### Transformations and Invariants Connected with Linear
Homogeneous Difference Equations and other Functional Equations

American Journal of Mathematics, Vol. 35, No. 4 (October 1913), 431-452

Introduction

The subject of Difference Equations has had an extended development during the last few years. Contributions of importance have been made by Guichard, Nörlund, Galbrun, Carmichael, Birkhoff, Horn, Ford, Perron, Bôcher, and others.

The object of the present paper is to discuss for the difference equation primarily (Part I), and also for a general type of functional equation (Part II), the question of functions that remain invariant for a certain broad type of transformations. The results assume a very simple and elegant form.

In section 1, it is shown that the most general point transformation that changes every linear homogeneous difference equation into a difference equation that is linear, homogeneous, and of the same order as the given equation, is of the form

*x* = *u*(ξ), y = λ(ξ)η_{ξ},
(I)

where *u*(ξ) satisfies one of the two relations

*u*(ξ + 1) – *u*(ξ) = ±1.

In section 2, the group character of the transformations (I) is verified and the existence of a certain subgroup is shown.

In section 3, a list of definitions of terms used in the remaining part of the paper is given.

In sections 4-7, certain fundamental sets of seminvariants, invariants, semi-covariants and covariants are determined.

In sections 8-9, a certain general type of functional equation is dealt with, the procedure being along the same lines as that in the discussion of the difference equation in the preceding articles. In section 8, the most general point transformation that changes every linear homogeneous functional equation of the type under consideration into another of the same type and order is determined, the group character of these transformations is verified, and the existence of a certain subgroup is shown. In section 9, fundamental sets of semi-variants, invariants, semi-covariants and covariants are determined.