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 linearly independent solutions which are *singular*
are said to be "of the second kind." Common examples of functions of the
second kind defined in this way include the Bessel
function of the second kind, Chebyshev
polynomial of the second kind, confluent
hypergeometric function of the second kind, Hankel
function of the second kind, and so on.

The term "second 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 second kind, elliptic integral of the second kind, Fredholm integral equation of the second kind, Stirling number of the second kind, Volterra integral equation of the second kind, and so on.