Abstract The two linearly independent solutions of Legendre equation are called the first and second kind of Legendre functions, respectively. When the differential equation takes the eigenvalue, the first kind of Legendre function is interpreted as a polynomial, so the independent variable can take any value (except infinity). The second kind of Legendre function is still infinite series, diverging when the independent variable is equal to ±1 and converging when the absolute value is greater than 1. Since Legendre equation belongs to the type of hypergeometric equation, we give the expression of arbitrary order derivatives of different special functions of this type equation. Therefore, the hypergeometric expression of the first kind of Legendre function and its theoretical relationship with other special functions are given directly. In view of the complexity of solving the second kind of Legendre function, the hypergeometric expression of the second kind of Legendre function is directly given by using the series expansion method.