Special functions which arise as solutions to second order ordinary differential equations are commonly said to be "of the first kind" if they are nonsingular at the origin, while the corresponding linearly independent solutions which are singular are said to be "of the second kind." Common examples of functions of the first kind defined in this way include the Bessel function of the first kind, Chebyshev polynomial of the first kind, confluent hypergeometric function of the first kind, Hankel function of the first kind, and so on.
The term "first kind" is also used in a more general context to distinguish between two or more types of mathematical objects which, however, all satisfy some common overall property. Examples of objects of this kind include the Christoffel symbol of the first kind, elliptic integral of the first kind, Fredholm integral equation of the first kind, Stirling number of the first kind, Volterra integral equation of the first kind, and so on.
