ABSTRACT
Series solutions for second order ordinary differential equations (ODE) with non-constant coefficients occupy an essential place in the development of the solution techniques for differential equations. A technique called the Frobenius series has been found useful in solving it. Many special functions were defined as the solutions for these second order ODEs with non-constant coefficients. The solutions for the Bessel equation are called Bessel functions of the first and second kinds. The chapter considers some classical differential equations that fall into this category, including the Bessel equation, modified Bessel equation, Legendre equation, associated Legendre equation, hypergeometric equation, and generalized hypergeometric equation. It introduces the gamma function, which is essential for obtaining solutions of these important differential equations. The chapter shows that the gamma function becomes infinity at zero and negative integers. It discusses an alternative factorial function which was proposed by French mathematician Hadamard.
