations are indicated. Finally, numerical computations are made to show the superiority of our formulas over other known formulas.

Groupoid of orientational variants

Daughter crystals in orientation relationship with a parent crystal are called variants. They can be created by a structural phase transition (Landau or reconstructive), by twinning or by precipitation. Internal and external classes of transformations defined from the point groups of the parent and daughter phases and from a transformation matrix allow the orientations of the distinct variants to be determined. These are algebraically identified with left cosets and their number is given by the Lagrange formula. A simple equation links the numbers of variants of the direct and inverse transitions. The equivalence classes on the transformations between variants are isomorphic to the double cosets (operators) and their number is given by the Burnside formula. The orientational variants and the operators constitute a groupoid whose composition table acts as a crystallographic signature of the transition. A general method that determines if two daughter variants can be inherited from more than one parent crystal is also described. A computer program has been written to calculate all these properties for any structural transition; some results are given for Burgers transitions and for martensitic transitions in steels. The complexity, irreversibility and entropy of fractal systems constituted by orientational variants generated by thermal cycling are briefly discussed.

An application of the duality between representations and classes

Starting from the principle of duality between representations and classes of groups of finite order a 3 k-symbol is defined in close analogy to the well-known 3 j-symbols. For these 3 k-symbols a relation is derived which may be considered as a supplement to Burnside's formula for the dimensions of the irreducible representations.

On enumerative equivalence of group elements

Let G be a finite group acting on a finite set S. Burnside's formula expresses the number of orbits of S under G in terms of the numbers I(g), where I(g) is the number of fixed points of S under the group element g. This paper characterizes the circumstances under which two group elements, g 1 and g 2, satisfy I(g 1)= I(g 2) for all actions of G. Several related questions are also discussed.