Bille C. Carlson (b. 1924 in Cambridge, Massachusetts) is Professor Emeritus in the Department of Mathematics and Associate of the Ames Laboratory (U.S. Department of Energy) at Iowa State University, Ames, Iowa.
The main theme of Carlson’s research has been to expose previously hidden permutation symmetries that can eliminate a set of transformations and thereby replace many formulas by a few. In 1963 he defined the R-function, a multivariate hypergeometric function that is homogeneous in its variables, each variable being paired with a parameter. If some of the parameters are equal, then the R-function is symmetric in the corresponding variables. This symmetry led to the development of symmetric elliptic integrals, which are free from the transformations of modulus and amplitude that complicate the Legendre theory. Symmetric integrals and their degenerate cases allow greatly shortened integral tables and improved algorithms for numerical computation. Also, the homogeneity of the R-function has led to a new type of mean value for several variables, accompanied by various inequalities.
The foregoing matters are discussed in Carlson’s book Special Functions of Applied Mathematics, published by Academic Press in 1977. In 2004 he found a previously hidden symmetry in relations between Jacobian elliptic functions, which can now take a form that remains valid when the letters c, d, and n are permuted. This invariance usually replaces sets of twelve equations by sets of three equations and applies also to the relation between the first symmetric elliptic integral and the Jacobian functions. In 2009 he found an analogous hidden symmetry between theta functions.
Carlson was elected a Fellow of the American Physical Society in 1971.
Carlson is author of the following DLMF chapter