# Glossary

• An ansatz is an expression for the general solutions of a problem, or for most of them, which typically involves undetermined coefficients, and is used for an educated guess.
• A Chebyshev expansion is a series of the form $$f(x)=\frac{c_0}{2}+\sum_{n=1}^{+\infty} c_nT_n(x)$$ where $$T_n(x)$$ denotes the $$n$$th Chebyshev polynomial of the first kind.
• A formal power series $$f(x)$$ is an infinite sum of the form $$\sum_{n = 0}^\infty f_n x^n$$ where the $$f_n$$ are in some common ring $$R$$. The coefficient $$f_n$$ of $$x^n$$ in $$f(x)$$ is also denoted $$[x^n] \, f(x)$$. The ring of formal power series is denoted $$R[[x]]$$.
• A formal logarithmic sum is a finite sum of the form $$\sum_{\alpha,j} x^\alpha f_{\alpha,j}(x) \log(x)^j$$ where the $$\alpha$$ are in some common ring $$R$$ and the $$f_{\alpha,j}(x)$$ are formal power series in $$R[[x]]$$.
• The indicial equation (or indicial polynomial) of a linear differential equation with polynomial coefficients is the coefficient of the power of_$$x$$ with lowest exponent in the expression that is obtained by the substitution $$y(x) = x^\alpha$$ into the equation. Its roots are exactly the exponents of $$x$$ that can appear as lowest exponents in a series solution of the differential equation.
• A series $$\sum_{n\geq0} g_n z^n$$ with nonnegative coefficients is a majorant series of a series $$\sum_{n\geq0} f_n z^n$$ with complex coefficients if $$|f_n| \leq g_n$$ for all $$n$$.
• A complex number $$z_0$$ is an ordinary point of a linear ordinary differential equation with polynomial coefficients, if the leading coefficient of the equation does not vanish at $$z_0$$.
• The ramification index is the smallest positive integer $$r$$ such that the exponent of an exponential factor can be expressed as a polynomial in $$x^{-1/r}$$.
• A complex number $$z_0$$ is a singular point (or singularity of a linear ordinary differential equation with polynomial coefficients, if the leading coefficient of the equation vanishes at $$z_0$$.
• The Laplace transform of a function $$f$$ of the variable $$z$$ is the function $$F$$ defined by the integral $$F(s) = \int_0^\infty e^{-sz} f(z) \,dz$$.