# Why was Indicial equations named so?

In ODE, in Frobenius method, there's an equation called "Indicial Equation." Is there any particular contextual/historical reason that it is named so?

## 1 Answer

"Indicial" is derived from "indices", in this case indices of the coefficients. The Latin word indicium originally meant a sign, indicator. "Indicial equation" is most often used in the context of solving equations of the form $$a_nx^n\frac{d^n}{dx^n}+\dots+a_1x\frac{dy}{dx}+a_0y=0$$. A substitution, known already to Euler, reduces it to a linear equation with constant coefficients, and the characteristic equation of the latter is called the "indicial equation" of the original.

Euler, who gave a general method for solving these types of equations, used neither the term nor the notation. Most notably, his coefficients did not have the subscripts, like $$a_n$$, making the arguments somewhat cumbersome. In modern notation, the indicial equation is formed directly from the indexed coefficients. In What′s in a Name: Why Cauchy and Euler Share the Cauchy–Euler Equation Parker quotes Sandifer's view of the significance of that (from his Some Facets of Euler’s work on Series, in the volume Leonhard Euler: Life, Work and Legacy volume):

"Indeed, C. E. Sandifer noticed [19, p. 281] "...the many circumlocutions of notation Euler used to do what modern mathematicians would do with indicial notations like subscripts and superscripts. This will demonstrate how the lack of a convenient and powerful notation distorted certain kinds of mathematical developments of the era...". The argument above [i.e. Euler's solution] could be simplified by using subscripts in the coefficients of the differential equation. Sandifer claims that Euler’s lack of indices hindered the development of other areas such as Bernoulli numbers, Gamma functions, and approximations of integrals."